Skip to main content
Log in

Locality of Quadratic Forms for Point Perturbations of Schrödinger Operators

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper we study point perturbations of the Schrödinger operators within the framework of Krein's theory of self-adjoint extensions. A locality criterion for quadratic forms is proved for such perturbations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, New-York, 1988.

    Google Scholar 

  2. B. S. Pavlov, “The theory of extensions and solvable models,” Uspekhi Mat. Nauk [Russian Math.Surveys], 42 (1987), no. 6, 99–131.

    Google Scholar 

  3. S. P. Novikov, “Schrödinger operators in periodic fields in dimension two” in: Modern Problems of Mathematics. New Advances [in Russian], 23, VINITI, Moscow (1983), 3–32.

    Google Scholar 

  4. S. Albeverio and V. A. Geiler, “The band structure of the general periodic Schrödinger operator with point interactions,” Comm. Math. Phys., 210 (2000), no. 1, 29–48.

    Google Scholar 

  5. V. A. Geiler and K. V. Pankrashkin, “On fractal structure of the spectrum for periodic point perturbations of the Schrödinger operator with a uniform magnetic field,” Operator Theory: Adv. and Appl., 108 (1999), 259–265.

    Google Scholar 

  6. Yu. G. Shondin, “Semibounded local Hamiltonians for perturbations of the Laplacian on curvres with corner points in ℝ4,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 106 (1996), no. 2, 179–199.

    Google Scholar 

  7. B. Simon, “Schrödinger semigroups,” Bull. Amer. Math. Soc., 7 (1982), no. 3, 447–526.

    Google Scholar 

  8. M. G. Krein and G. K. Langer, “On deficiency subspaces and generalized resolvents of an Hermitian operator in Пκ-space,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 5 (1971), no. 2, 59–71, no. 3, 54–69.

    Google Scholar 

  9. V. A. Geiler, V. A. Margulis, and I. I. Chuchaev, “Zero range operators and Carleman operators,” Sibirsk. Mat. Zh. [Siberian Math. J.], 36 (1995), no. 4, 628–641.

    Google Scholar 

  10. L. Bers, F. John, and M. Schehter, Partial Derivatives Equations, John Wiley and Sons Inc., New York-London-Sydney, 1957.

    Google Scholar 

  11. V. B. Korotkov, Integral Operators [in Russian], Nauka, Novosibirsk, 1983.

  12. M. Sh. Birman, “On the theory of self-adjoint extensions of positive definite operators,” Mat. Sb. [Math. USSR-Sb.], 38 (1956), no. 4, 431–450.

    Google Scholar 

  13. V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps,” J. Funct. Anal., 95 (1991), no. 1, 1–95.

    Google Scholar 

  14. A. Teta, “Quadratic forms for singular pertubations of the Laplacian,” Publ. RIMS Kyoto Univ., 26 (1990), 803–817.

    Google Scholar 

  15. V. D. Koshmanenko, Singular Bilinear Forms in Perturbation Theory of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev, 1993.

    Google Scholar 

  16. V. A. Geiler, “The Schrödinger operator in two dimensions with homogeneous magnetic field and its perturbations by zero range potential,” Algebra i Analiz [St. Petersburg Math. J.], 3 (1991), no. 3, 1–48.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pankrashkin, K.V. Locality of Quadratic Forms for Point Perturbations of Schrödinger Operators. Mathematical Notes 70, 384–391 (2001). https://doi.org/10.1023/A:1012352029965

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012352029965

Navigation