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.Comment: 29 pages, 1 figur
