Fast-convergent resummation algorithm and critical exponents of φ4-theory in three dimensions

Journal of Mathematical Physics

Volumn 42, Issue 1, 2001, Pages 52-73

  Jasch, Florian ^a   Kleinert, Hagen ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035586169     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1289377     Document Type: Article

References (74)

1. D. B. Murray and B. G. Nickel (unpublished).
2. H. Kleinert, J. Neu, V. Schulte-Frohlinde, K. G. Chetyrkin, and S. A. Larin, Phys. Lett. B 272, 39 (1991).
3. H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995).
4. R. Guida and J. Zinn-Justin, "Critical exponents of the N-vector model," J. Phys. A 31, 8130 (1998).
5. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
6. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
7. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
8. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
9. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
10. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
11. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
12. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
13. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
14. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
15. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
16. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
17. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
18. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
19. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H.
20. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
21. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
22. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
23. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
24. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
25. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
26. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
27. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
28. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
29. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
30. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
31. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
32. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
33. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
34. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From
35. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
36. 4-Theories (World Scientific, Singapore, 2000) (http://www.physik.fu-berlin.de/˜kleinert/b8). There were two main predecessors to variational perturbation theory coming from two different directions. From the mathematical side, the seminal paper was by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). From the physical side, inspiration came from R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.fu-berlin.de/˜kleinert/159), and its systematic extension in H. Kleinert, Phys. Lett. A 173, 332 (1993). For the contributions of numerous other authors see Notes and References of Chaps. 5 and 17 in the first textbook. A small excerpt is given below. The crucial breakthrough which opened up the previous quantum mechanical variational approaches to quantum field theory came in three steps. First, still in quantum mechanics, by exploiting previously unused even approximants which do not have an extremum, as explained in Chap. 5 of the textbook. For applications to quantum field theory, two more ingredients were important, as pointed out in Refs. 8 and 9: the determination of the exponent ω by the leading power behavior in the strong-coupling limit, and an extrapolation procedure to infinite order on the basis of the theoretically known analytic order dependence. These developments were essential in obtaining accurate critical exponents rivaling the powerful combination of renormalization group and Borel-type resummation methods. Variational perturbation theory also yields directly ε-expansions for the critical exponents without the renormalization group formalism, as shown in Ref. 10. A selected list of historic contributions is T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); B. S. Shaverdyan and A. D. Usherveridze, Phys. Lett. B 123, 316 (1983); P. M. Stevenson, Phys. Rev. D 30, 1712 (1985); 32, 1389 (1985); P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); 36, 2415 (1987); W. Namgung, P. M. Stevenson, and J. F. Reed, Z. Phys. C 45, 47 (1989); U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51, 469 (1991); M. H. Thoma, ibid. 44, 343 (1991); I. Stancu and P. M. Stevenson, Phys. Rev. D 42, 2710 (1991); R. Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A. N. Sissakian, I. L. Solivtosv, and O. Y. Sheychenko, Phys. Lett. B 313, 367 (1993). A. Duncan and H. F. Jones, Phys. Rev. D 47, 2560 (1993). For the anharmonic oscillator, the highest accuracy in the strong-coupling limit was reached with exponentially fast convergence in Ref. 18. That paper contains references to earlier less accurate calculations of strong-coupling expansion coefficients from weak-coupling perturbation theory, in particular F. M. Fernández and R. Guardiola, J. Phys. A 26, 7169 (1993); F. M. Fernńdez, Phys. Lett. A 166, 173 (1992); R. Guardiola, M. A. Solís, and J. Ros, Nuovo Cimento 107, 713 (1992); A. V. Turbiner and A. G. Ushveridze, J. Math. Phys. 29, 2053 (1988); B. Bonnier, M. Hontebeyrie, and E. H. Ticembal, ibid. 26, 3048 (1985). Those works were unable to extract the exponential law of convergence from their data. This was shown to be related to the convergence radius of the strong-coupling expansion by H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) and in the second paper in Ref. 7. Predecessors of these works which did not yet explain the exponentially fast convergence in the strong-couplings limit are I. R. C. Buckley, A. Duncan, and H. F. Jones, Phys. Rev. D 47, 2554 (1993); C. M. Bender, A. Duncan, and H. F. Jones, ibid. 49, 4219 (1994); A. Duncan and H. F. Jones, ibid. 47, 2560 (1993); C. Arvanitis, H. F. Jones, and C. S. Parker, ibid. 52, 3704 (1995); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. (Paris) 241, 152 (1995).
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52. J. J. Loeffel, in Large-Order Behaviour of Perturbation Theory, edited by J. C. Le Guillou and J. Zinn-Justin (North-Holland, Amsterdam, 1990); J. C. Le Guillou and J. Zinn-Justin, J. Phys. (France) Lett. 46, L137 (1985); J. Phys. (Paris) 48, 19 (1987); 50, 1365 (1987).
53. s(1-1/ω)-β0 dx.
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59. J. A. Lipa, D. R. Swanson, J. Nissen, T. C. P. Chui, and U. E. Israelson, Phys. Rev. Lett. 76, 944 (1996). In the preprint version of the present paper, we have quoted in Eq. (94) the originally published value α=0.01285±0.00038, in quite good agreement with our result (96). After our paper was accepted, a new analysis of the experimental date changed the critical exponent to (94), now in superb agreement with our result (96). The new experimental value is published in Ref. [15] of J. A. Lipa, D. R. Swanson, J. Nissen, Z. K. Geng, P. R. Williamson, D. A. Stricker, T. C. P. Chui, U. E. Israelson, and M. Larson, Phys. Rev. Lett. 84, 4894 (2000).
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