engleski

Temple-Kato inequalities and applications in Mathematical Physics

The standard method to assess the accuracy of the Ritz value approximations to the eigenvalues of a self adjoint operator is to apply the variants of Temple-Kato inequalities. In the standard setting one requires a test vector(s) from the domain of the self adjoint operator and a bound on the part of the spectrum one is not interested in. We will present a new class of Temple-Kato like estimates that are particularly suited to the study of nonnegative self adjoint operators. The new estimates are sharper than the standard ones (they do not give a negative lower estimate to the eigenvalue of a positive definite operator) and apply to the broader class of test spaces. Namely, we allow a test space from the form (weak) domain of the operator. In particular, this extension has favorable consequences when one considers finite element approximations of the spectrum (think of Laplace operator and linear finite elements). In this talk we will present the new results and discuss their applicability on the several examples from Mathematical Physics.

eigenvalues; eigenvectors; Ritz values Ritz vectors

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