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Wave Operators for Nonlocal Sturm-Liouville Operators with Trivial Potential

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Book cover Methods of Spectral Analysis in Mathematical Physics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 186))

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Abstract

The wave operators for the pair of Sturm-Liouville operators corresponding to different nonlocal boundary conditions are considered. By analogy with an absolutely continuous subspace in the selfadjoint case a so-called essential subspace is defined. This definition is given for a pair of operators (not one operator only). In the case of the same finite sets of spectral singularities of two operators the similarity of corresponding semigroups is proved. The domain of definition of the semigroup has finite codimension.

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References

  1. Birman M.Sh. On perturbation of the spectrum of singular operator under the variation of the boundary condition// Dokl. Akad. SSSR, 1961, n. 137, 761–763 (in Russian).

    MathSciNet  Google Scholar 

  2. Sachnovich L.A. Dissipative operators with absolutely continuous spectrum, Trudy Moscow Math. Soc., 1968, v. 19, 211–269 (in Russian).

    Google Scholar 

  3. Pavlov B.S. On the conditions of separability of spectral components for dissipative operators// Izv. Akad. SSSR, ser. math., 1975, v. 39, n.1, 123–148 (in Russian).

    MATH  Google Scholar 

  4. Naboko S.N. Functional model of perturbation theory and its applications to the scattering theory// Trudy Moscow Math. Soc., 1980, n. 147, 86–114 (in Russian).

    MATH  MathSciNet  Google Scholar 

  5. Pavlov B.S. Spectral analysis of differential operators with “spreaded” boundary condition// Probl. Math. Phys., 1973, n. 6, 101–119 (in Russian).

    Google Scholar 

  6. Kato T. Wave operators and similarity for some non-selfadjoint operators// Math. Ann., 1966, n. 162, 258–279.

    Article  MATH  MathSciNet  Google Scholar 

  7. Kako T., Yajima K. Spectral and scattering theory for a class of non-selfadjoint operators// Sci. Par. Coll. Gen. Educ. Univ. Tokyo, 1976, v. 26, n. 2, 73–89.

    MATH  MathSciNet  Google Scholar 

  8. Cheremnikh E.V. Nonselfadjoint Friedrichs’ model and Weyl function// Dokl. Akad. Ukraine, 2001, n. 8, 22–29 (in Ukrainian).

    MathSciNet  Google Scholar 

  9. Cheremnikh E.V. Friedrichs’ model and the problems with non-local boundary conditions// Math. Meth. and Phys.-Mech. Fields, 2000, v. 43, n. 3, 146–156 (in Ukrainian).

    MathSciNet  Google Scholar 

  10. Alsholm Preben Inverse scattering theory for perturbation rank one// Duke Math. J., 1980, v. 47, n. 2, 11–23.

    Google Scholar 

  11. Diaba F., Cheremnikh E. On asymptotic time for an evolution with non-local boundary condition// J. Dynam. Syst. and Geom. Theories, 2006, v. 4, n. 1, 61–77.

    MATH  MathSciNet  Google Scholar 

  12. Kato T. Perturbation theory for linear operators, Springer-Verlag, 1966.

    Google Scholar 

  13. Yosida K. Functional analysis, Springer-Verlag, 1965.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Gnatjuka Str. 8, Flat 12A, Lviv, 79007, Ukraine

    E. GV. Cheremnikh

Authors
  1. E. GV. Cheremnikh
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Editor information

Editors and Affiliations

  1. Institut of Mathematics, Polish Academy of Sciences, ul. Sw. Tomasza 30/7, 31-027, Krakow, Poland

    Jan Janas

  2. Department of Mathematics, Lund Institute of Technology, 221 00, Lund, Sweden

    Pavel Kurasov

  3. Department of Mathematical Physics, St. Petersburg State University, Ulyanovskaya 1, 198904, St. Petersburg, Russia

    Sergei Naboko

  4. Department of Mathematics South Kensington Campus, Imperial College London, SW7 2AZ, London, UK

    Ari Laptev

  5. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Ari Laptev

  6. Department of Mathematics, University of Alabama, 452 Campbell Hall, Birmingham, AL, 35294-1170, USA

    Günter Stolz

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© 2008 Birkhäuser Verlag Basel/Switzerland

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Cheremnikh, E.G. (2008). Wave Operators for Nonlocal Sturm-Liouville Operators with Trivial Potential. In: Janas, J., Kurasov, P., Naboko, S., Laptev, A., Stolz, G. (eds) Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, vol 186. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8755-6_3

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