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A high-precision study of anharmonic-oscillator spectra. (English) Zbl 0974.81015

Summary: High-precision methods are developed to evaluate eigenvalues for all states \(|m,N) \) of the \(x^{2m}\) anharmonic oscillators with \(m\) from 2 to 6, for all values of the anharmonicity parameter. There are three basic steps: rescaling to introduce a length scale natural to the problem; use of fifth-order JWKB to generate an accurate starting estimate of the rescaled energy; and shifted (resolvent-based) Lanczos algorithm to sharpen the initial JWKB estimate. JWKB itself gives 33-figure accuracy for \(N\) greater than 1500 (for \(m=2)\) to 3500 (for \(m=6)\). With the JWKB starting energy, the shifted Lanczos algorithm converges to 33 figures in 3 iterations or less for all states. These methods are used in a study of the systematics of anharmonic-oscillator spectra and of the physical effects of the rescaling transformation.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
46A50 Compactness in topological linear spaces; angelic spaces, etc.
81Q15 Perturbation theories for operators and differential equations in quantum theory
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