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

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.