J-Selfadjoint Extension Problems Of J-Symmetric Differential Operators With Transmission Conditions

The J-selfadjoint differential operator is a class of non-self-adjoint differential operator which is very important in operator theory,and has a wide range of application.In recent years,with the continuous extension of the theoretical knowledge of non-selfadjoint differential operators,the problems of differential operators with transmission conditions have stimulated the interest of many researchers.In this paper,we investigate the J-selfadjointness for the complex-valued coefficients differential operators with transmission conditions.Firstly,we discuss a class of complex-valued second-order differential operators with transmission conditions.Using the definition of J-selfadjoint operator,we prove that this class of second-order differential operators with separate boundary conditions and transmission conditions are J-selfadjoint.Secondly,we investigate three classes of fourth-order differential operators with complex-valued coefficients and transmission conditions.We determine an inner product according to the coefficient matrix of the transmission condition,and use this inner product to establish a Hilbert space H.By applying the definition of J-selfadjoint operator and the Ptching Lemma,we prove these fourth-order J-symmetric differential operators with different boundary conditions and transmission conditions are J-selfadjoint in the space H.At last,we investigate a class of 2nth-order differential operators with complexvalued coefficients and transmission conditions.While the coefficient matrices of boundary conditions and transmission conditions satisfy certain conditions,by the definition of J-selfadjoint operator and the method of matrix representation,we prove the operator is J-selfadjoint.