Finite rank singular perturbations and distributions with discontinuous test functions

Authors:
P. Kurasov and J. Boman

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1673-1683

MSC (1991):
Primary 34L40, 46F10, 47A55, 81Q15

DOI:
https://doi.org/10.1090/S0002-9939-98-04291-9

MathSciNet review:
1443392

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Abstract: Point interactions for the -th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint extension of the -th derivative operator restricted to the set of functions vanishing in a neighborhood of the singular point. It is proven that the selfadjoint extensions can be described by the planes in the space of boundary values which are Lagrangian with respect to the symplectic form determined by the adjoint operator. A distribution theory with discontinuous test functions is developed in order to determine the selfadjoint operator corresponding to the formal expression

representing a finite rank perturbation of the -th derivative operator with the support at the origin.

**[AGHKH]**S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, ``Solvable Models in Quantum Mechanics'', Springer-Verlag, 1988. MR**90a:81021****[AlKu]**S. Albeverio, P. Kurasov, Rank one perturbations, approximations and selfadjoint extensions,*J. Func. Anal.***148**(1997), 152-169.**[AlKu2]**S. Albeverio, P. Kurasov, Rank one perturbations of not semibounded operators,*Itegr. Eq. Oper. Theory***27**(1997), 379-400.**[AlKu3]**S. Albeverio, P. Kurasov, Finite rank perturbations and distributions theory, SFB 237 - Preprint No. 368, Bochum, Germany (1997).**[BeFad]**F. Berezin, L. Faddeev, A remark on Schrödinger's equation with a singular potential,*Dokl. Akad. Nauk. SSSR***137**, N5 (1961), 1011-1014. MR**23:B2345****[GeSi]**F. Gesztesy, B. Simon, Rank one perturbations at infinite coupling,*J. Func. Anal.***128**(1995), 245-252. MR**95m:47014****[H]**L. Hörmander, ``The Analysis of Linear Partial Differential Operators I'', Springer-Verlag, Berlin, 1983. MR**85g:35002a****[KiSi]**A. Kiselev, B. Simon, Rank one perturbations with infinitesimal coupling,*J. Func. Anal.***130**(1995), 345-356. MR**96e:47012****[Ku]**P. Kurasov, Distribution theory for discontinuous test functions and differential operators with generalized coefficients,*J. Math. Anal. Appl.***201**(1996), 297-323. CMP**96:15****[ReSi]**M. Reed, B. Simon, ``Methods of modern mathematical physics'', Vol. II, Academic Press, New York, 1975. MR**58:12429b****[Si]**B. Simon, Spectral analysis of rank one perturbations and applications,*CRM Proceedings and Lectures Notes*Vol. 8 (1995), 109-149. MR**97c:47008**

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Additional Information

**P. Kurasov**

Affiliation:
(P. Kurasov) Department of Mathematics, Stockholm University, S-10691 Stockholm, Sweden; Department of Mathematics, Luleå$ $ University, S-97187 Luleå, Sweden; Department of Mathematical Physics, St.Petersburg University,198904 St.Petersburg, Russia

Email:
pak@matematik.su.se

**J. Boman**

Affiliation:
(J.Boman) Department of Mathematics, Stockholm University, S-10691 Stockholm, Sweden

Email:
jabo@matematik.su.se

DOI:
https://doi.org/10.1090/S0002-9939-98-04291-9

Received by editor(s):
November 7, 1996

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society