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Spectral measures and dynamics : typical behaviors / Moacir Aloisio, Silas L. Carvalho, César R. de Oliveira.

9783031382895 (electronic bk.)
3031382897 (electronic bk.)
9783031382888
3031382889

Cham : Springer, 2023.

English

1 online resource (xi, 246 pages) : color illustrations.

10.1007/978-3-031-38289-5 doi

QA320 .A46 2023

515/.7222

This book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors. A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years. A valuable example of such subtleties is found in the so-called "Wonderland theorem" first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singular continuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research. Graduate students with a specific interest in the spectral properties of spectral measure are the primary target audience for this work, while researchers benefit from a selection of important results, many of them presented in the book format for the first time.

Includes bibliographical references and index.

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Online resource; title from PDF title page (SpringerLink, viewed October 30, 2023).

Latin American Mathematics Series. UFSCar subseries, 2524-6763

Print version: 9783031382888

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