Abstract
The main purpose of this chapter is to develop in the context of obstacles, a collection of methods and quantitative bounds, relating the Dirichlet-Schrödinger semigroups of Chapter 1 and the bottom of their spectrum. General methods and some first examples are discussed in Section 1. Section 2 presents two instances of the profound link between capacity and bottom of the spectrum. In Section 3 we develop certain one-dimensional bounds on principal Dirichlet eigenvalues of the Laplacian in the presence of obstacles. We apply these bounds to study the asymptotic behavior of a certain random variational problem for Poissonian obstacles. This is a caricature of the ‘pinning effect’ we shall discuss in Chapter 6.
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Sznitman, AS. (1998). Some Principal Eigenvalue Estimates. In: Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11281-6_3
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DOI: https://doi.org/10.1007/978-3-662-11281-6_3
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