Semiclassical functional calculus for $h$-dependent functions

TL;DR

This paper develops a semiclassical functional calculus for $h$-dependent functions of pseudodifferential operators, providing an explicit trace formula with remainder estimates for spectral analysis in small windows.

Contribution

It introduces a new semiclassical trace formula for $f_h(P(h))$ with explicit remainder estimates, suitable for spectral analysis in narrow spectral windows.

Findings

Derived an explicit trace formula with remainder estimates

Applicable to operators on Euclidean space and closed manifolds

Facilitates spectral analysis in windows of size $h^elta$

Abstract

We study the functional calculus for operators of the form $f_h(P(h))$ within the theory of semiclassical pseudodifferential operators, where $\{f_h\}_{h\in (0,1]}\subset C^\infty_c(\mathbb{R})$ denotes a family of $h$-dependent functions satisfying some regularity conditions, and $P(h)$ is either an appropriate self-adjoint semiclassical pseudodifferential operator in $L^2(\mathbb{R}^n)$ or a Schr\"odinger operator in $L^2(M)$, $M$ being a closed Riemannian manifold of dimension $n$. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of $P(h)$ in spectral windows of width of order $h^\delta$, where $0\leq \delta <\frac{1}{2}$.