메뉴 건너뛰기




Volumn 196, Issue 3-4, 2004, Pages 265-310

The problem of two fixed centers: Bifurcations, actions, monodromy

Author keywords

Action variables; Integrable systems; Monodromy; Two center problem

Indexed keywords

APPROXIMATION THEORY; HAMILTONIANS; INTEGRATION; MOTION ESTIMATION; PHASE SPACE METHODS; PROBLEM SOLVING; QUANTUM THEORY; TOPOLOGY;

EID: 4344650952     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.05.006     Document Type: Article
Times cited : (76)

References (57)
  • 2
    • 0003319234 scopus 로고
    • Mathematical methods of classical mechanics
    • Springer, New York
    • V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, Springer, New York, 1978.
    • (1978) Graduate Texts in Mathematics , vol.60
    • Arnold, V.I.1
  • 3
    • 0039490410 scopus 로고    scopus 로고
    • Quantal two-center Coulomb problem treated by means of the phase-integral method. I. General theory
    • N. Athavan, P.O. Fröman, N. Fröman, M. Lakshmanan, Quantal two-center Coulomb problem treated by means of the phase-integral method. I. General theory, J. Math. Phys. 42 (11) (2001) 5051-5076.
    • (2001) J. Math. Phys. , vol.42 , Issue.11 , pp. 5051-5076
    • Athavan, N.1    Fröman, P.O.2    Fröman, N.3    Lakshmanan, M.4
  • 4
    • 0035641395 scopus 로고    scopus 로고
    • Quantal two-center Coulomb problem treated by means of the phase-integral method. II. Quantization conditions in the symmetrical case expressed in terms of complete elliptic integrals. Numerical illustration
    • N. Athavan, M. Lakshmanan, N. Fröman, Quantal two-center Coulomb problem treated by means of the phase-integral method. II. Quantization conditions in the symmetrical case expressed in terms of complete elliptic integrals. Numerical illustration, J. Math. Phys. 42 (11) (2001) 5077-5095.
    • (2001) J. Math. Phys. , vol.42 , Issue.11 , pp. 5077-5095
    • Athavan, N.1    Lakshmanan, M.2    Fröman, N.3
  • 5
    • 0035605470 scopus 로고    scopus 로고
    • Quantal two-center Coulomb problem treated by means of the phase-integral method. III. Quantization conditions in the general case expressed in terms of complete elliptic integrals. Numerical illustration
    • N. Athavan, M. Lakshmanan, N. Fröman, Quantal two-center Coulomb problem treated by means of the phase-integral method. III. Quantization conditions in the general case expressed in terms of complete elliptic integrals. Numerical illustration, J. Math. Phys. 42 (11) (2001) 5096-5115.
    • (2001) J. Math. Phys. , vol.42 , Issue.11 , pp. 5096-5115
    • Athavan, N.1    Lakshmanan, M.2    Fröman, N.3
  • 9
    • 21344447640 scopus 로고    scopus 로고
    • Topology of energy surfaces and existence of transversal Poincaré sections
    • A. Bolsinov, H.R. Dullin, A. Wittek, Topology of energy surfaces and existence of transversal Poincaré sections, J. Phys. A 29 (1996) 4977-4985.
    • (1996) J. Phys. A , vol.29 , pp. 4977-4985
    • Bolsinov, A.1    Dullin, H.R.2    Wittek, A.3
  • 10
    • 0001986404 scopus 로고
    • La mécanique ondulatoire de Schrödinger; une méthode générale de résolution par approximations successives
    • L. Brillouin, La mécanique ondulatoire de Schrödinger; une méthode générale de résolution par approximations successives, C. R. Acad. Sci. 183 (1926) 24-26.
    • (1926) C. R. Acad. Sci. , vol.183 , pp. 24-26
    • Brillouin, L.1
  • 11
    • 0034158611 scopus 로고    scopus 로고
    • Semiclassical energies of low-lying states of one-electron diatoms
    • E.J. Buehler, E.E. Gooch, J.L. Dial, S.K. Knudson, Semiclassical energies of low-lying states of one-electron diatoms, Chem. Phys. 253 (2000) 219-230.
    • (2000) Chem. Phys. , vol.253 , pp. 219-230
    • Buehler, E.J.1    Gooch, E.E.2    Dial, J.L.3    Knudson, S.K.4
  • 12
    • 0009023852 scopus 로고
    • Über einige in Aussicht genommene Berechnungen betreffend einen Spezialfall des Dreikörper-Problems
    • C. Burrau, Über einige in Aussicht genommene Berechnungen betreffend einen Spezialfall des Dreikörper-Problems, Vierteljahresschrift Astron. Ges. 41 (1906) 261-265.
    • (1906) Vierteljahresschrift Astron. Ges. , vol.41 , pp. 261-265
    • Burrau, C.1
  • 13
    • 84974766439 scopus 로고
    • Compotment semi-classique du spectre conjoint d'opérateur pseudo-différentiels qui commutent
    • A.-M. Charbonel, Compotment semi-classique du spectre conjoint d'opérateur pseudo-différentiels qui commutent, Asymptotic Anal. 1 (1988) 227-261.
    • (1988) Asymptotic Anal. , vol.1 , pp. 227-261
    • Charbonel, A.-M.1
  • 15
    • 18344377769 scopus 로고
    • Semiclassical theory of tunneling and curve-crossing problems: Adiagrammatic approach
    • M.S. Child, Semiclassical theory of tunneling and curve-crossing problems: adiagrammatic approach, J. Mol. Spectrosc. 53 (1974) 280-301.
    • (1974) J. Mol. Spectrosc. , vol.53 , pp. 280-301
    • Child, M.S.1
  • 16
    • 0032535923 scopus 로고    scopus 로고
    • Quantum states in a champagne bottle
    • M.S. Child, Quantum states in a champagne bottle, J. Phys. A 31 (1998) 657-670.
    • (1998) J. Phys. A , vol.31 , pp. 657-670
    • Child, M.S.1
  • 17
    • 85034507044 scopus 로고    scopus 로고
    • 2O and other systems: New insight into the level structure of quasi-linear molecules
    • 2O and other systems: new insight into the level structure of quasi-linear molecules, Mol. Phys. 96 (1999) 371-379.
    • (1999) Mol. Phys. , vol.96 , pp. 371-379
    • Child, M.S.1    Weston, T.2    Tennyson, J.3
  • 18
    • 84995178433 scopus 로고
    • A constant of the motion for the two-centre Kepler problem
    • C.A. Coulson, A. Joseph, A constant of the motion for the two-centre Kepler problem, Int. J. Quant. Chem. 1 (1967) 337-347.
    • (1967) Int. J. Quant. Chem. , vol.1 , pp. 337-347
    • Coulson, C.A.1    Joseph, A.2
  • 21
    • 0033153845 scopus 로고    scopus 로고
    • Monodromy in perturbed Kepler systems: Hydrogen atom in crossed fields
    • R.H. Cushman, D.A. Sadovskií, Monodromy in perturbed Kepler systems: hydrogen atom in crossed fields, Europhys. Lett. 47 (1999) 1-7.
    • (1999) Europhys. Lett. , vol.47 , pp. 1-7
    • Cushman, R.H.1    Sadovskií, D.A.2
  • 22
    • 0347450519 scopus 로고    scopus 로고
    • Monodromy in the hydrogen atom in crossed fields
    • R.H. Cushman, D.A. Sadovskií, Monodromy in the hydrogen atom in crossed fields, Physica D 142 (2000) 166-196.
    • (2000) Physica D , vol.142 , pp. 66-196
    • Cushman, R.H.1    Sadovskií, D.A.2
  • 23
    • 4344627211 scopus 로고
    • Le problème des deux centres fixes
    • A. Deprit, Le problème des deux centres fixes, Bull. Soc. Math. Belg. 14 (1962) 12-45.
    • (1962) Bull. Soc. Math. Belg. , vol.14 , pp. 12-45
    • Deprit, A.1
  • 24
    • 0033143339 scopus 로고    scopus 로고
    • Periodic orbits of the hydrogen molecular ion
    • Y. Duan, J.M. Yuan, Periodic orbits of the hydrogen molecular ion, Eur. Phys. J. D 6 (1999) 319-326.
    • (1999) Eur. Phys. J. D , vol.6 , pp. 319-326
    • Duan, Y.1    Yuan, J.M.2
  • 25
    • 0000104961 scopus 로고
    • Periodic orbits of the hydrogen molecular ion and their quantization
    • Y. Duan, J.M. Yuan, C. Bao, Periodic orbits of the hydrogen molecular ion and their quantization, Phys. Rev. A 52 (5) (1995) 3497-3502.
    • (1995) Phys. Rev. A , vol.52 , Issue.5 , pp. 3497-3502
    • Duan, Y.1    Yuan, J.M.2    Bao, C.3
  • 26
    • 84980167278 scopus 로고
    • On global action-angle coordinates
    • J.J. Duistermaat, On global action-angle coordinates, Commun. Pure Appl. Math. 33 (1980) 687-706.
    • (1980) Commun. Pure Appl. Math. , vol.33 , pp. 687-706
    • Duistermaat, J.J.1
  • 27
    • 21844511778 scopus 로고
    • Action integrals and energy surfaces of the Kovalevskaya top
    • H.R. Dullin, M. Juhnke, P.H. Richter Action integrals and energy surfaces of the Kovalevskaya top, Bifurcation and Chaos 4 (6) (1994) 1535-1562.
    • (1994) Bifurcation and Chaos , vol.4 , Issue.6 , pp. 1535-1562
    • Dullin, H.R.1    Juhnke, M.2    Richter, P.H.3
  • 28
    • 0002485626 scopus 로고
    • Zum Quantensatz von Sommerfeld und Epstein
    • A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verb. DPG 19 (1917) 82-92.
    • (1917) Verb. DPG , vol.19 , pp. 82-92
    • Einstein, A.1
  • 29
    • 4243172371 scopus 로고
    • A note on the one-electron states of diatomic molecules
    • H.A. Erikson, E.L. Hill, A note on the one-electron states of diatomic molecules, Phys. Rev. 75 (1) (1949) 29-31.
    • (1949) Phys. Rev. , vol.75 , Issue.1 , pp. 29-31
    • Erikson, H.A.1    Hill, E.L.2
  • 30
    • 0001410014 scopus 로고
    • Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom
    • T. Ratiu (Ed.), Springer, New York
    • A.T. Fomenko, Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom, in: T. Ratiu (Ed.), The Geometry of Hamiltonian Systems, Springer, New York, 1991, pp. 131-339.
    • (1991) The Geometry of Hamiltonian Systems , pp. 131-339
    • Fomenko, A.T.1
  • 31
    • 36749116209 scopus 로고
    • A method for determining a stochastic transition
    • J.M. Greene, A method for determining a stochastic transition, J. Math. Phys. 20 (1979) 1183-1201.
    • (1979) J. Math. Phys. , vol.20 , pp. 1183-1201
    • Greene, J.M.1
  • 32
    • 0010986618 scopus 로고
    • Problem of two fixed centers and a finite dipole: A unified treatment
    • J.E. Howard, T.D. Wilkerson, Problem of two fixed centers and a finite dipole: a unified treatment, Phys. Rev. A 52 (6) (1995) 4471-4492.
    • (1995) Phys. Rev. A , vol.52 , Issue.6 , pp. 4471-4492
    • Howard, J.E.1    Wilkerson, T.D.2
  • 34
    • 34250932038 scopus 로고
    • Zur Theorie des Wasserstoffmolekülions
    • G. Jaffé, Zur Theorie des Wasserstoffmolekülions, Z. Phys. 87 (1933) 535-544.
    • (1933) Z. Phys. , vol.87 , pp. 535-544
    • Jaffé, G.1
  • 35
    • 85030827169 scopus 로고    scopus 로고
    • Diplomarbeit, Universität Bremen
    • ++, Diplomarbeit, Universität Bremen, 2001.
    • (2001) ++
    • Junge, A.1
  • 36
    • 0000724502 scopus 로고
    • Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems
    • J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys. (NY) 4 (1958) 180-188.
    • (1958) Ann. Phys. (NY) , vol.4 , pp. 180-188
    • Keller, J.B.1
  • 37
    • 0031321234 scopus 로고    scopus 로고
    • Semiclassical electronic eigenvalues for charge asymmetric one-electron diatomic molecules: General method and sigma states
    • S.K. Knudson, I.C. Palmer, Semiclassical electronic eigenvalues for charge asymmetric one-electron diatomic molecules: general method and sigma states, Chem. Phys. 224 (1997) 1-18.
    • (1997) Chem. Phys. , vol.224 , pp. 1-18
    • Knudson, S.K.1    Palmer, I.C.2
  • 38
    • 34250915737 scopus 로고
    • Wellenmechanik und halbzahlige Quantisierung
    • H.A. Kramers, Wellenmechanik und halbzahlige Quantisierung, Z. Phys. 39 (1926) 828-840.
    • (1926) Z. Phys. , vol.39 , pp. 828-840
    • Kramers, H.A.1
  • 39
    • 0001083274 scopus 로고
    • Semiclassical treatment of multiple turning-point problems - Phase shifts and eigenvalues
    • W.H. Miller, Semiclassical treatment of multiple turning-point problems - phase shifts and eigenvalues, J. Chem. Phys. 48 (4) (1968) 1651-1658.
    • (1968) J. Chem. Phys. , vol.48 , Issue.4 , pp. 1651-1658
    • Miller, W.H.1
  • 42
    • 0037525498 scopus 로고    scopus 로고
    • Quantum monodromy in integrable systems
    • S. Vu Ngoc, Quantum monodromy in integrable systems, Commun. Math. Phys. 203 (1999) 465-479.
    • (1999) Commun. Math. Phys. , vol.203 , pp. 465-479
    • Vu Ngoc, S.1
  • 43
    • 0038026787 scopus 로고    scopus 로고
    • Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type
    • S. Vũ Ngoc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Commun. Pure Appl. Math. 53 (2000) 143-217.
    • (2000) Commun. Pure Appl. Math. , vol.53 , pp. 143-217
    • Vũ Ngoc, S.1
  • 44
    • 0000295567 scopus 로고
    • + as a tunneling problem; application of first-order and third-order JWKB approximation
    • + as a tunneling problem; application of first-order and third-order JWKB approximation, Mol. Phys. 43 (4) (1981) 753-771.
    • (1981) Mol. Phys. , vol.43 , Issue.4 , pp. 753-771
    • Pajunen, P.1
  • 45
    • 84967537850 scopus 로고
    • Über das Modell des Wasserstoffmolekülions
    • W. Pauli, Über das Modell des Wasserstoffmolekülions, Ann. Phys. (Leipzig) 68 (1922) 177-240.
    • (1922) Ann. Phys. (Leipzig) , vol.68 , pp. 177-240
    • Pauli, W.1
  • 47
    • 0000707901 scopus 로고    scopus 로고
    • Monodromy, diabolic points, and angular momentum coupling
    • D.A. Sadovskií, B.I. Zhilinskií, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256 (1999) 235-244.
    • (1999) Phys. Lett. A , vol.256 , pp. 235-244
    • Sadovskií, D.A.1    Zhilinskií, B.I.2
  • 49
    • 84979134645 scopus 로고
    • Recherches numériques concernant des solutions périodiques d'un cas spécial du problème des trois corps
    • T.N. Thiele, Recherches numériques concernant des solutions périodiques d'un cas spécial du problème des trois corps, Astron. Nachr. 138 (1896) 1-10.
    • (1896) Astron. Nachr. , vol.138 , pp. 1-10
    • Thiele, T.N.1
  • 50
    • 0037061391 scopus 로고    scopus 로고
    • Quantum monodromy in trapped Böse condensates
    • H. Waalkens, Quantum monodromy in trapped Böse condensates, Europhys. Lett. 58 (2002) 162-168.
    • (2002) Europhys. Lett. , vol.58 , pp. 162-168
    • Waalkens, H.1
  • 51
    • 0037050349 scopus 로고    scopus 로고
    • Quantum monodromy in prolate ellipsoidal billiards
    • H. Waalkens, H.R. Dullin, Quantum monodromy in prolate ellipsoidal billiards, Ann. Phys. (NY) 295 (2002) 81-112.
    • (2002) Ann. Phys. (NY) , vol.295 , pp. 81-112
    • Waalkens, H.1    Dullin, H.R.2
  • 52
    • 0038515348 scopus 로고    scopus 로고
    • Quantum monodromy in the two-center problem
    • H. Waalkens, A. Junge, H.R. Dullin, Quantum monodromy in the two-center problem, J. Phys. A 36 (2003) L307-L314.
    • (2003) J. Phys. A , vol.36
    • Waalkens, H.1    Junge, A.2    Dullin, H.R.3
  • 54
    • 34250917346 scopus 로고
    • Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik
    • G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Z. Phys. 38 (1926) 518-529.
    • (1926) Z. Phys. , vol.38 , pp. 518-529
    • Wentzel, G.1
  • 55
    • 0038971596 scopus 로고    scopus 로고
    • Energy surfaces of ellipsoidal billiards
    • J. Wiersig, P.H. Richter, Energy surfaces of ellipsoidal billiards, Z. Naturforsch. 51a (1996) 219-241.
    • (1996) Z. Naturforsch. , vol.51 A , pp. 219-241
    • Wiersig, J.1    Richter, P.H.2
  • 57
    • 9344225119 scopus 로고    scopus 로고
    • A note on focus-focus singularities
    • Nguyen Tien Zung, A note on focus-focus singularities, Differential Geom. Appl. 7 (2) (1997) 123-130.
    • (1997) Differential Geom. Appl. , vol.7 , Issue.2 , pp. 123-130
    • Zung, N.T.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.