Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions

TL;DR

This paper develops heat kernel asymptotics for the Hodge Laplacian on manifolds with edge singularities, focusing on algebraic self-adjoint extensions and revealing unique phenomena in heat trace behavior.

Contribution

It introduces a microlocal heat kernel construction for algebraic boundary conditions on edge manifolds, extending previous methods to a more complex geometric setting.

Findings

Established heat kernel asymptotics for algebraic extensions of the Hodge operator on edges

Identified exotic phenomena in heat trace asymptotics for non-Friedrichs extensions

Extended microlocal analysis techniques to edge singularities

Abstract

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setup of edges. We then establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in case of a non-Friedrichs extension.