Gaussian-weighted normal operators on Euclidean space

TL;DR

This paper studies the spectral properties of a Gaussian-weighted normal operator related to the X-ray transform in Euclidean space, revealing its eigenfunctions and spectrum connection to elliptic operators.

Contribution

It characterizes eigenfunctions of the Gaussian-weighted normal operator as joint eigenfunctions of harmonic oscillator and spherical Laplacian, linking spectrum to elliptic operators.

Findings

Eigenfunctions are joint eigenfunctions of harmonic oscillator and spherical Laplacian.

Spectrum relates to elliptic operators in the 1-cusp pseudodifferential calculus.

Provides spectral analysis for Gaussian-weighted X-ray transform operators.

Abstract

We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space with dimension at least 3. We show the eigenfunctions of the normal operator are joint eigenfunctions of the harmonic oscillator and the spherical Laplacian, and we relate the spectrum to that of elliptic operators in the 1-cusp pseudodifferential calculus.