Complex symmetric differential operators on Fock space

TL;DR

This paper characterizes complex symmetric differential operators on Fock space, a key space in quantum mechanics and harmonic analysis, and explores their spectral properties and self-adjoint structures.

Contribution

It provides a complete characterization of complex symmetric differential operators on Fock space and analyzes their spectral properties, including point spectrum computation.

Findings

Characterization of complex symmetric differential operators on Fock space

Analysis of self-adjoint differential operators on Fock space

Explicit computation of the point spectrum for certain operators

Abstract

The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modelling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail.