Geodesic
On the nature of the spectrum of self-adjoint extensions of operators admitting separation of variables
Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 577-584 
V. A. Mikhailets. On the nature of the spectrum of self-adjoint extensions of operators admitting separation of variables. Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 577-584. http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a9/
@article{MZM_1974_16_4_a9,
     author = {V. A. Mikhailets},
     title = {On the nature of the spectrum of self-adjoint extensions of operators admitting separation of variables},
     journal = {Matemati\v{c}eskie zametki},
     pages = {577--584},
     year = {1974},
     volume = {16},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a9/}
}
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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.