Self-Adjointness of Toeplitz Operators on the Segal-Bargmann Space

TL;DR

This paper introduces a new criterion for establishing the self-adjointness of Toeplitz operators with unbounded, operator-valued symbols on the Segal-Bargmann space, extending existing estimates and applying to quantum mechanics contexts.

Contribution

It presents a novel criterion for self-adjointness of Toeplitz operators with unbounded symbols, extending the Berger-Coburn estimate to vector-valued spaces and applying to Schrödinger operators.

Findings

Established a new self-adjointness criterion for Toeplitz operators.

Extended Berger-Coburn estimate to vector-valued Segal-Bargmann spaces.

Proved self-adjointness for quadratic forms in Schrödinger representation.

Abstract

We prove a new criterion that guarantees self-adjointness of Toeplitz operator with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schr\"odinger representation.