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A New Formula Relating Localisation Operators to Time Operators

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Spectral Analysis of Quantum Hamiltonians

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 224))

Abstract

We consider in a Hilbert space a self-adjoint operator H and a family Φ ≡ (Φ1,…,Φd) 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.

Mathematics Subject Classification (2010). 46N50, 81Q10, 47A40.

S. Richard

On leave from Université de Lyon, Université Lyon 1, CNRS, UMR5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK

    S. Richard

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

    R. Tiedra de Aldecoa

Authors
  1. S. Richard
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  2. R. Tiedra de Aldecoa
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Correspondence to S. Richard .

Editor information

Editors and Affiliations

  1. , Departamento de Física, P. Universidad Católica de Chile, Santiago, 22, Chile

    Rafael Benguria

  2. , Departamento de Matemáticas, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Eduardo Friedman

  3. , Departamento de Matemática, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Marius Mantoiu

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Richard, S., de Aldecoa, R.T. (2012). A New Formula Relating Localisation Operators to Time Operators. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_14

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