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| HAL: hal-00411376, version 1 |
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| On a new formula relating localisation operators to time operators |
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| Serge Richard 1, 2Rafael Tiedra De Aldecoa 3 |
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| (2009-08-27) |
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| 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. |
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| 1: | Institut Camille Jordan (ICJ) |
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CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon |
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| 2: | DPMMS/CMS |
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University of Cambridge |
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| 3: | Facultad de Matemáticas |
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Pontificia Universidad Catolica |
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| Subject | : | Physics/Mathematical Physics Mathematics/Mathematical Physics |
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| Attached file list to this document: | |||||
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| hal-00411376, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00411376 | |
| oai:hal.archives-ouvertes.fr:hal-00411376 | |
| From: Aurélie Reymond | |
| Submitted on: Thursday, 27 August 2009 11:17:33 | |
| Updated on: Sunday, 30 August 2009 10:04:54 | |
