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Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the self--adjoint extensions of H can have inside the gap J and provide methods how to construct self--adjoint extensions of H with prescribed spectral properties inside J . Under some weak assumptions about the operator H which are satisfied, e. g., provided the deficiency indices of H are infinite and the operator (H \Gamma ) \Gamma1 is compact for one regular point of H, we shall show that for every (auxiliary) self--adjoint operator M 0 in the Hilbert space H and every open subset J 0 of the gap J of H there exists a self--adjoint extension ~ H of H such that inside J the self--adjoint extension ~ H of H has the same absolutely continuous and the same point spectrum as the given operator M 0 and the singular continuous spectrum of ~ H in J equals the closure of J 0 in J . Moreover we shall present a method how to ...

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre-Dirichlet form E \Gamma ß oe of the Poisson measure ß oe in terms of the extrinsic one E \Gamma ß oe ;H X oe . More precisely, replacing ß oe by a Gibbs measure ¯ on the configuration space \Gamma X we derive a relation between the intrinsic pre-Dirichlet form E \Gamma ¯ of the measure ¯ and the extrinsic one E P ¯;H X oe . As a consequence we prove the closability of E \Gamma ¯ on L 2 (\Gamma X ; ¯) under very general assumptions on the interaction potential of the Gibbs measures µ.

We discuss the Schrödinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the one-dimensional Laplacian on a segment.

We discuss the Schr#odinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the one-dimensional Laplacian on a segment. 1 1 Introduction The idea to perturb the Schr#odinger operator by an object living on a set of vanishing Lebesgue measure originated in physics. Atypical problem treated in Quantum Mechanics is to describe the motion of a non-relativistic particle moving in the #eld of external potential forces. The corresponding Hamiltonian is then given by H = ,#+V #1.1# where # is the Laplace operator in IR n with the Dirichlet boundary condition at in#nity and V a measurable function. If one deals with the short-range forces around say point 0, then V is supported by a small vicinityof0.Itis then natural to think of idealization leading to zero-range potential supported by a point. Symbolically one would then have H = ,#+## 0 #1.2# or more...

For a large class of operators H a formula for the resolvent will be derived and sufficient conditions will be given in order that the resolvent difference (H \Gamma z) is compact resp. the wave (H; H 0 ) exist and are complete; here H 0 denotes the free quantum mechanical Hamiltonian. The mentioned class of operators contains, a.o., the generator of a Brownian motion with killing, the generator of the superposition of a Brownian motion and a diffusion process on a submanifold and Hamiltonians describing the interaction of a quantum mechanical particle with a potential supported by a set with classical capacity zero.