Semiclassical functional calculus for $h$-dependent functions

Abstract

We study the functional calculus for operators of the form $f_h(P(h))$ within the theory of semiclassical pseudodifferential operators, where $\{f_h{\}}_{h\in (0,1]}\subset C_c^{\infty }(\mathbb{R})$ 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 Schrödinger 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 $0\le \delta <\frac{1}{2}$.

v2: minor corrections, 33 pages