Strong time operators associated with generalized Hamiltonians

TL;DR

This paper constructs a new class of symmetric operators related to generalized Hamiltonians that satisfy the weak Weyl relation, expanding the framework of time operators in quantum mechanics.

Contribution

It introduces a method to generate symmetric operators obeying the weak Weyl relation from functions of Hamiltonians, broadening the understanding of time operators.

Findings

Established conditions for constructing symmetric operators from functions of Hamiltonians.

Extended the weak Weyl relation to a wider class of operators.

Provided a framework for analyzing time operators associated with generalized Hamiltonians.

Abstract

Let the pair of operators, $(H, T)$, satisfy the weak Weyl relation: $Te^{-itH} = e^{-itH}(T + t)$, where $H$ is self-adjoint and $T$ is closed symmetric. Suppose that g is a realvalued Lebesgue measurable function on $\RR$ such that $g \in C^2(R K)$ for some closed subset $K \subset \RR$ with Lebesgue measure zero. Then we can construct a closed symmetric operator $D$ such that $(g(H), D)$ also obeys the weak Weyl relation.