Geodesic
Correct and selfadjoint problems for quadratic operators
Eurasian mathematical journal, Tome 1 (2010) no. 2, pp. 122-135

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In this paper we present a simpler method of proving the correctness and selfadjointtness of the operators of the form $B^2$ corresponding to some boundary value problems. We also give explict representations for the unique solution of these problems. 
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I. N. Parasidis; P. C. Tsekrekos. Correct and selfadjoint problems for quadratic operators. Eurasian mathematical journal, Tome 1 (2010) no. 2, pp. 122-135. http://geodesic.mathdoc.fr/item/EMJ_2010_1_2_a8/