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Published in:
2012 | OriginalPaper | Chapter
A New Formula Relating Localisation Operators to Time Operators
Authors : S. Richard, R. Tiedra de Aldecoa
Published in: Spectral Analysis of Quantum Hamiltonians
Publisher: Springer Basel
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We consider in a Hilbert space a self-adjoint operator
H
and a family Φ
≡
(Φ1,…,Φ
d
) of mutually commuting self-adjoint operators.Unde r some regularity properties of
H
with respect to Φ, we propose two new formulae for a time operator for
H
and prove their equality.O ne of the expressions is based on the time evolution of an abstract localisation operator defined in terms of Φ 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.