Perturbations of elliptic operators on high codimension subsets and the extension theory on an indefinite metric space

Yu. G. Shondin

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 23, Tome 222 (1995), pp. 246-292

Résumé

The spectral aspect of the problem of perturbations supported on thin sets of codimension $\theta\ge2m$ in $\mathbb R^n$ is considered for elliptic operators of order $m$. The problem of realization of such perturbations is formulated as a problem of self-adjoint extension of a linear symmetric relation in a space with indefinite metric. It is shown how to construct such a relation for a given elliptic operator and a family of distributions. Its functional model is obtained in terms of $Q$-fiunctions. Self-adjoint extensions and their resolvents are described. The theory developed is applied to quantum models of point interactions in high dimensions and high moments. Bibliography: 35 titles.