C0-Groups, Commutator Methods and Spectral Theory of N-Body HamiltoniansThe relevance of commutator methods in spectral and scattering theory has been known for a long time, and numerous interesting results have been ob tained by such methods. The reader may find a description and references in the books by Putnam [Pu], Reed-Simon [RS] and Baumgartel-Wollenberg [BW] for example. A new point of view emerged around 1979 with the work of E. Mourre in which the method of locally conjugate operators was introduced. His idea proved to be remarkably fruitful in establishing detailed spectral properties of N-body Hamiltonians. A problem that was considered extremely difficult be fore that time, the proof of the absence of a singularly continuous spectrum for such operators, was then solved in a rather straightforward manner (by E. Mourre himself for N = 3 and by P. Perry, 1. Sigal and B. Simon for general N). The Mourre estimate, which is the main input of the method, also has consequences concerning the behaviour of N-body systems at large times. A deeper study of such propagation properties allowed 1. Sigal and A. Soffer in 1985 to prove existence and completeness of wave operators for N-body systems with short range interactions without implicit conditions on the potentials (for N = 3, similar results were obtained before by means of purely time-dependent methods by V. Enss and by K. Sinha, M. Krishna and P. Muthuramalingam). Our interest in commutator methods was raised by the major achievements mentioned above. |
Contents
Chapter 2 | 29 |
Chapter 3 | 67 |
CoGroups and Functional Calculi | 73 |
Chapter 4 | 164 |
Some Examples of CoGroups | 171 |
Chapter 5 | 192 |
Differentiability Properties of OperatorValued | 231 |
The Conjugate Operator Method | 249 |
Remarks on the Functional Calculus Associated | 264 |
The Gronwall Lemma | 349 |
Chapter 8 | 358 |
Chapter 9 | 401 |
Remarks on the 11 Property | 431 |
Bibliography | 445 |
Notations | 453 |
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Common terms and phrases
absolutely continuous adjoint admissible norm algebra associated assume B-spaces B-subspace Banach space belongs Besov scale Borel bounded operator C*-algebra class Ck closed graph theorem Co-group compact compact operator consider constant continuous function continuously embedded convergence Corollary definition denote densely defined domain dominated convergence theorem embedding estimate example fact finite Fk)m Fourier transform Friedrichs couple Fs,p functional calculus H₁ hamiltonian hence Hilbert space identity implies inequality infinity integral interpolation space K-functional Lemma Let F linear operator lower semicontinuous measure multi-index N-body neighbourhood notation observable affiliated obtain operator in H preceding PROOF properties Proposition prove real number reflexive result satisfied Section self-adjoint operator sequence sesquilinear form Sobolev space F spectral strongly subset subspace symmetric Theorem topology translation group vector space weakly zero λο