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Birkhäuser - Birkhäuser Physics | Intermediate Spectral Theory and Quantum Dynamics

Intermediate Spectral Theory and Quantum Dynamics

de Oliveira, César R.

2009, XIII, 410p.

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  • A detailed discussion on the quantum dynamics consequences of point, absolutely continuous and singular continuous spectra; some of these have not appeared in book form yet
  • Novel discussions on boundary triples, "fractal" alpha-Hölder spectral measures and the Wonderland theorem
  • No knowledge of quantum mechanics is supposed; the required quantum concepts are introduced when appropriate and rigorously approached
  • Large numbers of worked examples illustrate the developed theory, some based on recently published papers in the mathematical and physical literature
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.

Content Level » Research

Keywords » Potential - Quantum Dynamics - Quantum return probability - Schrödinger Operators - Spectral Theory - mathematical physics - mechanics - quantum mechanics - self-adjoint operators - spectral decomposition

Related subjects » Birkhäuser Physics

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