Determining Spectra in Quantum Theory

1. Front Matter

   Pages i-xi

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2. Measures and Transforms

   Pages 1-27

3. Selfadjointness and Spectrum

   Pages 29-58

4. Criteria for Identifying the Spectrum

   Pages 59-109

5. Operators of Interest

   Pages 111-151

6. Applications

   Pages 153-201

7. Back Matter

   Pages 203-219

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Themainobjectiveofthisbookistogiveacollectionofcriteriaavailablein the spectral theory of selfadjoint operators, and to identify the spectrum and its components in the Lebesgue decomposition. Many of these criteria were published in several articles in di?erent journals. We collected them, added some and gave some overview that can serve as a platform for further research activities. Spectral theory of Schr¨ odinger type operators has a long history; however the most widely used methods were limited in number. For any selfadjoint operatorA on a separable Hilbert space the spectrum is identi?ed by looking atthetotalspectralmeasureassociatedwithit;oftenstudyingsuchameasure meant looking at some transform of the measure. The transforms were of the form f,?(A)f which is expressible, by the spectral theorem, as ?(x)dµ (x) for some ?nite measureµ . The two most widely used functions? were the sx ?1 exponential function?(x)=e and the inverse function?(x)=(x?z) . These functions are “usable” in the sense that they can be manipulated with respect to addition of operators, which is what one considers most often in the spectral theory of Schr¨ odinger type operators. Starting with this basic structure we look at the transforms of measures from which we can recover the measures and their components in Chapter 1. In Chapter 2 we repeat the standard spectral theory of selfadjoint op- ators. The spectral theorem is given also in the Hahn–Hellinger form. Both Chapter 1 and Chapter 2 also serve to introduce a series of de?nitions and notations, as they prepare the background which is necessary for the criteria in Chapter 3.

• Potential
• disordered system
• quantum theory
• scattering theory
• spectral theory
• wavelet
• partial differential equations

“In my opinion, the basic idea of the monograph is to help graduate students working on spectral theory and beginning researchers in the field to build a toolkit. The book definitely has more than enough material for this purpose, some of which is quite advanced, and it is very up to date. ... In conclusion, I believe that this book will prove extremely useful for its target audience (advanced graduate students with an interest in this area). Moreover, it will also very much feel at home on the bookshelf of an expert.”(MATHEMATICAL REVIEWS)

• Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zellerfeld, Germany

  Michael Demuth

• Institute of Mathematical Sciences, CIT Campus — Taramani, Chennai, India

  Maddaly Krishna

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