Resolvents and complex powers of semiclassical cone operators

TL;DR

This paper provides a uniform microlocal analysis framework for resolvents and complex powers of semiclassical cone operators, especially near the semiclassical limit, with applications to Laplacians on conic manifolds.

Contribution

It introduces a constructive geometric microlocal approach to analyze resolvents and complex powers of semiclassical cone operators as the parameter approaches zero.

Findings

Constructed Schwartz kernels as conormal distributions on a resolution space.

Characterized domains of fractional powers of the semiclassical Laplacian.

Proved propagation of semiclassical regularity through cone points.

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 $h^2\Delta_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ 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 $[0,1)_h\times X\times X$ 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 $(h^2\Delta_g+1)^{w/2}$ for $\mathrm{Re}\,w\in(-\frac{n}{2},\frac{n}{2})$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.