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Few-body Krein’s Formula

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Operator Theory and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 118))

Abstract

Selfadjoint extensions of symmetric operators with infinite deficiency indices are discussed. In particular the operators describing the system of several quantum particles with generalized point interactions are investigated in detail and a few-body analog of Krein’s formula for generalized resolvents is proven. New conditions for semiboundedness of M-body quantum Hamiltonian with generalized point interactions in the three-dimensional space are derived.

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References

  1. N.I. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert space, vol. II, Pitman, Boston, 1981.

    MATH  Google Scholar 

  2. S. Albeverio, W. Hunziker, W. Schneider and R. Schrader, A note on L.D. Faddee’s three-particle theory, Helv. Phys. Acta 40 (1967), 745–748.

    MathSciNet  MATH  Google Scholar 

  3. S. Albeverio, R. Høegh-Krohn and L. Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys. 18 (1977), 907–917.

    Article  MATH  Google Scholar 

  4. S. Albeverio, R. Høegh-Krohn and T.T. Wu, A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior, Phys. Lett. A 83(1981), 105–109.

    Article  MathSciNet  Google Scholar 

  5. S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable models in quantum mechanics, Springer-Verlag, New York, 1988.

    Book  MATH  Google Scholar 

  6. S. Albeverio and P. Kurasov, Singular perturbations of differential operators, Cambridge Univ. Press, to appear.

    Google Scholar 

  7. B.L. Altshuler, P.A. Lee and R.A. Webb, eds., Mesoscopic Phenomena in Solids, North Holland, New York, 1991.

    Google Scholar 

  8. W. Amrein, J. Jauch and K. Sinha, Scattering theory in quantum mechanics, Benjamin, Reading, Mass, 1977.

    MATH  Google Scholar 

  9. F.V. Atkinson, H. Langer, R. Mennicken and A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5–20.

    Article  MathSciNet  MATH  Google Scholar 

  10. F.A. Berezin and L.D. Faddeev, Remark on the Schrödinger equation with singular potential, Dokl. Akad. Nauk SSSR 137 (1961), 1011–1014.

    MathSciNet  Google Scholar 

  11. Yu.N. Demkov and V.N. Ostrovsky, Zero-range potentials and their applications in atomic physics, Plenum Press, New York and London, 1988.

    Book  Google Scholar 

  12. V.A. Derkach and M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1–95.

    Article  MathSciNet  MATH  Google Scholar 

  13. L.D. Faddeev and S.R Merkuriev, Quantum scattering theory for several particle systems, Kluwer, Dordrecht, 1993.

    MATH  Google Scholar 

  14. E. Fermi, Sul moto dei neutroni nelle sostanze idrogenate, Ricerce Scientifica, 7: 13–52, 1936 (in Italian), English translation in E. Fermi, Collected papers, vol. I, Italy 1921–1938, Univ. of Chicago Press, Chicago-London, 1962, pp. 980-1016.

    Google Scholar 

  15. V.I. Gorbachuk and M.L. Gorbachuk, Boundary value problems for operator differential equations, Kluwer, Dordrecht, 1991.

    Book  Google Scholar 

  16. S. Hassi and H. de Snoo, On rank one perturbations of self adjoint operators, Integr. Eq. Oper. Theory 29 (1997), 288–300.

    Article  MATH  Google Scholar 

  17. S. Hassi, H. Langer and H. de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1, 1). In Topics in operator theory, operator algebras and applications (Timisoara, 1994), pp. 115-145. Rom. Acad., Bucharest, 1995.

    Google Scholar 

  18. G.R. Kirchoff, Gesammelte Abhandlungen Publ. Leipzig: Barth, 1882, p. 641.

    Google Scholar 

  19. M. Krein, Concerning the resolvents of an Hermitian operator with the deficiency — index (m, m), Comptes Rendue (Doklady) Acad. Sci. URSS (N.S.) 52 (1946), 651–654.

    MathSciNet  MATH  Google Scholar 

  20. M. Naimark. Self-adjoint extensions of the second kind of a symmetric operatots Bull. Acad. Sci. URSS., Ser. Math. 4 (1940), 53–104.

    MATH  Google Scholar 

  21. V. Kostrykin and R. Schrader Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen. 32 (1999), 595–630.

    Article  MathSciNet  MATH  Google Scholar 

  22. Yu.A. Kuperin and S.R Merkurev, Selfadjoint extensions and scattering theory for several-body systems, in: Spectral theory of operators (Novgorod, 1989), Amer. Math. Soc. Transl. Ser. 2 150, Amer. Math. Soc., Providence, RI, 1992, 141–176.

    Google Scholar 

  23. P.B. Kurasov and B.S. Pavlov, Scattering problem withphysical behavior of scattering matrix and operator relations, to be published in Vinogradov’s memorial volume.

    Google Scholar 

  24. P.B. Kurasov and B.S. Pavlov, Generalized perturbations and operator relations, to be published in Proc. Conf. “Hyperfunctions, Operator and Dynamical Systems”, Brussel, 1997.

    Google Scholar 

  25. P.B. Kurasov, Energy dependent boundary conditions and few-body scattering problem, Rev. Math. Phys. 9 (1997), 853–906.

    Article  MathSciNet  MATH  Google Scholar 

  26. K.A. Makarov, Semiboundedness of the energy operator of a system of three particles with paired interactions of δ-function type, Algebra i Analiz 4 (1992), 155–171.

    MATH  Google Scholar 

  27. K.A. Makarov, V.V. Melezhik and A.K. Motovilov, Point interactions in the problem of three quantum particles with internal structure, Teoret. Mat. Fiz. 102 (1995), 258–282.

    MathSciNet  Google Scholar 

  28. K.A. Makarov and V.V. Melezhik, Two sides of a coin: the Efimov effect and collapse in a three-body system with point interactions. I, Teoret. Mat. Fiz. 107 (1996), 415–432.

    Article  MathSciNet  Google Scholar 

  29. R. Mennicken and A. Shkalikov, Spectral Decomposition of Symmetric Operator Matrices. Mathematische Nachrichten. 179 (1996), 259–273.

    Article  MathSciNet  MATH  Google Scholar 

  30. S.P. Merkuriev and S.L. Yakovlev, On the quantum scattering problem for four identical particles interacting in the S state, Soviet J. Nuclear Phys. 39 (1984), 1002–1006.

    MathSciNet  Google Scholar 

  31. S.P. Novikov, Schr”odinger operators on graphs and symplectic geometry, in: The Arnol’dfest (Proceedings of the Fields Institute Conference in Honour of the 60th Birthday of Vladimir I. Arnol’d), eds. E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden, to appear in the Fields Institute Communications Series.

    Google Scholar 

  32. B. Pavlov, A model of zero-radius potential with internal structure, Teor. Mat. Fiz. vol. 59, 3 (1984), 345–354 (English translation: Theoret, and Math. Phys 59 (1984), no. 3, 544-550).

    Article  Google Scholar 

  33. B.S. Pavlov, Operator Extension Theory and explicitly solvable models, Uspekhi Mat. Nauk 42 (1987), 99–131, 247.

    MathSciNet  MATH  Google Scholar 

  34. B.S. Pavlov, Boundary conditions on thin manifolds and the semiboundedness of the three-body Schrödinger operator with point potential, Mat. Sb. (N.S.) 136(178) (1988), 163–177, 301.

    Google Scholar 

  35. I.Yu. Popov, Theory of extensions and the localization of resonances for domains of trap type, Mat. Sb. 181 (1990), 1366–1390.

    MATH  Google Scholar 

  36. I.Yu. Popov, The Helmholtz resonator and operator extension theory in a space with indefinite metric, Mat. Sb. 183 (1992), 3–27.

    Google Scholar 

  37. Yu.G. Shondin, On the three-particle problem with δ-potentials, Teoret. Mat. Fiz. 51 (1982), 181–191.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Pavel Kurasov

  2. Dept. of Mathematics, Auckland University, Private Bag, 92019, Auckland, New Zealand

    Boris Pavlov

Authors
  1. Pavel Kurasov
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  2. Boris Pavlov
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, University of Odessa, 270026, Odessa, Ukraine

    V. M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    I. Gohberg

  3. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    M. Gorbachuk  & V. Gorbachuk  & 

  4. Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081, HV Amsterdam, The Netherlands

    M. A. Kaashoek

  5. Department of Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/1411, 1040, Vienna, Austria

    H. Langer

  6. Institute of Mathematics, Economics and Mechanics, Odessa State University, 270057, Dvoryanskaya str. 2, Odessa, Ukraine

    G. Popov

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© 2000 Springer Basel AG

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Kurasov, P., Pavlov, B. (2000). Few-body Krein’s Formula. In: Adamyan, V.M., et al. Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol 118. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8413-6_13

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  • DOI: https://doi.org/10.1007/978-3-0348-8413-6_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9557-6

  • Online ISBN: 978-3-0348-8413-6

  • eBook Packages: Springer Book Archive

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