Completeness of one two-interval boundary value problem with transmission conditions

Abstract

Abstract : This paper presents a new approach to the two-interval Sturm-Liouville eigenfunction expansions, based essentially on the method of integralequations. We consider the Sturm-Liouville problem together with two sup-plementary transmission conditions at one interior point. Further we developGreen’s function method for spectral analysis of the considered problem in mod-ified Hilbert space.Keywords : Sturm-Liouville problems, transmission conditions, expansionstheorem.AMS subject classifications : 34L10, 34L15 1 Introduction The development of classical, rather then the operatoric, Sturm-Liouville theoryin the years after 1950 can be found in various sources; in particular in the textsof Atkinson [2], Coddington and Levinson [4], Levitan and Sargsjan [9]. The op-erator theoretic development is given in the texts by Naimark [10] and Akhiezerand Glazman [1]. The subject of eigenfunction expansions is as old as operatortheory. The Sturm-Liouville problem is also important because the solutions to ahomogeneous BVP with homogeneous BCs produce a set of orthogonal functions.Such functions can be used to represent functions in Fourier series expansions.The completeness of classical systems of eigenfunction expansions was originallyrelated to mechanical problems and boundary value problems for differential op-erators. Later the study of eigenfunctions expansions has gained an independentand abstract status. The expansion has an integral operator form whose kernelis a spectral function, the representation of which is the Green function of theoperator. For the method of treating such problems see [8]. The method of Sturm