Spectral properties of the scaled quartic anharmonic oscillator. (English) Zbl 0646.58038

The analytic properties of the eigenvalues of the scaled quartic anharmonic oscillator are studied for complex values of the quadratic coupling constant \(\gamma\). The spectral crossing points that are square root singularities of the eigenvalues are located in the complex \(\gamma\)-plane by numerical methods. The quantum numbers associated with the crossing points are determined and a suitable Riemann surface for the energy is chosen in each parity that contains an infinite number of sheets with one for each quantum number n. The distribution of crossing points is studied and an accumulation is expected on the asymptotic sheet of the Riemann surface. Qualitative arguments that a relation exists between the lack of smoothness in crossing a separatrix orbit in a classical problem and the phenomena of level crossing in the corresponding quantum problem are presented. This relation gives a new insight into the divergence of perturbation theory for the quartic anharmonic oscillator and also may have implications for problems exhibiting quantum chaos.
Reviewer: P.E.Shanley


58Z05 Applications of global analysis to the sciences
81S99 General quantum mechanics and problems of quantization
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI


[1] von Neumann, J.; Wigner, E., Phys. Z., 30, 467, (1929)
[2] Bender, C.M.; Wu, T.T., Phys. rev., 184, 1231, (1969)
[3] Bender, C.M.; Wu, T.T., Phys. rev. D, 7, 1620, (1973)
[4] Simon, B., Ann. phys. (N.Y.), 58, 76, (1970)
[5] Bender, C.M.; Happ, H.J.; Svetitsky, B., Phys. rev. D, 9, 2324, (1974)
[6] Hunter, C.; Guerrieri, B.; Hunter, C.; Guerrieri, B.; Blanch, G.; Clemm, D.S.; Maluendes, S.A.; Fernandez, F.M.; Castro, E.A., Stud. appl. math., SIAM J. appl. math., Math. comp., J. chem. phys., 83, 4599, (1985)
[7] Hunter, C.; Gurrieri, B., Stud. appl. math., 66, 217, (1982)
[8] Reik, H.G.; Doucha, M., Phys. rev. lett., 57, 787, (1986)
[9] Reinhardt, W.P.; Noid, D.W.; Koszykowski, M.L.; Marcus, R.A.; Stechel, E.B.; Heller, E.J.; Rice, S.A., (), Annu. rev. phys. chem., Annu. rev. phys. chem., Adv. chem. phys., 47, 117, (1981)
[10] Shanley, P.E., Ann phys. (N.Y.), 186, 325, (1988)
[11] Kato, T., (), 365
[12] Abrahamowitz, M.; Stegun, I.A., (), 882
[13] Friedberg, S.H.; Insel, A.J.; Spence, L.E., (), 302
[14] Kato, T., (), 186
[15] Arnold, V.I.; Avron, J.E.; Simon, B., (), Ann. phys. (N.Y.), 110, 85, (1978)
[16] Smith, K., (), 68
[17] Loeffel, J.J.; Martin, A.; Loeffel, J.J.; Martin, A., (), CERN report no. TH-1167, (1970), also
[18] Rauch, H.E.; Lebowitz, A., (), 21
[19] Heins, M., Ann. math., 55, 296, (1952)
[20] Shanley, P.E., Phys. lett. A, 117, 161, (1986)
[21] Di Prima, R.C.; Hall, P., (), 75
[22] Rotenberg, M.; Scadron, M.; Weinberg, S.; Wright, J., Ann. phys. (N.Y.), Phys. rev. B, 135, 202, (1964)
[23] Chirikov, B.V., Phys. rep., 52, 263, (1979)
[24] Cary, J.R.; Escande, D.F.; Tennyson, J.L.; Hannay, J.H.; Vasilev, A.N.; Guzev, M.A., Phys. rev. A, J. phys. A, Theor. math. phys., 68, 907, (1987)
[25] Arnold, V.I., (), 297
[26] Messiah, A., (), 744
[27] Berry, M.V.; Mount, K.E., Rep. prog. phys., 35, 315, (1972)
[28] Peres, A., Phys. scr., 34, 736, (1986)
[29] Coleman, S.; Norton, R.E.; Rubin, M.; Sugar, R.; Tiktopoulos, G.; Amado, R.D., (), 146, 112, (1966)
[30] Bender, C.M., J. math. phys., 11, 796, (1970)
[31] Banks, T.I.; Bender, C.M., J. math. phys., 13, 1320, (1972)
[32] Kohn, W.; Avron, J.E.; Simon, B.; Ferrari, M.; Grecchi, V.; Zironi, F., Phys. rev., Ann. phys. (N.Y.), J. phys. C, 18, 5825, (1985)
[33] Kean, H.P.; van Moerbeke, P.; McKean, H.P.; Trubowitz, E.; McKean, H.P.; Trubowitz, E., Inventiones math., Comm. pure appl. math., Bull. amer. math. soc., 84, 1042, (1978)
[34] Lichtenberg, A.J.; Lieberman, M.A., (), 24
[35] Coleman, S.; Wightman, A.S., (), 805-983
[36] Bender, C.M.; Orszag, S.A., (), 537
[37] Fröman, N.; Paulsson, R.; Fröman, N., Ark. fys., Ann. phys. (N.Y.), 163, 227, (1985)
[38] Hille, E.; Hille, E., (), 185
[39] Arold, V.I.; Avez, A., ()
[40] Percival, I.C., J. phys. B, 6, L229, (1973)
[41] Noid, D.W.; Koszykowski, M.L.; Tabor, M.; Marcus, R.A.; Noid, D.W.; Koszykowski, M.L.; Marcus, R.A.; Ramaswamy, R.; Marcus, R.A.; Marcus, R.A.; Noid, D.W.; Koszykowski, M.L.; Marcus, R.A.; Uzer, T.; Noid, D.W.; Marcus, R.A., J. chem. phys., Chem. phys. lett., J. chem. phys., Faraday discuss. chem. soc., J. chem. phys., J. chem. phys., 79, 4412, (1983)
[42] Graffi, S.; Paul, T.; Silverstone, H.J., Phys. res. lett., 59, 255, (1987)
[43] Ceschino, F.; Kuntzmann, J., (), 143
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.