Complete Differentiable Semiclassical Spectral Asymptotics

TL;DR

This paper develops a comprehensive and differentiable semiclassical spectral asymptotic analysis for operators with decaying potentials, providing detailed insights into the spectral projector kernel behavior.

Contribution

It introduces a complete and differentiable asymptotic framework for the spectral projector kernel of semiclassical operators with decaying potentials.

Findings

Established complete asymptotics of the spectral projector kernel.

Proved differentiability with respect to spectral parameter $ au$.

Applicable under certain decay and regularity assumptions.

Abstract

For an operator $A:= A_h= A^0(hD) + V(x,hD)$ with a "potential" $V$ decaying as $|x|\to \infty$ we establish under certain assumptions the complete and differentiable with respect to $\tau$ asymptotics of $e_h(x,x,\tau)$ where $e_h(x,y,\tau)$ is the Schwartz kernel of the spectral projector.