Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions

Abstract

We consider the Hodge-Laplace operator on manifolds with incomplete edge singularities and an intricate elliptic boundary value theory. We single out the class of algebraic self-adjoint extensions for the Hodge Laplacian. 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 setting of edges. We 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 the case of a non-Friedrichs extension.