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Nodal properties of the scaled quartic anharmonic oscillator. (English) Zbl 0646.58039

The motions in the complex x-plane of the nodes, extrema, and turning points associated with the wavefunction \(\psi_ n(\gamma,x)\) of the scaled quartic anharmonic oscillator are studied when the quadratic coupling constant \(\gamma\) takes on complex values so as to produce level crossing. Nodal and extremal position functions are defined and their analytic properties in \(\gamma\) are studied together with those of the turning points. All functions are found to have a Riemann surface structure that may be related to that of the eigenvalues. A discussion is given of the possible application of the ideas developed to problems associated with quantum chaos.
Reviewer: P.E.Shanley

MSC:

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
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