Spectral Measures and Dynamics: Typical Behaviors

This book explores spectral measures, which are essential in quantum mechanics, and presents classic and advanced topics. It highlights recent discoveries about singular continuity and generalized dimensions, with implications for dynamics, ergodicity, and evolution semigroups. Graduate students and researchers in the field will find it valuable.

Format: Hardback
Length: 246 pages
Publication date: 28 October 2023
Publisher: Springer International Publishing AG

This book brings together and expands on existing knowledge about spectral measures, which have been scattered across various literature. It begins with fundamental topics such as the Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, and then delves into more advanced material, some of which was developed by the authors themselves. The spectral measure is a crucial concept in the mathematical theory of quantum mechanics, connecting the components of the quantum state to the energy levels of the Hamiltonian operator and the dynamics of the state. However, these connections are not straightforward, and recent years have revealed many nuances and subtleties. One notable example is the "Wonderland theorem" first published by B. Simon in 1995, which demonstrates that, for certain metric spaces 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 further revealed that spectral measures exhibit other generic properties, including upper and lower dimensions, generalized dimensions, dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. These properties have opened up new avenues of research and have the potential to inspire graduate students and researchers alike. The primary target audience for this work is graduate students with a specific interest in the spectral properties of spectral measure, while researchers will benefit from a selection of important results, many of which are presented in the book format for the first time.

Dimension: 235 x 155 (mm)
ISBN-13: 9783031382888
Edition number: 1st ed. 2023