Description:

Title: Gaussian-weighted normal operators on Euclidean space

Abstract: We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space. 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, considered recently by Jia, Vasy, Zachos et al. in considering integral geometry questions on asymptotically Euclidean spaces. Time permitting, we discuss potential future projects studying the SVD and range characterization of the corresponding Gaussian-weighted X-ray transform, which will require a better understanding of the geometry of the space of lines.

The techinques used in this work are elementary, with related projects potentially suitable for undergraduate/early graduate research--these are aspects I hope to convey in the talk.