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Journal of Mathematical Physics

Tight lower bounds to eigenvalues of the Schrödinger equation

Robert Nyden Hill

Published Aug 1, 1980• 24 citations
Publisher

Abstract

A new tight lower bound to eigenvalues of the Schrödinger equation, which tends to be better than the well-known Temple lower bound, is presented. The new bound works (as does the Temple bound) for both ground and excited states. Optimization of both the Temple bound and the new tight bound with respect to a variational trial function is discussed. Numerical results are given for the anharmonic oscillator.

DOI
10.1063/1.524700

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A new tight lower bound to eigenvalues of the Schrödinger equation, which tends to be better than the well-known Temple lower bound, is presented. The new bound works (as does the Temple bound) for both ground and excited states. Optimization of both the Temple bound and the new tight bound with respect to a variational trial function is discussed. Numerical results are given for the anharmonic oscillator.
A new variational principle is presented which provides a lower bound to the eigenvalues of a general quantum Hamiltonian. Applications are made to ground and excited states in Schr\"odinger theory. The variational principle leads naturally to localized (classical-like) states that nevertheless contain effects due to quantum fluctuations in a variational manner. Except for possible exceptional end-point solutions, these "classical" or "semiquantum" configurations provide lower bonds to the true eigenvalue.