Lower bounds to eigenvalues

DIPC Seminars

Eli Pollak, Weizmann Institute of Science, Israel
Donostia International Physics Center
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Lower bounds to eigenvalues The determination of eigenvalues of operators is fundamental to different fields of science. In his famous papers of 1908, 1909, Ritz invented the "Ritz method" for the convergent computation of upper bounds of eigenvalues. Twenty years later, Temple presented a method for obtaining convergent lower bounds. Ninety years later, the Ritz method is a staple of university courses while the Temple lower bound remains relatively hidden, for good reason. Its convergence is slow, and too often, it is orders of magnitude less accurate that the Ritz upper bound. Finding “good” lower bounds remained a challenge. I have recently shown how one may improve upon the Temple formula, leading to lower bounds whose convergence rate is comparable to the upper bound Ritz based evaluation. The new method is based on a - improvement in evaluating overlaps squared of exact eigenfunctions with approximate ones. b - constructing a tridiagonal representation of the Hamiltonian, especially by employing the Lanczos method. Results for oscillators with increasing and decreasing level spacing with increasing energy will be presented. Further examples will include Heisenberg chains with increasing complexity where the distance of the lower and upper bounds from the exact answer decrease by fifteen orders of magnitude as the basis set size is increased.