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On time development of a quasi-quantum particle in quartic potential \((x^2-a^2)^2/2g\). (English) Zbl 0894.47060

It has been known for some time that the semiclassical approximation of the time evolution kernel for the Schrödinger operator with quartic interaction behaves oddly: it does not seem to approach the well-known harmonic oscillator result in a suitable limit, even though the classical motion, which is readily solved by elliptic functions, behaves correctly in that limit. Matsutani claims that the observed discrepancy should be blamed on avoiding elliptic function theory in the semiclassical quantum case and sets out to use that theory consistently for a particle moving in a one-dimensional potential of the form \(V(x)= (2g)^{-1}(x^2- a^2)^2\). He finds that fluctuations around the classical path are governed by the Lamé equation, and, by solving it exactly, he demonstrates agreement with solutions of the quantum harmonic oscillator for energies that are small compared to \((2g)^{-1}a^4\).

MSC:

47N50 Applications of operator theory in the physical sciences
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81S40 Path integrals in quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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