Ultra-Weak Time Operators of Schroedinger Operators

TL;DR

This paper introduces the concept of ultra-weak time operators for Schrödinger operators, establishing their existence under certain conditions and expanding the understanding of time operators in quantum mechanics.

Contribution

It defines ultra-weak time operators, proves their existence for Schrödinger operators with specific potentials, and identifies functions of these operators that also admit such operators.

Findings

Ultra-weak time operators exist for Schrödinger operators with certain potentials.

The Hamiltonian of the hydrogen atom admits an ultra-weak time operator.

Certain Borel measurable functions of Schrödinger operators have ultra-weak time operators.

Abstract

In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schroedinger operators H with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions $f$ such that $f(H)$ has an ultra-weak time operator is found.