Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

Abstract

It is purpose of this paper to investigate Sturm-Liouville equation on many-interval with the eigenvalue parameter appearing linearly in the boundary conditions and with two supplementary transmission conditions. The classical Sturmian theory did not cover such type of many-interval boundary value transmission problems. For the classical Sturm-Liouville problems it is guaranteed that the problem is self-adjoint with compact resolvent, the spectrum is disctrete and consist of eigenvalues and the corresponding eigenfunctions form an orthogonal basis in the well-known Hilbert space . But the boundary-value-transmission problems are not self-adjoint and the system of eigenfunctions did not form a basis in the classical Hilbert space in general. Taking in view this fact we suggest a new approach for self-adjoint realization of such type transmission problems. Moreover, we define some new Hilbert spaces to establish positiveness of corresponding operator-pencil. At first we define a concept of generalized eigenfunctions for this kind of spectral problems. In particular it is shown that if the potential is continuous then the generalized eigenfunctions satisfies the considered problem is the classical sense. Then we introduce to the consideration some compact operators such a way that the considered boundary-value-transmission problem can be reduced to the appropriate operator-pencil equation. Finally, we prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. It is important to note that the obtained results extends classical results associated with regular Sturm-Liouville problems.