Finite rank perturbations and distribution theory

Albeverio, S.; Kurasov, P.

doi:10.1090/S0002-9939-99-04992-8

Finite rank perturbations and distribution theory
HTML articles powered by AMS MathViewer

by S. Albeverio and P. Kurasov

Proc. Amer. Math. Soc. 127 (1999), 1151-1161

DOI: https://doi.org/10.1090/S0002-9939-99-04992-8

PDF | Request permission

Abstract:

Perturbations $A_T$ of a selfadjoint operator $A$ by symmetric finite rank operators $T$ from $\mathcal {H}_2 (A)$ to $\mathcal {H}_{-2} (A)$ are studied. The finite dimensional family of selfadjoint extensions determined by $A_T$ is given explicitly.

References

Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
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, DOI 10.1007/978-3-642-88201-2
Sergio Albeverio, Witold Karwowski, and Vladimir Koshmanenko, Square powers of singularly perturbed operators, Math. Nachr. 173 (1995), 5–24. MR 1336950, DOI 10.1002/mana.19951730102
S.Albeverio, V.Koshmanenko, Some remarks on the Gesztesy-Simon version of rank one perturbations, Kiev Preprint (1996).
S. Albeverio and P. Kurasov, Rank one perturbations, approximations, and selfadjoint extensions, J. Funct. Anal. 148 (1997), no. 1, 152–169. MR 1461497, DOI 10.1006/jfan.1996.3050
S.Albeverio, P.Kurasov, Rank one perturbations of not semibounded operators, Integr. Eq. Oper. Theory 27, 379-400 (1997).
S.Albeverio, P.Kurasov, "Solvable Schrödinger type operators. Singular perturbations of differential operators", Cambridge Univ. Press, to appear.
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, DOI 10.1090/S0002-9939-98-04291-9
F. Gesztesy and B. Simon, Rank-one perturbations at infinite coupling, J. Funct. Anal. 128 (1995), no. 1, 245–252. MR 1317717, DOI 10.1006/jfan.1995.1030
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
S.Hassi, H. de Snoo, On rank one perturbations of selfadjoint operators, Integr. Eq. Oper. Theory 29, 288–300 (1997).
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, DOI 10.1007/978-3-642-96750-4
A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), no. 2, 345–356. MR 1335385, DOI 10.1006/jfan.1995.1074
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, DOI 10.1007/BF01067286
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, DOI 10.1006/jmaa.1996.0256
P.Kurasov, Rank one perturbations of nonsemibounded operators, Research report 1995-15, Luleå University of Technology, Luleå, Sweden, 1995.
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
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, DOI 10.1090/crmp/008/04