Semiclassical analysis of a nonlocal boundary value problem related to magnitude

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.

arXiv comment54 pages, Appendices shortened, detailed computations available in v1. This revised version appears in Journal d'Analyse Mathématique

References in corpus (14)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Large N expansion for the 2D Dyson gas
• Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators
• On the asymptotic magnitude of subsets of Euclidean space
• Spectral results for mixed problems and fractional elliptic operators
• The magnitude of a metric space: from category theory to geometric measure theory
• Positive definite metric spaces
• On the magnitude of spheres, surfaces and other homogeneous spaces
• Integration by parts and Pohozaev identities for space-dependent fractional-order operators
• Magnitude, diversity, capacities, and dimensions of metric spaces
• On the magnitude function of domains in Euclidean space
• Persistent Magnitude
• The Willmore energy and the magnitude of Euclidean domains
• On the magnitude and intrinsic volumes of a convex body in Euclidean space

Cited by in corpus (1)

• The Laplace-Beltrami Operator on the Surface of the Ellipsoid