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Secular equations through the exact WKB analysis. (English) Zbl 0834.34068

Boutet de Monvel, Louis (ed.), Analyse algébrique des perturbations singulières. I. Méthodes résurgentes. Conférences du symposium franco-japonais sur l’analyse algébrique des perturbations singulières, CIRM, Marseille-Luminy, France, October 20-26, 1991. Paris: Hermann. Trav. Cours. 47, 85-102 (1994).
A formal derivation of the secular equation in the form conjectured by C. M. Bender and T. T. Wu [Phys. Rev. 184, 1231-1260 (1969)] for the one-dimensional Schrödinger equation \(({d^2\over dr^2}- {1\over 4} (\varepsilon- r^2+ \rho r^4)) \psi(r)= 0\) is carried out. The derivation implies finding a transformation which brings the above equation to the Weber equation if \(r\ll 1\) and another transformation which brings it to the Airy equation if \(r\gg 1\), while WKB-solutions of these equations are used to find matching conditions on the intermediate domain. The secular equation was also derived for non- symmetric double well potentials. The theory by A. Voros [Ann. Inst. Henri Poincaré Sect. A 39, 211-338 (1983; Zbl 0526.34046)] and the Bender-Wu type analyzes give the same secular equation for the anharmonic oscillators discussed. In the appendix the correspondence of some contour integrals under Weber type transformations is studied.
For the entire collection see [Zbl 0824.00035].
Reviewer: V.Burjan (Praha)

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Citations:

Zbl 0526.34046
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