PT-symmetric quantum mechanics

Journal of Mathematical Physics

Volumn 40, Issue 5, 1999, Pages 2201-2229

  Bender, Carl M ^a   Boettcher, Stefan ^b,c   Meisinger, Peter N ^a  

^a WASHINGTON UNIVERSITY   (United States)

^b CENTER FOR NONLINEAR STUDIES   (United States)

^c CLARK ATLANTA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033435556     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532860     Document Type: Article

References (19)

1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). Note that in this reference the notation N represents the quantity 2 + ∈.
2. Several years ago, D. Bessis and J. Zinn-Justin conjectured that the spectrum of H in Eq. (1.1) for the special case ∈ = 1 is real (private communication). A partial proof for the reality of the spectrum in this special case has been given by M. P. Blencowe, H. Jones, and A. P. Korte, Phys. Rev. D 57, 5092 (1998) using the linear delta expansion and by E. Delabaere and F. Pham, Ann. Phys. 261, 180 (1997) using WKB methods.
3. Several years ago, D. Bessis and J. Zinn-Justin conjectured that the spectrum of H in Eq. (1.1) for the special case ∈ = 1 is real (private communication). A partial proof for the reality of the spectrum in this special case has been given by M. P. Blencowe, H. Jones, and A. P. Korte, Phys. Rev. D 57, 5092 (1998) using the linear delta expansion and by E. Delabaere and F. Pham, Ann. Phys. 261, 180 (1997) using WKB methods.
4. T. J. Hollowood, Nucl. Phys. B 386, 166 (1992).
5. D. R. Nelson and N. M. Shnerb, Phys. Rev. E 58, 1383 (1998).
6. N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996); Phys. Rev. B 56, 8651 (1997).
7. N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996); Phys. Rev. B 56, 8651 (1997).
8. C. M. Bender and S. Boettcher, J. Phys. A 31, L273 (1998).
9. C. M. Bender and K. A. Milton, Phys. Rev. D 55, R3255 (1997).
10. 4 field theory has been argued using analytic continuation techniques [see K. Gawadzki and A. Kupiainen, Nucl. Phys. B 257, 474 (1985)] and using Gaussian approximation [see B. Rosenstein and A. Kovner, Phys. Rev. D 40, 504 (1989)].
11. 4 field theory has been argued using analytic continuation techniques [see K. Gawadzki and A. Kupiainen, Nucl. Phys. B 257, 474 (1985)] and using Gaussian approximation [see B. Rosenstein and A. Kovner, Phys. Rev. D 40, 504 (1989)].
12. C. M. Bender and K. A. Milton, Phys. Rev. D 57, 3595 (1998).
13. C. M. Bender, S. Boettcher, H. F. Jones, and P. N. Meisinger, in preparation.
14. C. M. Bender and K. A. Milton. J. Phys. A 32, L87 (1999).
15. Equation (2.3) is a complex version of the statement that the velocity is the time derivative of the position (v = dx/dt). Here, the time is real but the velocity and position are complex.
16. C. M. Bender and J. P. Vinson, J. Math. Phys. 37, 4103 (1996).
17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
18. C. M. Bender and A. Turbiner, Phys. Lett. A 173, 442 (1993).
19. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.