Letters in Mathematical Physics

, Volume 45, Issue 1, pp 33–47 | Cite as

Symmetries of Schrödinger Operator with Point Interactions

  • S. Albeverio
  • L. Dabrowski
  • P. Kurasov
Article

Abstract

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.

Schrödinger operators symmetries extension theory point interactions exactly solvable models. 

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References

  1. 1.
    Albeverio, S., Brzeźniak, Z. and Dąbrowski, L.: Time dependent propagator for point interactions J. Phys. A 27 (1994), 4933–4943.Google Scholar
  2. 2.
    Albeverio, S., Brzeźniak, Z. and Dąbrowski, L.: Fundamental solution of the heat and Schrödinger equation with point interactions, J. Funct. Anal. 128 (1995), 220–254.Google Scholar
  3. 3.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988.Google Scholar
  4. 4.
    Albeverio, S., Gesztesy, F. and Holden, H.: Comment on a recent note on the Schrödinger equation with a δ′-interaction, J. Phys. A: Math. Gen. 26 (1993), 3903–3904.Google Scholar
  5. 5.
    Albeverio, S. and Kurasov, P.: Rank one perturbations, approximations and selfadjoint extensions, J. Funct. Anal. 148 (1997), 152–169.Google Scholar
  6. 6.
    Albeverio, S. and Kurasov, P.: Rank one perturbations of not semibounded operators, Integral Equations Operator Theory 27 (1997), 349–400.Google Scholar
  7. 7.
    Berezin, F. A. and Faddeev, L. D.: A remark on Schrödinger equation with a singular potential, Soviet Math. Dokl. 2 1961, 372–375 (Math. USSR Dokl., 137 (1961), 1011–1014).Google Scholar
  8. 8.
    Cheon, T. and Shigehara, T.: Realizing discontinuous wave function with renormalized shortrange potentials, quant-ph/9709035.Google Scholar
  9. 9.
    Chernoff, P. and Hughes, R.: A new class of point interactions in one dimension, J. Funct. Anal. 111 (1993), 97–117.Google Scholar
  10. 10.
    Coutinho, F. A. B., Nogami, Y. and Fernando Perez, J.: Generalized point interactions in onedimensional quantum mechanics, J. Phys. A: Math. Gen. 30 (1997), 3937–3945Google Scholar
  11. 11.
    Demkov, Yu. N., Kurasov, P. B. and Ostrovsky, V. N.: Doubly periodical in time and energy exactly soluble system with two interacting systems of states, J. Phys. A: Math. Gen. 28 (1995), 4361–4380.Google Scholar
  12. 12.
    Demkov, Yu. N. and Ostrovskii, V. N.: Zero-range Potentials and their Applications in Atomic Physics, Leningrad Univ. Press, Leningrad, 1975; English translation: Plenum Press, New York-London, 1988.Google Scholar
  13. 13.
    Gesztesy, F. and Holden, H.: A new class of solvable models in quantum mechanics describing point interactions on the line, J. Phys. A: Math. Gen. 20 (1987), 5157.Google Scholar
  14. 14.
    Gesztesy, F. and Kirsch, W.: One-dimensional Schrödinger operators with interactions singular on a discrete set, J. Reine Angew. Math. 362 (1985), 28–50.Google Scholar
  15. 15.
    Griffiths, D. J.: Boundary conditions at the derivative of a delta function, J. Phys. A: Math. Gen. 26 (1993), 2265–2267.Google Scholar
  16. 16.
    Hassi, S., Langer, H. and de Snoo, H.: Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1), 15th OT Conference Proc., 1995, pp. 115–145.Google Scholar
  17. 17.
    Hassi, S. and de Snoo, H.: On rank one perturbations of selfadjoint operators, Integral Equations Operator Theory 29 (1997), 288–300.Google Scholar
  18. 18.
    Kiselev, A. and Simon, B.: Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), 345–356.Google Scholar
  19. 19.
    Kurasov, P.: Distribution theory for the discontinuous test functions and differential operators with the generalized coefficients, J. Math. Anal. Appl. 201 (1996), 297–323.Google Scholar
  20. 20.
    Kurasov, P. and Boman, J.: Finite rank singular perturbations and distributions with discontinuous test functions, accepted for publication in Proc. Amer. Math. Soc. Google Scholar
  21. 21.
    Kurasov, P. and Elander, N.: On the δ′-potential in one dimension, Preprint MSI 93-7, ISSN-1100-214X (1993).Google Scholar
  22. 22.
    Kurasov, P., Scrinzi, A. and Elander, N.: On the δ′-potential arising in exterior complex scaling, Phys. Rev. A 49 (1994) 5095–5097.Google Scholar
  23. 23.
    Krein, M. G.: On the resolvents of a Hermitian operator with deficiency indices (m, m), Dokl. Akad. Nauk SSSR 52 (1946), 657–660 (in Russian).Google Scholar
  24. 24.
    Pavlov, B. S.: The theory of extensions and explicitly-soluble models, Russian Math. Surveys 42(6) (1987), 127–168.Google Scholar
  25. 25.
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, vol. IV, Academic Press, New York, 1975.Google Scholar
  26. 26.
    Seba, P.: The generalized point interaction in one dimension, Czech. J. Phys. B 36 (1986), 667–673.Google Scholar
  27. 27.
    Seba, P.: Some remarks on the δ′-interaction in one dimension, Rep. Math. Phys. 24 (1986), 111–120.Google Scholar
  28. 28.
    Shubin Christ, C. and Stolz, G.: Spectral theory of one-dimensional Schrödinger operators with point interactions, J. Math. Anal. Appl. 184 (1994), 491–516.Google Scholar
  29. 29.
    Simon, B.: Spectral analysis of rank one perturbations and applications, in: CRM Proc. Lectures Notes 8, Amer. Math. Soc., Providence, 1995, pp. 109–149.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • S. Albeverio
    • 1
  • L. Dabrowski
    • 2
  • P. Kurasov
    • 3
  1. 1.Institute of MathematicsRuhr-UniversitätBochum; SFB 237; BiBoS; Cerfim (Locarno); Acc.Arch, USI (Mendrisio)
  2. 2.SISSATriesteItaly
  3. 3.Alexander von Humboldt fellow, Institute of Mathematics, Ruhr-Universität, Bochum Department of MathematicsStockholm University; Department of Mathematical and Computational Physics, St. Petersburg UniversityItaly

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