Selfadjointness and Positiveness of the Differential Operators Generated by New Type Sturm-Liouville Problems
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.
Keywords
Boundary value problems,eigenfunctions,boundary and transmission conditions,positive operators
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