Semiclassical analysis of a nonlocal boundary value problem related to magnitude

Heiko Gimperlein, Magnus Goffeng*, Nikoletta Louca

*Corresponding author for this work

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Abstract

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.

Original languageEnglish
JournalJournal d'Analyse Mathematique
Early online date12 Dec 2023
DOIs
Publication statusE-pub ahead of print - 12 Dec 2023

ASJC Scopus subject areas

  • Analysis
  • General Mathematics

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