Resolvent expansions and trace regularizations for Schrödinger operators

Abstract

We provide a direct approach to a study of regularized traces for long range Schrödinger operators and small time asymptotics of the heat kernel on the diagonal. The approach does not depend on multiple commutator techniques and improves upon earlier treatments by Agmon and Kannai, Melin, and the authors.

The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that

“…Both Propositions 4.1 and 4.2 can be verified straightforwardly (see [13]) by induction on n. Let us use identity (4.2) to find the AE of the integral kernel R(x, x ; z) of R(z) as |z| → ∞, z ∈ Π θ . According to (4.3), the operator T n (z) = X n R n+1 0 (z) with n 1 has the integral kernel…”

“…We proceed from a modification (see [13]) of the iterated resolvent identity. It can be regarded as a special case of the noncommutative Taylor formula in [16].…”

High Energy Asymptotics and Trace Formulas for the Perturbed Harmonic Oscillator

“…In [7], the coefficients of the asymptotic eigenvalue expansion are calculated by means of recurrence formulae developed from expansions of the trace of the resolvent. In [12], it is shown how similar expansions for the resolvent together with further combinatorial techniques produces the explicit formula (3.1) for the computation of the heat invariants.…”

“…The discussion here of the existence of (D.1), describes the method as shown in [12], but different approaches to this can be found, for example see [2].…”

On the High-Energy Asymptotics of the Integrated Density of States

“…The connection between the iterated resolvent identity and the the expansion (3) has been pointed out in [5]; the identity itself has been proved in [7], but various versions have probably been used many times in the literature. A construction very similar to our proof is used in a recent paper [6].…”