Convergent expansions for tunneling. (English) Zbl 0579.47050

Authors’ abstract: A new method to compute effects of tunneling in one-dimensional multiple well is developed. A tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling. These splittings are given by convergent series in term of this tunneling parameter for a wide class of double well.
Reviewer: Josef Wloka (Kiel)


81Q15 Perturbation theories for operators and differential equations in quantum theory
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
47E05 General theory of ordinary differential operators
47N50 Applications of operator theory in the physical sciences
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35J10 Schrödinger operator, Schrödinger equation
Full Text: DOI


[1] Akhiezer, N.I., Glazman, I.M.: Theory of linear operators in Hilbert space. New York: Frederick Ungar Publishing Co. Inc. 1961 · Zbl 0098.30702
[2] Combes, J.M., Duclos, P., Seiler, R.: The Born Oppenheimer approximation, pp. 185-212 of Rigorous Atomic and Molecular Physics, Proceedings (1980). Wightman, A.S., Velo, G. (eds). New York: Plenum Press 1981
[3] Combes, J.M., Duclos, P., Seiler, R.: Krein’s formula and one dimensional multiple well. J. Funct. Anal.52, 257-301 (1983) · Zbl 0562.47002
[4] Dieudonné, J.: Calcul infinitésimal, Chap. VIII.7. Paris: Herman 1968
[5] Davies, E.B.: Double well hamiltonians. Preprint 1983 · Zbl 0546.34021
[6] Harrell, E.M.: Double wells. Commun. Math. Phys.75, 239-261 (1980) · Zbl 0445.35036
[7] Harrell, E.M.: The band structure of a one-dimensional, periodic system in a scaling limit. Ann. Phys. (NY)119, 351-369 (1979) · Zbl 0412.34013
[8] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: New approach to the semiclassical limit of quantum mechanics. I. Multiple tunnelings in one dimension. Commun. Math. Phys.80, 223-254 (1981) · Zbl 0483.60094
[9] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601
[10] Krein, M.: Über Resolventen hermitescher Operatoren mit Defektindex (m, m). Dokl. Akad. Nauk SSSR52, 657-660 (1946)
[11] Polyakov, A.N.: Quark confinement and topology of gauge theories. Nucl. Phys. B120, 429-458 (1977)
[12] Richard, J.L., Rouet, A.: Complex saddle points versus dilute-gas in the double well anharmonic oscillator. Nucl. Phys. B185, 47-60 (1981)
[13] Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. Henri Poincaré38, 295-307 (1983)
[14] Simon, B.: Instantons, double wells, and large deviations. Bull. Am. Math. Soc.8, 323-326 (1983) · Zbl 0529.35059
[15] Witten, E.: Supersymmetry and Morse theory. J. Differential Geometry17, 661 (1982) · Zbl 0499.53056
[16] Zinn-Justin, J.: Multi-instanton contributions in quantum mechanics. Nucl. Phys. B192, 125-140 (1981); and, The principles of instanton calculus: a few applications, Les Houches Lecture Notes 1982, Preprint DPH-T/80-82 (Oct. 1982)
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.