Semiclassical analysis of a nonlocal boundary value problem related to magnitude
We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential bo...
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Published in | Journal d'analyse mathématique (Jerusalem) Vol. 153; no. 2; pp. 401 - 487 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Jerusalem
The Hebrew University Magnes Press
01.09.2024
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Subjects | |
Online Access | Get full text |
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Summary: | We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels. |
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ISSN: | 0021-7670 1565-8538 |
DOI: | 10.1007/s11854-023-0310-3 |