  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.
