Self-adjointness of unbounded time operators

TL;DR

This paper constructs a self-adjoint, unbounded time operator for a semi-bounded self-adjoint operator with discrete spectrum, expanding the understanding of time operators in quantum mechanics.

Contribution

It introduces a new construction of a self-adjoint, unbounded time operator for operators with purely discrete spectrum, extending previous bounded cases.

Findings

Constructed a self-adjoint, unbounded time operator for $H$.

Extended the class of known time operators beyond bounded cases.

Provides mathematical framework for unbounded time operators in quantum theory.

Abstract

Time operators for an abstract semi-bounded self-adjoint operator $H$ with purely discrete spectrum is considered. The existence of a bounded self-adjoint time operator $T$ for $H$ is known as Galapon time operator. In this paper, a self-adjoint but unbounded time operator $T$ of $H$ is constructed.