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

A New Formula Relating Localisation Operators to Time Operators