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Domain perturbations, Brownian motion, capacities, and ground states of Dirichlet Schrödinger operators. (English) Zbl 0791.60072
Let $$\Omega \subset \mathbb{R}^ d$$, $$d \in \mathbb{N}$$, be open and connected and denote by $$-\Delta_ \Omega$$ the Dirichlet Laplacian in $$L^ 2(\Omega)$$. If $$\Lambda \subset \Omega$$ is another open connected set, then monotonicity of the spectrum $$\sigma (-\Delta_ \Omega)$$ of $$- \Delta_ \Omega$$ with respect to $$\Omega$$ yields the well known inequality $\inf [\sigma (-\Delta_ \Omega)] \leq \inf [\sigma (- \Delta_ \Lambda)]. \tag{*}$ We prove that if $$\inf [\sigma (-\Delta_ \Lambda)]$$ is actually an eigenvalue of $$-\Delta_ \Lambda$$, then the inequality (*) is strict if and only if the capacity of $$\Omega \backslash \Lambda$$ is strictly positive. This result is in fact proven for Dirichlet Schrödinger operators of the type $$H_ \Omega= - \Delta_ \Omega+q$$ under very general assumptions on the potential function $$q$$ by purely probabilistic techniques. (It is also shown that this result does not extend to higher eigenvalues of $$H_ \Lambda$$, $$H_ \Omega$$ in general).
Reviewer: F.Gesztesy

##### MSC:
 60J65 Brownian motion 60J45 Probabilistic potential theory
##### Keywords:
domain perturbations; Dirichlet Schrödinger operators
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##### References:
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