The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. Many examples and exercises are included that focus on quantum mechanics.

The spectral theory of linear operators in Hilbert spaces is the most important tool in the mathematical formulation of quantum mechanics; in fact, linear ope- tors and quantum mechanics have had a symbiotic relationship. However, typical physicstextbooks on quantum mechanics givejust a roughsketch of operator t- ory,occasionallytreating linear operatorsas matricesin ?nite-dimensional spaces; the implicit justi?cation is that the details of the theory of unbounded operators are involved and those texts are most interested in applications. Further, it is also assumed that mathematical intricacies do not show up in the models to be d- cussedorareskippedby heuristicarguments. Inmanyoccasionssomequestions, such as the very de?nition of the hamiltonian domain, are not touched, leaving an open door for controversies, ambiguities and choices guided by personal tastes and ad hoc prescriptions. All in all, sometimes a blank is left in the mathematical background of people interested in nonrelativistic quantum mechanics. Quantum mechanics was the most profound revolution in physics; it is not natural to our common sense (check, for instance, the wave-particle duality) and the mathematics may become crucial when intuition fails. Even some very simple systemspresentnontrivialquestionswhoseanswersneedamathematicalapproach. For example, the Hamiltonian of a quantum particle con?ned to a box involves a choice of boundary conditions at the box ends; since di?erent choices imply di?erentphysicalmodels,studentsshouldbeawareofthebasicdi?cultiesintrinsic tothis(inprinciple)verysimple model,aswellasinmoresophisticatedsituations. The theory of linear operators and their spectra constitute a wide ?eld and it is expected that the selection of topics in this book will help to ?ll this theoretical gap. Ofcoursethisselectionisgreatlybiasedtowardthepreferencesofthe author.

A detailed discussion on the quantum dynamics consequences of point, absolutely continuous and singular continuous spectra, some of them have not appeared in book form yet Discussions on boundary triples, fractal alpha-Hölder spectral measures and the Wonderland theorem are also interesting novelties No knowledge of quantum mechanics is supposed. The needed quantum concepts are introduced when appropriate in the text and rigorously approached The large number of worked examples that illustrate the developed theory, some based on rather recent published papers in the mathematical and physical literature, is also an appealing factor Includes supplementary material: sn.pub/extras

Autorentext

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).

Inhalt
A Glance at Quantum Mechanics.- Linear Operators and Spectra.- Adjoint Operator.- Fourier Transform and Free Hamiltonian.- Operators via Sesquilinear Forms.- Unitary Evolution Groups.- Kato-Rellich Theorem.- Boundary Triples and Self-Adjointness.- Spectral Theorem.- Applications of the Spectral Theorem.- Convergence of Self-Adjoint Operators.- Spectral Decomposition I.- Spectral Decomposition II.- Spectrum and Quantum Dynamics.- Some Quantum Relations.