On a new formula relating localisation operators to time operators

On a new formula relating localisation operators to time operators S. Richard1? and R. Tiedra de Aldecoa2 1 Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sci- ences, University of Cambridge, Cambridge, CB3 0WB, United Kingdom 2 Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Av. Vicun˜a Mackenna 4860, Santiago, Chile E-mails: , Abstract We consider in a Hilbert space a self-adjoint operator H and a family ? ? (?1, . . . ,?d) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to ?, 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 ? 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. 2000 Mathematics Subject Classification: 46N50, 81Q10, 47A40.

• morphism property


• a?? ?


• let assumptions


• main results


• e?ix·?h ?j


• quantum time


• commutation relation


• self- adjoint operator


• position operators


• self-adjoint operators