The generalized $\partial$-complex on the Segal Bargmann space

TL;DR

This paper investigates the properties of a generalized complex Laplacian on the Segal-Bargmann space, extending the understanding of operators related to quantum mechanics within this functional analytic framework.

Contribution

It introduces a generalized $ar{ ext{partial}}$-complex and analyzes the associated complex Laplacian on the Segal-Bargmann space, linking operator theory with quantum mechanics.

Findings

Characterization of the complex Laplacian $ ilde ox_D$ on the Segal-Bargmann space

Analysis of properties of unbounded differential operators of polynomial type

Extension of the $ar{ ext{partial}}$-complex framework to the Segal-Bargmann setting

Abstract

We study certain densely defined unbounded operators on the Segal-Barg\-mann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding $D$-complex and study properties of the corresponding complex Laplacian $\tilde \Box_D = D D^* + D^* D,$ where $D$ is a differential operator of polynomial type.