Enclosures for the eigenvalues of self-adjoint operators and applications to Schrodinger operators

Abstract

This thesis concerns how to compute upper and lower bounds for the eigenvalues of self-adjoint operators. We discuss two different methods: the so-called quadratic method and the Zimmermann-Mertins method. We know that the classical methods of computing the spectrum of a self-adjoint operator often lead to spurious eigenvalues in gaps between two parts of the essential spectrum. The methods to be examined have been studied recently in connection with the phenomenon of spectral pollution. In the first part of the thesis we show how to obtain enclosures of the eigenvalues in both the quadratic method and the Zimmermann-Mertins method. We examine the convergence properties of these methods for computing corresponding upper and lower bounds in the case of semi-definite self-adjoint operators with compact resolvent. In the second part of the thesis we find concrete asymptotic bounds for the size of the enclosure and study their optimality in the context of one-dimensional Schr¨odinger operators. The effectiveness of these methods is then illustrated by numerical experiments on the harmonic and the anharmonic oscillators. We compare these two methods, and establish which one is better suited in terms of accuracy and efficiency.