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- W2068491680 abstract "An important problem of nonrelativistic quantum theory is: Given a bound-state energy spectrum, determine the space dependence of a potential function that will produce that spectrum. This is sometimes called the ‘‘spectral inverse’’ problem. Here, by use of the first-order WKB energy quantization condition, an approximate analytic solution of this problem is given for the case of one-dimensional symmetric oscillators and for a class of radial oscillators. This is done by reformulating the WKB quantization condition as an Abel integral equation that relates the quantum number to the potential’s space dependence. The resultant integral equation can be solved exactly. The harmonic oscillator and the hydrogen atom problems are used to demonstrate the method. The solution for the case of one-dimensional asymmetric oscillators is also briefly discussed and the Morse oscillator is used as an example. Finally, the history of the spectral inverse problem is reviewed and the conditions necessary for its exact solution are discussed. In particular, the conditions required for the inversion process to be unique are considered." @default.
- W2068491680 created "2016-06-24" @default.
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- W2068491680 date "1989-04-01" @default.
- W2068491680 modified "2023-10-16" @default.
- W2068491680 title "An approximate analytic solution of the spectral inverse problem" @default.
- W2068491680 doi "https://doi.org/10.1119/1.16043" @default.
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