Pseudodifferential Operators on Asymptotically Conic Manifolds

On Euclidean space there are numerous pseudodifferential calculi that are adapted to the analysis of elliptic (or hypoelliptic) operators. They generally fit under the umbrella of the Hörmander-Weyl calculus. One particularly relevant pseudodifferential calculus on Euclidean space is the SG-calculus, which is characterized by requiring product-type symbolic estimates separately on the variables and covariables of the symbols of the operators, and SG-operators therefore have two separate orders. The SG-calculus has found many applications in the analysis of operators, including in Fredholm and index theory for elliptic operators, scattering theory, as well as to analyze the propagation of singularities and decay of solutions to PDEs using SG-wavefront sets. The SG-calculus makes sense on asymptotically Euclidean manifolds (compact perturbations of Euclidean space, i.e., manifolds that look like Euclidean space at infinity), and more generally on asymptotically conic manifolds, and is also known as the scattering calculus in that case after suitably compactifying the noncompact end to a manifold with boundary.

Notably, the harmonic oscillator, and more general anharmonic oscillators, are not fully elliptic in the SG-calculus on Euclidean space, and thus the SG-calculus does not prove the expected behavior of these operators, such as absence of essential spectrum or decay of eigenfunctions. For the harmonic oscillator the so-called Shubin calculus accomplishes this. More recently, Chatzakou, Delgado, and Ruzhansky studied a class of anharmonic oscillators using the Hörmander-Weyl calculus on Euclidean space, in part motivated by the appearance of such operators in the representation theory of certain nilpotent Lie groups. Their pseudodifferential calculus is an adaptation of the Shubin calculus.

In this talk we explain that one can define a version of the Shubin calculus on asymptotically conic manifolds. The calculus comes with attendant Sobolev spaces, and provides parametrices in the calculus for fully elliptic operators modulo integral operators with rapidly decreasing kernels. The Laplacian with polynomially growing potentials is fully elliptic in the calculus, and we obtain sharp elliptic regularity results, absence of essential spectrum, and rapid decay of eigenfunctions at infinity as consequences of the properties of this calculus.