On the nature of the spectrum of self-adjoint extensions of operators admitting separation of variables

We consider the operators: $L_0=\overline{M_0\otimes E''+E'\otimes Q}$, acting in the tensor product of the infinite-dimensional Hilbert spaces $H'$ and $H''$, where the operator $M_0$ is symmetric in $H'$ and $Q$ is self-adjoint in $H''$. We study the problem concerning the existence of self-adjoint extensions, the spectrum of which possesses certain preassigned properties. In particular, we obtain necessary and sufficient conditions under which the operator $L_0$ admits self-adjoint extensions with a discrete spectrum.