Schrödinger Operators

This chapter can be viewed as a crash course on Schrödinger operators. We discuss their definitions and some of their spectral properties. We illustrate the abstract theory of Chapter I by the concrete examples of the generalized Laplacian operators in Section II.1 and their perturbations by multiplication operators in Section II.2. The latter are of special importance since they are the Schrödinger operators we want to study. We consider the problem of the essential self-adjointness of these perturbations. We devote a subsection to the properties of the trace class perturbation theory in connection with the stability of the absolutely continuous spectrum. Obviously, the choice of the models we consider and of the techniques we use is strongly biased by the applications we contemplate. We consider only the so-called two-body Hamiltonians and we choose to give a probabilistic proof whenever such a proof is available. It turns out that, once properly defined as self-adjoint operators, Schrödinger operators generate semigroups (when they are bounded below) and unitary groups (when they are defined on the lattice ℤd) which can be written using path integrals. We discuss the relevant formulas in Section II.3 of this chapter. They will be very useful in the investigation of the so-called integrated density of states in Chapter VI. Finally, we use these path integral methods to show the regularity properties of the semigroups and the decay of the L2-eigenfunctions.