| The paper mainly include two parts. We mainly investigates a class of Sturm-Liouville problems with transmission conditions and indefinite weight function in first part, i.e. indefinite Sturm-Liouville problems. We find an important fact:the sign ofθ/p, which is the ratio of determinant about coefficient in transmission conditions, will in-fluence the choice of study methods of boundary-value. Whenθ/p is positive and weight function is indefinite, a Krein space K and a new operator A related to the boundary-value problem are constructed to make sure the eigenvalues of the operators A and T are same, and a Hilbert space H related to K and self-adjoint operator S in it are also constructed. By using spectrum theory of self-adjoint operator in Krein space and the properties of operator S. we prove that the eigenvalues of A are real. Thus the eigen-values of the boundary-value problem are real. Whenθ/p is negative and weight function is indefinite, we use classical method in inner product to study it. We get properties of eigenvalues of the operator. In second part we mainly investigate the discontinuous Sturm-Liouville operator L with boundary condition depending on spectral parameter and indefinite weight function and indefinite leading coefficient, i.e. "the indefinite S-L problem "with indefinite weight function and indefinite leading coefficient. Because of the boundary condition depends on the spectral parameterλ, so the operator also depends on the spectral parameterλ. We construct a Krein space and a new operator A that related to the boundary-value problem and not depended on the spectral parameter A. And we study the boundary-value problem by the method that has been used in-is positive in first part. We prove that the eigenvalues of the boundary-value problem are real. Finally, by means of studying the operator L itself, we obtain the necessary and sufficient conditions for A is it's eigenvalue, and give construct the Green's function of the new operator A. |
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