On a new formula relating localisation operators to time operators

TL;DR

This paper introduces two new formulas for a time operator in quantum mechanics, relating localisation and time operators, and applies spectral analysis methods to various classes of operators.

Contribution

It proposes and proves the equivalence of two novel formulas for a time operator under regularity conditions, expanding the theoretical framework for quantum operator analysis.

Findings

Derived two new formulas for time operators and proved their equality.

Applied the theory to various operators including Hamiltonians, Dirac, and graph operators.

Conducted spectral analysis using conjugate operator methods.

Abstract

We consider in a Hilbert space a self-adjoint operator H and a family Phi=(Phi_1,...,Phi_d) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to Phi, we propose two new formulae for a time operator for H and prove their equality. One of the expressions is based on the time evolution of an abstract localisation operator defined in terms of Phi while the other one corresponds to a stationary formula. Under the same assumptions, we also conduct the spectral analysis of H by using the method of the conjugate operator. Among other examples, our theory applies to Friedrichs Hamiltonians, Stark Hamiltonians, some Jacobi operators, the Dirac operator, convolution operators on locally compact groups, pseudodifferential operators, adjacency operators on graphs and direct integral operators.