On the 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}_{hin (0,1]}subset mathrm{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 (mathrm{L}^2({{mathbb {R}}}^n)) or a Schrödinger operator in (mathrm{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 (0le delta .