to define the singular Schrödinger operator in the present case: it acts as $$\label{cod2Ham}
H_{\alpha,\Gamma}f = -\Delta f \quad\mathrm{for}
 \quad x \in {\mathbb{R}}^{3}\setminus \Gamma$$ on the domain $$\label{cod2bc}
D(H_{\alpha,\Gamma}) := \{\,g\in {\mathcal{D}}:\:2\pi \alpha \Xi
 (g)(s)=\Omega (g)(s)\}\,.$$ In this way we get a well-defined Hamiltonian which we seek [@EK2]:

\[cod2sa+res\] Under the stated assumptions the operator $H_{\alpha,\Gamma}$ is self-adjoint.

As in the case of codimension one and the conditions (\[2-1bc\]) the above definition has a natural meaning as a point interaction in the normal plane to $\Gamma$.

The proof of Theorem \[cod2sa+res\] is technically more involved and we restrict ourselves to a few remarks referring to [@EK2] for the full exposition. The argument is in a sense opposite to the previous considerations. It is based on an abstract analogue of Proposition \[BS\] proved in [@Pos1], see also [@Pos2], which shows existence of a self-adjoint operator with the resolvent of the appropriate form, after that one verifies that this operator coincides with the above $H_{\alpha,\Gamma}$.

\[r:3BS\] Since the mentioned resolvent formula analogous to that of Proposition \[BS\] will be used in the following, we will describe it at least briefly. It contains again traces of the free resolvent, which is now given 