Spectral inequalities for discrete and continuous differential operators

Abstract

In this thesis spectral inequalities and trace formulae for discrete and continuous differential operators are discussed. We first investigate spectral inequalities for Jacobi operators with matrix-valued potentials and present a new, direct proof of a sharp inequality corresponding to a Lieb–Thirring inequality for the power 3/2 using the commutation method. For the special case of a discrete Schrödinger operator we also prove new inequalities for higher powers of the eigenvalues and the potential and compare our results to previously established bounds. We then approximate a Schrödinger operator on L^2(\R) by Jacobi operators on \ell^2(\Z) and use the established inequalities to provide new proofs of sharp Lieb–Thirring inequalities for the powers \gamma=1/2 and \gamma=3/2. By means of interpolation we derive spectral inequalities for Jacobi operators that yield (non-sharp) Lieb–Thirring constants on the real line for powers 1/2<\gamma<3/2. We then consider Schrödinger operators on a finite interval [0,b] with matrix-valued potentials and establish trace formulae of the Buslaev–Faddeev–Zakharov type. The results link sums of powers of the negative eigenvalues to terms dependent on the potential and scattering functions. Finally, we discuss the Berezin inequality, which is well-known on sets of finite measure and find an analogous inequality for the magnetic operator with constant magnetic field on a set whose complement has finite measure. We obtain a similar bound for the Heisenberg sub-Laplacian.