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Lower bounds to eigenvalues of the Schrodinger equation by solution of a 90-y challenge.

ABSTRACT: The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple, Proc. R. Soc. A Math. Phys. Eng. Sci. 119, 276-293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple's lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple's result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz's upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

SUBMITTER: Martinazzo R

PROVIDER: S-EPMC7368311 | biostudies-literature | 2020 Jul

REPOSITORIES: biostudies-literature

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Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge.

Martinazzo Rocco R Pollak Eli E

Proceedings of the National Academy of Sciences of the United States of America 20200629 28

The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple, <i>Proc. R. Soc. A Math. Phys. Eng. Sci.</i> 119, 276-293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such ...[more]

PMID: 32601240

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Lower bounds to eigenvalues of the Schrodinger equation by solution of a 90-y challenge.

Publications

Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge.

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