Variational Bounds to Eigenvalues of Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum

Abstract

In the present paper a method by Lehmann-Maehly and Goerisch is extended to self-adjoint eigenvalue problems with arbitrary essential spectrum. This extension is obtained by consequently making use of the local character of the method. In this way, upper and lower bounds to all isolated eigenvalues are derived. In our proofs, the close relationship to Wielandt's inverse iteration becomes quite obvious.

Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

“…The Rayleigh-Ritz procedure provides upper bounds for those eigenvalues of H which are less than the bottom of the essential spectrum. Complementary lower bounds have been obtained by Temple, Lehmann and Goerisch; see [13], [28], [27], [33] for accounts of these methods. However the methods do not work as they stand for eigenvalues in a spectral gap.…”

“…The dependence upon ρ appears not to have been previously investigated from our point of view, but our theorems are confirmed by numerical data in earlier publications, as well as by our own computations for a test example. We mention particularly the recent papers of Zimmerman and Mertins [33], [27] who obtained enclosures for eigenvalues in the gap between two parts of the essential spectrum of a self-adjoint operator. This part of our analysis depends heavily upon a theorem of Kato [19] which has been little appreciated outside the numerical analysis community [17].…”

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

“…The main result is stated below, under the assumption that A is bounded. In the article on spectral pollution [7] in 2004, the above method was linked with various techniques due to Lehmann [16], Behnke et al [3], Zimmerman et al [22]. The problem of spurious eigenvalues in a spectral gap was addressed by considering the following function.…”

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

“…These are strongly linked with the classical complementary bounds for eigenvalues by Temple and Lehmann [19, theorem 4.6.3], which played a prominent role in the early days of quantum mechanics (see [20,21]). The so-called quadratic method, developed by Davies [22], Shargorodsky [23] and others [24,25], is an alternative to these approaches.…”