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Brackets to the eigenvalues of the Schrödinger equation, part 1. Tridiagonal matrices
Author(s) -
Weltin E.
Publication year - 1970
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560040304
Subject(s) - tridiagonal matrix , eigenvalues and eigenvectors , mathematics , eigenvalue perturbation , matrix differential equation , mathematical analysis , matrix (chemical analysis) , harmonic oscillator , diagonal , upper and lower bounds , pure mathematics , quantum mechanics , physics , differential equation , geometry , chemistry , chromatography
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.