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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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 153; no. 2; pp. 401 - 487
Main Authors Gimperlein, Heiko, Goffeng, Magnus, Louca, Nikoletta
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.09.2024
Subjects
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Functional Analysis
Matematik
Matematisk analys
Mathematical Analysis
Mathematics
Mathematics and Statistics
Natural Sciences
Naturvetenskap
Partial Differential Equations
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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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-023-0310-3