The determinants of fourth order dissipative operators with transmission conditions

In this paper, discontinuous non-self-adjoint differential operators in Weylʼs limit circle are studied. We give the determinant of perturbation connected with the dissipative operator L generated by fourth order differential expression in L2(I), where discontinuity of operator is dealt with transmission conditions. We obtain the Greenʼs function, then, using characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions of this dissipative operator.