Semiclassical analysis of a nonlocal boundary value problem related to magnitude

TL;DR

This paper investigates a boundary value problem involving the fractional Laplacian on manifolds, linking the magnitude invariant of metric spaces to geometric properties through semiclassical analysis.

Contribution

It introduces a semiclassical approach to analyze the magnitude invariant via a nonlocal boundary problem, connecting geometric invariants with spectral properties.

Findings

Asymptotic expansion of magnitude in terms of curvature invariants

Structure of the solution operator for the boundary problem

Relation between magnitude and heat kernel invariants

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.