On the discrete spectrum of partial differential and integral operators

This thesis consists of four papers on various topics of concrete spectral analysis of partial differential and integral operators. Its main object is the investigation of the discrete spectrum, appearing under the perturbation of a partial differential operator in spectral gaps of the unperturbed ones. The following results are obtained.

A. The sum of the square roots of the absolute values of the negative eigenvalues of all one-dimensional Schrodinger operator is bounded by the $L\sb1$-norm of the potential. This proves the remaining "limit" case in the scale of the Lieb-Thirring estimates.

B. It will be shown, that the behavior of the singular numbers of compact integral operators a(x)b(D) is dominated by the behavior of the distribution function of the non-smooth symbol $a(x)b(\xi)$ in the corresponding phase space. The respective estimate generalizes and unifies a scale of Cwikel type inequalities.

C. An abstract criteria for the appearance of negative spectrum for elliptic operators, perturbed by non-signdefined potentials with arbitrary small coupling constants, will be given. We apply this criteria to Schrodinger type operators, and generalize some results of Simon and Pinchover.

D. Second order differential operators with magnetic and periodic electric fields, perturbed by a strong local impurity, shall be considered. We show, that the main term of the asymptotics of the number of bound states, as the coupling constant tends to infinity, does not depend on the magnetic or periodic electric field. This holds for non-regular potentials with non-Weyl asymptotics.

In the papers B and D interpolation techniques and results on operator ideals will be exploited. The author uses here results of two former papers, which are not included in the thesis.