A PONTRYAGIN SPACE APPROACH TO SINGULAR PERTURBATIONS OF DIFFERENTIAL OPERATORS


HEINZ LANGER

hlanger@mail.zserv.tuwien.ac.at

Institute for Analysis and Technical Mathematics
Technical University of Vienna
Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria






In the study of singularly perturbed differential expressions operators of the form $ A+\langle\,\cdot\,,\chi\rangle\chi$ arise, where $ A$ is a self-adjoint operator in some Hilbert space $ \mathcal H$ and $ \chi$ is some element of a space with negative norm ( $ \chi\in{\mathcal H}_{-k-1}\setminus{\mathcal H}_{-k}$) associated with $ A$. Recently, by Yu. Shondin and J.F. van Diejen/A.Tip a method was developed which associates with the problem a family of operators in some Pontryagin space with negative index $ k$ which are extensions of a symmetric operator in the given Hilbert space. In the lecture an analytic approach to these extensions using the $ Q$-function or Titchmarsh-Weyl function will be explained. The method applies also e.g. to the Bessel operator $ -\frac{d^2}{dx^2}+\frac{\nu^2-\frac 1 4}{x^2}$ in the self-adjoint case $ \nu>1$, if the $ Q$-function from the non-self-adjoint case $ 0<\nu<1$ is considered for such values $ \nu>1$. Joint work with Y. Shondin and A. Dijksma.


EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts