Resolvents and complex powers of semiclassical cone operators

Abstract

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter h tends to 0. An example of such an operator is the shifted semiclassical Laplacian h 2 Δ g + 1 $h^2\Delta _g+1$ on a manifold ( X , g ) $(X,g)$ of dimension n ≥ 3 $n\ge 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space [ 0 , 1 ) h × X × X $[0,1)_h\times X\times X$ of h-dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of ( h 2 Δ g + 1 ) w / 2 ${\big (h^2\Delta _g+1\big )}^{w/2}$ for Re w ∈ − n 2 , n 2 $\operatorname{Re}w\in \left(-\tfrac{n}{2},\tfrac{n}{2}\right)$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.