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Quantum Dynamics and Decompositions of Singular Continuous Spectra (1995)
| Venue: | J. Funct. Anal |
| Citations: | 105 - 10 self |
BibTeX
@ARTICLE{Last95quantumdynamics,
author = {Y. Last},
title = {Quantum Dynamics and Decompositions of Singular Continuous Spectra},
journal = {J. Funct. Anal},
year = {1995},
volume = {142},
pages = {406--445}
}
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Abstract
. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,15--18,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
Keyphrases
quantum dynamic singular continuous spectrum time evolution spectral measure dimensional hausdorff measure schrodinger picture various property spectral decomposition hausdorff measure singular continuous measure self adjoint operator introduction let fractal property several recent paper spectral property separable hilbert space quantum mechanic
