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Brackets to the eigenvalues of the Schrödinger equation, part 1. Tridiagonal matrices

✍ Scribed by E. Weltin

Publisher
John Wiley and Sons
Year
1970
Tongue
English
Weight
525 KB
Volume
4
Category
Article
ISSN
0020-7608

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✦ Synopsis

Abstract

The problem of upper and lower bounds to the first few eigenvalues of a very large or infinite tridiagonal matrix H is studied. Those eigenvalues of a comparison‐matrix M~n~ which are lower than a characteristic limit, together with the corresponding eigenvalues of the variational matrix H~n~ are shown to bracket exact eigenvalues of H. M~n~ differs from H~n~ only in the last off‐diagonal element and is easily obtained from H. Sufficient conditions for lower bounds are based on a low estimate of the characteristic limit. For increasing dimensions n, the lower bounds approach the exact eigenvalues from below. As a numerical illustration, brackets to the known eigenvalues of the harmonic oscillator with a linear perturbation are calculated.

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