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- W2069333164 abstract "In this paper we investigate Schrödinger operators L =− Δ g + a ( x ) on a compact Riemannian manifold ( M , g ), where the potential function a ( x ) is assumed to be continuous, but not necessarily bounded, outside of some closed set Σ ⊂ M of measure zero. Under certain geometric hypotheses on Σ and growth conditions on a ( x ) as x → Σ , we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a ( x ) is allowed to approach −∞ as x → Σ . We also consider conditions on Σ and a ( x ) under which the Sturm–Liouville theory of L is “singular” in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L ; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally “essentially Dirichlet” (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Σ is a k -dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets." @default.
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- W2069333164 date "2001-11-01" @default.
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- W2069333164 title "Singular Sturm–Liouville Theory on Manifolds" @default.
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- W2069333164 doi "https://doi.org/10.1006/jdeq.2001.3986" @default.
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