On the Abstract Inverse Scattering Problem for Trace Class Perturbations
DOI:
https://doi.org/10.15407/mag13.01.003Ключові слова:
матриця розсіювання, модель Фрідріхса-Фаддеєва, обернена задача розсіюванняАнотація
Вивчено задачу розсiювання для пари самоспряжених операторiв {L0, L}, де L - L0 - ядерний. Отримано явний вигляд матрицi розсiювання та встановлено її властивостi. Знайдено рiвняння оберненої задачi.
Mathematics Subject Classification: 47A45.
Посилання
Ì. Reed and B. Simon, Methods of modern mathematical physics. III: Scattering Theory. Academic Press, Inc., 1979.
D.R. Yafaev, Mathematical Scattering Theory: General Theory. Transl. of Math. Monographs, 105, AMS, Providence, RI, 1992.
H. Baumg artel and M. Wollenberg, Mathematical Scattering Theory. Math. Textbooks and Monogr., vol. 59, Part II; Mathem. Monogr., Akademie-Verlag, Berlin, 1983. https://doi.org/10.1007/978-3-0348-5440-5
V.A. Marchenko, SturmLiouville Operators and Applications. AMS Chelsea Publishing, Revised edition, 2011. https://doi.org/10.1090/chel/373
L.D. Faddeev, The Inverse Problem in the Quantum Theory of Scattering. J. Math. Phys. 4 (1963), 72104. https://doi.org/10.1063/1.1703891
C. Shadan and P. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, 1989.
B.M. Levitan, Inverse Sturm Liouville Problems. VMU Science Press, Utrecht, 1987.
J.F. Brasche, M.M. Malamud, and H. Neidhardt, Weyl Function and Spectral Properties of Self-Adjoint Extensions. Integr. Eqs., Oper. Theory 43 (2002), No. 3, 264289.
J.F. Brasche, M.M. Malamud, and H. Neidhardt, Scattering Theory for Open Quantum Systems with Finite Rang Coupling. Math. Phys., Anal., Geom. 10 (2007), 331358.
J.F. Brasche, M.M. Malamud, and H. Neidhardt, Scattering Matrices and Weyl Functions. Proc. London Math. Soc. 97 (2008), No. 3, 568598.
R. Tiedra de Aldecoa, Time Delay for Dispersive Systems in Quantum Scattering Theory. Rev. Math. Phys. 21 (2009), No. 5, 675708.
N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, 1965.
N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, vol. 2, 3rd ed. Boston, Mass. London, Pitman (Advanced Publishing Program), 1981.
J.B. Garnett, Bounded Analytic Functions (Graduate Texts in Mathematics). Springer, New York, 2006.
P. Koosis, Introduction to H p Spaces. Cambridge University Press, Vol. 40, Cambridge, 1980.
F.D. Gahov, Boundary Problems. Ì. Fiz.-mat. lit., 1977. (Russian)
I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Vol. 18. AMS, 1969.
M.G. Krein, On the Trace Formula in Perturbation Theory. Mat. Sb. (N.S.), 33(75):3 (1953), 597626. (Russian)
M. Anthea Grubb and D.B. Pearson, Derivation of the Wave and Scattering Operators for Interactions of Rank One. J. Math. Phis. 11 (1970), 24152424. https://doi.org/10.1063/1.1665405
J. Kellendonk and S. Richard, On the Structure of the Wave Operators in OneDimensional Potential Scattering. Math. Phys. Electron. J. 14 (2008), 1321.
S. Richard and R. Tiedra de Aldecoa, New Formulae for the Wave Operators for aRank One Interaction. Integr. Equation, Oper. Theory 66 (2010), No. 2, 283292.
