Overview

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 singularcontinuous 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.

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9783031382888

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​Moacir Aloisio is an Adjunct Professor at the Math department of the Federal University of Jequitinhonha and Mucuri Valleys, Brazil. He earned his PhD in mathematics from the Federal University of Minas Gerais, Brazil, in 2019. His research interests lie in Operator Theory, Mathematical Physics, and Dynamical Systems. Some allied areas include Schrödinger and Dirac operators, quantum (in)stability, dynamic localization, abstract differential equations and operator algebras. Silas L. Carvalho has earned his PhD in Physics at the University of São Paulo, Brazil, in 2010. From 2011 to 2013, he was an Adjunct Professor at the Federal University of São Paulo. Since then, Dr. Carvalho has been serving as an Adjunct Professor at the Math department of the Federal University of Minas Gerais. He develops research in Mathematical Physics, Ergodic Theory and Dynamical Systems, mainly in problems that involve fractal dimensions and measures. Some allied areas include Schrödinger and Dirac operators, quantum dynamics, and stability problems involving $C_0$-semigroups. César R. de Oliveira is a Full Professor at the Math department of the Federal University of São Carlos, Brazil. His field of research is Mathematical Physics; more specifically, spectral theory of Schrödinger operators, the Aharonov-Bohm effect, and effective operators in systems with reduction of dimensions. He has spent extended research visits at the University of British Columbia, Canada, and Milan University, Italy. Dr. Oliveira has authored three books, including “Intermediate Spectral Theory and Quantum Dynamics” (Birkhäuser, 2009, ISBN 978-3-7643-8794-5).

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