Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space

The problem of realization of nontrivial perturbations supported on thin sets of “codimension” $\nu$ in $R^n$ for elliptic operators of order $m$, when $\nu\geqslant 2m$, is formulated as one of construction of the self-adjoint extensions of some symmetric linear relation in an indefinite metric space. The self-adjoint extensions and their resolvents are described. It is found that the same extensions can be obtained as a result of extensions of some symmetric operator in $L_2(R^n)$ with outgoing to a larger indefinite metric space. But such operator is picked out already by the “nonlocal” boundary conditions. Applications to quantum models of point interactions are discussed.