Finite rank perturbations
and distribution theory
Authors:
S. Albeverio and P. Kurasov
Journal:
Proc. Amer. Math. Soc. 127 (1999), 1151-1161
MSC (1991):
Primary 34L40, 46F10, 47A55, 81Q15
DOI:
https://doi.org/10.1090/S0002-9939-99-04992-8
MathSciNet review:
1622761
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Perturbations of a selfadjoint operator
by symmetric finite rank operators
from
to
are studied. The finite dimensional family of selfadjoint extensions determined by
is given explicitly.
- 1. N.I.Akhiezer, I.M.Glazman, "Theory of linear operators in Hilbert Space", vol I, II, Pitman, London, 1981. MR 83i:47001
- 2. Sergio Albeverio, Friedrich Gesztesy, Raphael Høegh-Krohn, and Helge Holden, Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer-Verlag, New York, 1988. MR 926273
- 3. Sergio Albeverio, Witold Karwowski, and Vladimir Koshmanenko, Square powers of singularly perturbed operators, Math. Nachr. 173 (1995), 5–24. MR 1336950, https://doi.org/10.1002/mana.19951730102
- 4. S.Albeverio, V.Koshmanenko, Some remarks on the Gesztesy-Simon version of rank one perturbations, Kiev Preprint (1996).
- 5. S. Albeverio and P. Kurasov, Rank one perturbations, approximations, and selfadjoint extensions, J. Funct. Anal. 148 (1997), no. 1, 152–169. MR 1461497, https://doi.org/10.1006/jfan.1996.3050
- 6. S.Albeverio, P.Kurasov, Rank one perturbations of not semibounded operators, Integr. Eq. Oper. Theory 27, 379-400 (1997). CMP 97:10
- 7. S.Albeverio, P.Kurasov, "Solvable Schrödinger type operators. Singular perturbations of differential operators", Cambridge Univ. Press, to appear.
- 8. P. Kurasov and J. Boman, Finite rank singular perturbations and distributions with discontinuous test functions, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1673–1683. MR 1443392, https://doi.org/10.1090/S0002-9939-98-04291-9
- 9. F. Gesztesy and B. Simon, Rank-one perturbations at infinite coupling, J. Funct. Anal. 128 (1995), no. 1, 245–252. MR 1317717, https://doi.org/10.1006/jfan.1995.1030
- 10. Seppo Hassi, Heinz Langer, and Henk de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1), Topics in operator theory, operator algebras and applications (Timişoara, 1994) Rom. Acad., Bucharest, 1995, pp. 115–145. MR 1421120
- 11. S.Hassi, H. de Snoo, On rank one perturbations of selfadjoint operators, Integr. Eq. Oper. Theory 29, 288-300 (1997). CMP 98:03
- 12. Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
- 13. A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), no. 2, 345–356. MR 1335385, https://doi.org/10.1006/jfan.1995.1074
- 14. Y. D. Koshmanenko, A contribution to the theory of rank-one singular perturbations of selfadjoint operators, Ukrain. Mat. Zh. 43 (1991), no. 11, 1559–1566 (English, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 43 (1991), no. 11, 1450–1457 (1992). MR 1156066, https://doi.org/10.1007/BF01067286
- 15. P. Kurasov, Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl. 201 (1996), no. 1, 297–323. MR 1397901, https://doi.org/10.1006/jmaa.1996.0256
- 16. P.Kurasov, Rank one perturbations of nonsemibounded operators, Research report 1995-15, LuleåUniversity of Technology, Luleå, Sweden, 1995.
- 17.
M.G.Krein, Concerning the resolvents of an Hermitian operator with the deficiency-index
, C. R. (Dokl.) Akad. Sci. URSS 52, 651-654 (1946). MR 8:277a
- 18. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
- 19. Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34L40, 46F10, 47A55, 81Q15
Retrieve articles in all journals with MSC (1991): 34L40, 46F10, 47A55, 81Q15
Additional Information
S. Albeverio
Affiliation:
Department of Mathematics, Ruhr-University Bochum, 44780 Bochum, Germany; SFB 237 Essen-Bochum-Düsseldorf, Germany; BiBoS Research Center, D 33615 Bielefeld, Germany; CERFIM, Locarno, Switzerland
Address at time of publication:
Institute of Applied Mathematics, University of Bonn, Bonn, Germany
Email:
albeverio@uni-bonn.de
P. Kurasov
Affiliation:
Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden; Alexander von Humboldt fellow, Department of Mathematics, Ruhr-University Bochum, 44780 Bochum, Germany; Department of Mathematical and Computational Physics, St.Petersburg University, 198904 St.Petersburg, Russia; Department of Mathematics, Luleå\( \) University, 97187 Luleå, Sweden
DOI:
https://doi.org/10.1090/S0002-9939-99-04992-8
Received by editor(s):
August 1, 1997
Communicated by:
David R. Larson
Article copyright:
© Copyright 1999
American Mathematical Society