Geodesic
On multiple completeness of eigen functions of differential operator bunch generated by general boundary conditions
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1987), pp. 3-8 
G. V. Virabyan; G. A. Sargsian. On multiple completeness of eigen functions of differential operator bunch generated by general boundary conditions. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1987), pp. 3-8. http://geodesic.mathdoc.fr/item/UZERU_1987_3_a0/
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For (I), (II) spectral problem with general boundary conditions (when the $\Omega$ region is single circle) the multiple completeness of the eigenfunctions in Sobolev’s space $^0 W^1_2 (\Omega)$ has been proved.

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[4] S. N. Sklyar, “Spektralnye svoistva puchka differentsialnykh operatorov, svyazannogo s odnoi smeshannoi zadachei”, Dinamika sploshnoi sredy. Sibirskoe otd. AN SSSR (In-t gidrodinamiki), 1976, no. 27

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[6] G. V. Virabyan, “Ob $n$-kratnoi polnote dlya odnogo klassa polinomialnykh operatornykh puchkov”, DAN Arm. SSR, XLVII:2 (1969)