A Chebyshev polynomial method for computing analytic solutions to eigenvalue problems with application to the anharmonic oscillator

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Abstract
In this work, I present a two‐part algorithm which employs Chebyshev polynomials to compute the eigenvalues and eigenfunctions of an ordinary differential equation and to represent their dependence on a problem parameter λ. To illustrate these methods, I apply them to the anharmonic oscillator where the parameter λ is the coupling constant. I show that these techniques are simple, efficient, and easy to program for a computer. Unlike perturbation methods or numerical tables, the Chebyshev algorithms calculate explicit, highly uniform polynomial approximations to each function on any interval on which the function is analytic. Perhaps the most important result of this work is that although the two Chebyshev algorithms are numerical, all the results are analytic—a dramatic counterexample to the conventional wisdom which holds that to obtain nonnumerical solutions, one must use analytic, paper‐and‐pencil techniques. Thus, the two‐part Chebyshev approach, despite its numerical basis, should be regarded principally as a new and powerful method for computing analytic solutions which will succeed where ordinary (and even extraordinary) perturbation methods fail.