## Tokyo Journal of Mathematics

### On the Perturbation Theory of Self-Adjoint Operators

#### Abstract

We show that all types of self-adjoint perturbations of a semi-bounded operator $A$ (purely singular, mixed singular, and regular) can be described and studied from a unique point of view in the framework of the extension theory as well as in the framework of the additive perturbation theory. We also show that any singular finite rank perturbation $\widetilde{A}$ can be approximated in the norm resolvent sense by regular finite rank perturbations of $A$. An application is given to the study of Schr\"{o}dinger operators with point interactions.

#### Article information

Source
Tokyo J. Math., Volume 31, Number 2 (2008), 273-292.

Dates
First available in Project Euclid: 5 February 2009

https://projecteuclid.org/euclid.tjm/1233844052

Digital Object Identifier
doi:10.3836/tjm/1233844052

Mathematical Reviews number (MathSciNet)
MR2477872

Zentralblatt MATH identifier
1182.47013

#### Citation

ALBEVERIO, Sergio; KUZHEL, Sergei; NIZHNIK, Leonid P. On the Perturbation Theory of Self-Adjoint Operators. Tokyo J. Math. 31 (2008), no. 2, 273--292. doi:10.3836/tjm/1233844052. https://projecteuclid.org/euclid.tjm/1233844052

#### References

• S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin/New York, 1988; $2^{\mathrm{nd}}$ ed. (with an appendix by P. Exner), AMS Chelsea Publishing, Providence, RI, 2005.
• S. Albeverio and P. Kurasov, Singular perturbations of differential operators. Solvable Schrödinger type operators, London Math. Soc. Lecture Note Ser. 271, Cambridge Univ. Press, Cambridge, 2000.
• S. Albeverio and L. P. Nizhnik, A Schrödinger operator with point interactions on Sobolev spaces, Lett. Math. Phys., 70 (2004), 185–199.
• S. Albeverio and L. P. Nizhnik, Schrödinger operators with nonlocal point interactions, J. Math. Anal. Appl., 332 (2007) 884–895.
• T. Ando and K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J., 22 (1970), 65–75.
• Yu. M. Arlinskii and E. R. Tsekanovskii, Some remarks of singular perturbations of self-adjoint operators, Methods Funct. Anal. Topology, 9, No. 4 (2003), 287–308.
• Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Amer. Math. Soc. 17, Providence, Rhode Island, 1968.
• M. S. Birman, On the self-adjoint extensions of positive definite operators, Mat. Sbornik, 38 (1956), 431–450 (Russian).
• E. A. Coddington, Self-adjoint subspace extensions of nondensely defined symmetric operators, Advances in Math., 14, No. 3 (1974), 309–332.
• V. Derkach and S. Hassi, H. de Snoo, Singular perturbations of self-adjoint operators, Math. Physics, Analysis and Geometry, 6 (2003), 349–384.
• V. A. Derkach and M. M. Malamud, Characteristic functions of almost solwable extensions of Hermitian operators, Ukrainian Math. J., 44 (1992), 379–401.
• E. Fermi, Sul moto dei neutroni nelle sostanze idrogenate, Ricerca Scientifica, 7 (1936), 13–52. (English translation in E. Fermi, Collected papers, Vol. I, Univ. of Chicago Press, Chicago, 1962, pp. 980–1016.)
• M. L. Gorbachuk and V. I. Gorbachuk, Boundary-Value Problems for Operator-Differential Equations, Kluwer, Dordrecht, 1991.
• P. R. Halmos, Finite-Dimensional Vector Spaces, D. van Nostrand Company, Princeton, 1961.
• S. Hassi and S. Kuzhel, On symmetries in the theory of singular perturbations, Preprint University of Vaasa, 2006, (http://www.uwasa.fi/julkaisu/).
• T. Kato, Perturbation Theory of Linear Operators, Springer, Berlin, New-York, 1980.
• A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal., 130 (1995), 345–356.
• M. Krasnosel'skii, On self-adjoint extensions of Hermitian operators, Ukr. Mat. Zh., 1 (1949), 21–28.
• A. N. Kochubei, On extensions of nondensely defined symmetric operator, Sib. Math. J., 18, No. 2 (1977), 314–320.
• M. G. Krein, The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications I, Mat. Sbornik, 20 (1947), 431–495 (Russian).
• P. Kurasov and S. T. Kuroda, Krein's resolvent formula and perturbations theory, J. Oper. Theory, 51, No. 2 (2004), 321–334.
• S. T. Kuroda and H. Nagatani, $\mathcal{H}_2$-construction of general type and its application to point interactions, J. Evol. Equ., 1 (2001), 421–440.
• A. Kuzhel and S. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998.
• S. Kuzhel and L. Nizhnik, Finite rank self-adjoint perturbations, Meth. Func. Anal. Topology, 12 (2006), no. 3, 223–241.
• L. P. Nizhnik, On rank one singular perturbations of self-adjoint operators, Meth. Func. Anal. Topology, 7, No. 3 (2001), 54–66.
• L. P. Nizhnik, One-dimensional Schrödinger operators with point interactions in the Sobolev spaces, Funct. anal. appl., 40, No. 2 (2006), 74–79.
• A. Posilicano, Self-adjoint extensions by additive perturbations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Vol. II (5) (2003), 1–20.