The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A Rayleigh-Ritz variational approach to the problem, which might be useful in bridging the gap between perturbative and nonperturbative viewpoints, is here proposed. ...

Abstract. We investigate several generalizations of the harmonic and refined Rayleigh–Ritz method. These may be practical when one is interested in eigenvalues close to one of two targets (for instance, when the eigenproblem has Hamiltonian structure such that eigenvalues come in pairs or quadruples), or in rightmost eigenvalues close to (for instance) the imaginary axis. Our goal is to develop...

Eigenvalue estimates that are optimal in some sense have selfevident appeal and leave estimators with a sense of virtue and economy. So, it is natural that ongoing searches for effective strategies for difficult tasks such as estimating matrix eigenvalues that are situated well into the interior of the spectrum revisit from time to time methods that are known to yield optimal bounds. This artic...

In this article we further develop a perturbation approach to the Rayleigh– Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the operator is being approximated by the Ritz values an...

where the values of the constants Ai, A2 and Pi, P2 are not simultaneously zero. Several methods have been proposed to determine the proper elements (i.e. eigenvalues and eigenfunctions) of Sturm-Liouville equations. Most of them have been reviewed by Kopal [1], but we shall examine one of them, the so called Rayleigh-Ritz method, in order to explain the main defect they have in common and to j...