Scattering matrices and Weyl functions

For a scattering system {A Θ , A 0 } consisting of self-adjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrodinger operators with point interactions.

[1]  Yoshimi Saito,et al.  Eigenfunction Expansions Associated with Second-order Differential Equations for Hilbert Space-valued Functions , 1971 .

[2]  Some Applications of the Spectral Shift Operator , 1999, math/9903186.

[3]  G. Reuter.  LINEAR OPERATORS PART II (SPECTRAL THEORY) , 1969 .

[4]  A. Dijksma,et al.  SYMMETRIC AND SELFADJOINT RELATIONS IN KREIN SPACES .2. , 1987 .

[5]  J. Cooper.  MONOTONE MATRIX FUNCTIONS AND ANALYTIC CONTINUATION , 1976 .

[6]  Sergio Albeverio,et al.  Solvable Models in Quantum Mechanics , 1988 .

[7]  The Ξ operator and its relation to Krein's spectral shift function , 1999, math/9904050.

[8]  V. Derkach,et al.  Generalized resolvents and the boundary value problems for Hermitian operators with gaps , 1991 .

[9]  W. Müller.  The spectral shift function , 1987 .

[10]  H. Weyl.  Ueber gewöhnliche lineare Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen. (2. Note) , 1910 .

[11]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[12]  H. Weyl,et al.  Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen , 1910 .

[13]  E. C. Titchmarsh,et al.  Reviews , 1947, The Mathematical Gazette.

[14]  V. Marchenko.  Sturm-Liouville Operators and Applications , 1986 .

[15]  V. Derkach,et al.  The extension theory of Hermitian operators and the moment problem , 1995 .

[16]  H. Holden,et al.  ON TRACE FORMULAS FOR SCHRODINGER-TYPE OPERATORS , 1997 .

[17]  D. Yafaev.  Mathematical Scattering Theory: General Theory , 1992 .

[18]  F. Berezin,et al.  A Remark on Schrodinger's equation with a singular potential , 1960 .

[19]  J. Weidmann.  Lineare Operatoren in Hilberträumen , 2000 .

[20]  D. Yafaev.  Mathematical scattering theory , 1992 .

[21]  Sergio Albeverio,et al.  Singular Perturbations of Differential Operators , 2000 .

[22]  V. A. Marchenko,et al.  The Inverse Problem of Scattering Theory , 1963 .

[23]  G. Weiss,et al.  EIGENFUNCTION EXPANSIONS. Associated with Second-order Differential Equations. Part I. , 1962 .

[24]  H. Neidhardt,et al.  Weyl function and spectral properties of self-adjoint extensions , 2002 .

[25]  Spectral properties of the scattering matrix , 1989 .

[26]  V. Gorbachuk,et al.  Boundary Value Problems for Operator Differential Equations , 1990 .

[27]  B. Simon,et al.  INVERSE SPECTRAL ANALYSIS WITH PARTIAL INFORMATION ON THE POTENTIAL, I. THE CASE OF AN A.C. COMPONENT IN THE SPECTRUM , 1997 .

[28]  P. H. Müller,et al.  T. Kato, Perturbation theory for linear operators. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 132) XX + 592 S. m. 3 Fig. Berlin/Heidelberg/New York Springer-Verlag. Preis geb. DM 79,20 , 1967 .

[29]  W. Donoghue.  Monotone Matrix Functions and Analytic Continuation , 1974 .

[30]  J. Garnett,et al.  Bounded Analytic Functions , 2006 .

[31]  B. M. Levitan,et al.  Introduction to spectral theory : selfadjoint ordinary differential operators , 1975 .

[32]  H. Baumgärtel,et al.  Mathematical Scattering Theory , 1983 .

[33]  V. Potapov.  The multiplicative structure of J-contractive matrix functions , 1960 .

[34]  A relation for the spectral shift function of two self-adjoint extensions , 2001 .

[35]  Richard C. Gilbert.  Simplicity of linear ordinary differential operators , 1972 .

[36]  S. Albeverio,et al.  Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions , 2005 .