DISSIPATIVE STURM-LIOUVILLE OPERATORS WITH A SPECTRAL PARAMETER IN THE BOUNDARY CONDITION ON BOUNDED TIME SCALES
B Allahverdiev, A Eryilmaz, H Tuna, Bilender Allahverdiev, Aytekin Eryılmaz, Hüseyin Tuna
Electronic Journal of Differential Equations
In this article we consider a second-order Sturm-Liouville operator with a spectral parameter in the boundary condition on bounded time scales. We construct a selfadjoint dilation of the dissipative Sturm-Liouville operators. Using by methods of Pavlov [40, 41, 42], we prove the completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales. 2010 Mathematics Subject Classification. 47A20, 47A40, 47A45, 34B05, 34B10, 39A10.
... sciplines. The construction of functional models for dissipative operators, natural analogues of spectral decompositions for selfadjoint operators is based on Sz. Nagy-Foias dilation theory  and Lax-Phillips scattering theory . Pavlov's approach [40, 41, 42] to the model construction of dissipative extensions of symmetric operators was followed by Allahverdiev in his works [3, 4, 5, 6, 7] and some others [23, 38, 39, 46, 47]. The theory of the dissipative Schrödinger operator on a finite interval was applied to the problems arising in the semiconductor physics [9, 10, 11]. In [12, 13, 14, 15], Pavlov's functional model was extended to (general) dissipative operators which are finite dimensional extensions of a symmetric operator , and the corresponding dissipative and Lax-Phillips scattering problems were investigated in some detail. We extend the results [3, 4, 5, 6, 7, 38, 39, 46, 47] to the more general eigenvalues problem (2.2)-(2.4) on time scales. While proving our results, we use the machinery and method of [3, 4, 5, 6, 7]. The organization of this document is as follows: In Section 2, some time scale essentials are included for the convenience of the reader. In Section 3, we construct a selfadjoint dilation of dissipative Sturm-Liouville operator on bounded time scales. We present its incoming and outcoming spectral representations which makes it possible to determine the scattering matrix of the dilation according to the Lax and Phillips scheme . A functional model of this operator is constructed by methods of Pavlov [40, 41, 42] and define its characteristic functions. Finally, we proved a theorem on completeness of the system of eigenvectors and associated vectors of dissipative operators.