Heat asymptotics with spectral boundary conditions

Abstract

Let P be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type.

On Properties of the Asymptotic Expansion of the Heat Trace for the N/D Problem

“…There is one setting where such terms are known to arise. If instead of local boundary conditions, spectral conditions are imposed, one has a partial asymptotic expansion of the form, a(D, B)(t) = n<m a n (D, B)t (n−m)/2 + O(t Again the invariants are locally computable and we refer to [7,12] for formulae if n ≤ 3. However, the complete asymptotic expansion involves non-local and log terms [17,18].…”

Heat Content Asymptotics for Operators of Laplace Type with Spectral Boundary Conditions

“…Modulo terms in L aa , we can replace γ m ψ A γ m by ψ A # and ψ A by − ψ A # γ m . By Lemma 2.1 (5), γ m L aa can not appear. Furthermore, the invariants γ a ψ P γ a and γ a γ m ψ P γ a would violate Remark 2.5.…”

Heat trace asymptotics and the Gauss–Bonnet theorem for general connections

“…We then have {∆ n−1 Tr(E)}(x) = 0 for n ≥ 1. Theorem 3 now follows from Equation (9) and from Equation (10). ⊓ ⊔…”