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Demuth, M. (1991), On topics in spectral and stochastic analysis for Schrodinger operators, in "Recent Developments in Quantum Mechanics". Eds. A. Boutel de Monvel et. , Kluwer, 223-242. j Kirsch, W. (1992), Quantitative estimates of the rate of large coupling convergence for Schrodinger operators (to be published). j van Casteren, J. (1989), On spectral theory for selfadjoint Feller generators, Rev. Math. Phys. 1,325-414. j van Casteren, J. (1991), Perturbations for generalized Schrodinger operators in stochastic spectral analysis, Preprint Univ.

Denotes the conditional Wiener measure. ) are the trajectories of the Wiener process. e. by the Wiener density function (2) Therefore the proofs of spectral properties of Ho+ V using the Feynman-Kac-representation in (1) are mainly based on several characteristic features of the Wiener density function Pw. This density function has many "nice" properties. For instance it is symmetric with respect to any coordinate :Xi E R, i = 1, ... , n. It is decreasing if any l:Xil -+ 00, even with the same rate of convergence.

If V is bounded analytic on 5"" with complex threshold less or equal to IIVull. 4 Q' bigger than ~, then H has a Since the RFD(H) v given by theorem 4, is a union of half plane we can look for the boundary f of v by computing the envelop of the family of straight lines Df3. 3) with dirichlet boundary condition at zero. 4) Notice that V is bounded analytic on 5~. 4) it is straight forward to d~rive the parametric equations of the boundary rn of the RF D( H) given by theorem 4. We give in the figure 5 below, the graphs of r n for n = 1,2, ...