Semiclassical functional calculus for $h$-dependent functions

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 $0\leq \delta <\frac{1}{2}$.