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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1443392
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Abstract: Point interactions for the $n$-th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint extension of the $n$-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

\begin{displaymath}L=\left(i\frac d{dx}\right)^n+\sum^{n-1}_{l,m=0}c_{lm}\delta^{(m)}(\cdot) \delta^{(l)},\qquad c_{lm}=\overline{c_{ml}},\end{displaymath}

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


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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: http://dx.doi.org/10.1090/S0002-9939-98-04291-9
PII: S 0002-9939(98)04291-9
Received by editor(s): November 7, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society