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Contractivity properties of Schrödinger semigroups on bounded domains. (English) Zbl 0856.35031

The paper proves that intrinsic ultracontractivity (IUC) holds, in a variety of cases, for the Schrödinger operator \(H= -\Delta+ V\) with Dirichlet boundary conditions on bounded domains \(\Omega\) in \(\mathbb{R}^n\). The potential \(V\) is chosen to be positive, but it is neither assumed to belong to the Kato class, nor to be relatively form-bounded w.r.t. the Dirichlet Laplacian on \(\Omega\). The class of domains considered includes John domains and a class of Hölder domains. Finally, it is given an example of a bounded domain \(\Omega\) on which the Dirichlet Laplacian is not IUC, but on which \(- \Delta+ V\) is IUC for a positive potential \(V\) diverging sufficiently fast near the boundary of \(\Omega\).
Reviewer: G.Gabriele (Udine)

MSC:

35J10 Schrödinger operator, Schrödinger equation
47D07 Markov semigroups and applications to diffusion processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
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