  An inner closed (without boundary) smooth manifold of a lower dimension is
cut from a multidimensional ball. In this region, invertible restrictions of
the Laplace operator are well defined. In particular, the well-posed non-smooth
Bitsadze-Samarski\u{i} problem for the Laplace equation is defined. Moreover,
we obtain M.G. Krein's formula for the trace of the difference of resolvents of
the studied operators. We prove assertions on the spectrum of the non-smooth
Bitsadze-Samarski\u{i} problem.
