Mathematical Scattering Theory: General Theory
Preliminary facts Basic concepts of scattering theory Further properties of the WO Scattering for relatively smooth perturbations The general setup in stationary scattering theory Scattering for perturbations of trace class type Properties of the scattering matrix (SM) The spectral shift function (SSF) and the trace formula
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The General Scheme in Stationary Scattering Theory
Scattering for Perturbations of Trace Class Type
Properties of the Scattering Matrix SM
The Spectral Shift Function SSF and the Trace Formula
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absolutely continuous according to Theorem apply arbitrary assertion assumed assumptions Birman Borel set bounded interval bounded operator coincide conditions of Theorem consider constructed converges COROLLARY corresponding decomposition defined by equality definition denote dense set derivative differential direct integral eigenvalues elements equal to zero equation establish estimate example finite finite-dimensional follows directly full measure function f H0 and H H0-smooth hence Hilbert space Hilbert-Schmidt operators Holder continuous holds holomorphic identification inclusion inequality integral operator inverse isometric kernel Lebesgue measure left-hand side measure zero Moreover multiplication norm obtain operator H operator-valued function pair H0 possible proof of Theorem Proposition relation respect right-hand side satisfied scattering matrix scattering operator scattering theory selfadjoint operator semibounded sesquilinear form set of full singular spectral theorem subspace Suppose the operator tends to zero time-dependent trace class trace class perturbations trace formula 2.1 unitary operators valid weak
Page 8 - V(t) as the sum of the second and third terms on the right-hand side of (10), with signs reversed.
Page 335 - VS Buslaev, Trace formulas and some asymptotic estimates of the resolvent kernel for the Schrödinger equation in three-dimensional space, Problemy Mat.
Page 335 - A typical one-dimensional Schrodinger operator has pure point spectrum, Funktsional. Anal. i Prilozhen.
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