Unfolding the quartic oscillator. (English) Zbl 0977.34052

The paper is concerned with geometry of energy levels of the quartic oscillator \(-h^2\Psi''+V(q) \Psi=E\Psi\), where \(V\) is the polynomial \(V(q)=q^4+\alpha q^2+\beta q\). The authors introduce the “swallowtail surface” \((V'(q)=V(q)-E=0)\) in the parameter space \((\alpha, \beta,E)\) and discuss various regions (simple oscillator, double oscillator, etc.) in this space. The basic point of the paper is that the parameter space “can be covered by various analytic charts, in each of which the quantization condition is given exactly by an explicit analytic relation between the coordinates of the chart”. These relations, called model quantization conditions, look very similar to “what one would get by a lowest-order semiclassical approximation (for instance then ‘model equation’ in section 3.1 has exactly the same form as C. M. Bender and T. T. Wu’s ‘secular equation’ in [C. M. Bender and T. T. Wu, Phys. Rev., II. Ser. 184, 1231-1260 (1969) (section 4)])”. The coordinates of local charts are obtained by resumming WKB-type expansions in powers of \(h\), using resurgence theory. The paper is written on the “physical level” of rigor, but, according to the authors, most of the details can be found in [E. Delabaere, H. Dillinger and F. Pham, J. Math. Phys. 38, No. 12, 6126-6184 (1997; Zbl 0896.34051)].


34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34M37 Resurgence phenomena (MSC2000)


Zbl 0896.34051
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