Infinite Matrix Representations of Isotropic Pseudodifferential Operators

TL;DR

This paper characterizes isotropic pseudodifferential operators through their infinite matrix representations in Hermite functions, establishing decay properties and differences that relate to the operators' order, and proves an analogue of Beal's theorem.

Contribution

It introduces a novel characterization of isotropic pseudodifferential operators via their Hermite matrix entries and their decay properties, extending Beal's theorem to this class.

Findings

Matrix entries decay rapidly off the diagonal

Discrete difference operators reduce the order by one

Provides an analogue of Beal's theorem for isotropic pseudodifferential operators

Abstract

We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of order r if and only if its "matrix" $(K(A))_{m,n} := < A\phi_n,\phi_m>_{L^2(\mathbb{R})}$ is rapidly decreasing away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where applying the discrete difference operator along the diagonal decreases the order by one. Additionally, we use this result to prove an analogue of Beal's theorem for isotropic pseudodifferential operators.