Spectral Properties Of Sturm - Liouville Equations With Distribution Potential And Transition Conditions

In recent years of the math’s development,more and more people begin to pay close attention to research the Sturm-Liouville problem, and apply the discontinuous Sturm-Liouville problem with discontinu-ous points to the field of engineering technology and physics problem. In history we have researched more about the classical Sturm-Liouville problem and also come to some classical conclusions,such as:the so-lutions and quasi-derivatives of the classical Sturm-Liouville problem are absolutely continuous on the compact subset of the problem inter-val.But in fact not all the problems meet this conclusion,such as heat conduction problem,the problem of light diffraction,etc(see[19]). In particular, more and more scholars begin to study the discontinuous problemss and even the boundary conditions depending on eigenparam-eter,then study the asymptotic of eigenvalues and eigenfunctions and the completeness of the eigenfunction. Similar problems can be found in many literature([1]-[18]).In this paper,inspired by [20] and basing on the Weyl-Titchmarsh theory for Sturm-Liouville operators with distribu-tional potentials,we discuss the discontinuous Sturm-Liouville problem with boundary conditions depending and talk about the properties of the solutions and eigenvalues.In history,many mathematicians have made their own contributions to researching on the distribution potentials.About the differential ex-pression τf=1/r(-(f[1])’+sf[1]+qf),Bennewitz and Everitt have studied it in 1983,although their discussion is more general,their limited discus-sion on tight intervals and focus more the special left-definite case,refer to[35].After Weidmann the researching the high order op erators,these methods of discussion have caused everubodu for derivative research,in fact,in the process of research of high order differential operator,Wei-dmann has dealed with the distribution potential coefficient,refer to [34]. Baeteman and Chadan studied the strong singular and oscillating potentials of Schrodinger operator, see [36]. Until 1999, Savchuk and Shkalikov began to research Sturm-Liouville problems with distribution-al potential coefficients, involving many fields such as self-adjointness proofs, spectral and inverse spectral theory, oscillation properties, see [37] [38].In this article we discuss the discontinuous Sturm-Liouville problems containing the potentials and eigenparameters at two endpoints using the classical analysis techniques and spectral theory of linear operator to define a new self-adjoint operator associated with the Sturm-Liouville problem in a new Hilbert space, so that the eigenvalues of such a prob-lem is coincided with the related operator. By looking for the general solution of Sturm-Liouville equation and using boundary conditions and transmission conditions, obtain the fundamental solutions and the char-acteristic function of the Sturm-Liouville problem and using the nature of the defined boundary conditions and transmission conditions also get the property of the fundamental solutions and the characteristic function of the Sturm-Liouville problem.This paper can be divided into three chapters according to the con-tent.Chapter 1 Preference, we simply summarize the phylogeny of Sturm-Liouville theory and introduce the distribution potentials and transmis-sion conditions.Chapter 2 we will consider the discontinuous Sturm-Liouville equa-tion with distributional potentials and one discontinuous point.In this chapter, we consider the following discontinous eigenvalue problem which consist of Sturm-Liouville equation with distributional potentials at one interior point: where x∈I=[a,c)∪(c,b], c∈(a,b)(?)R, then the boundary condition at the endpoint a and the eigenparameter λ: where φa∈[0,π), the boundary condition at the endpoint b and the eigenparameter λ where φb∈[0,π), and two transmission conditions at the point of discontinuity x=cChapter 3 we will consider the discontinuous Sturm-Liouville equa-tion with distributional potentials and two discontinuous points.In this chapter,we consider the following discontinous eigenvalue problem which consist of Sturm-Liouville equation with distributional potentials at two interior points: where x∈I=[a,m)∪(m,n)∪(n,b],m<n,m,n∈(a,b)(?)R then the boundary condition at the endpoint a,b and the eigenparameter λ: where φa,φb∈[0,π], and two transmission conditions at the point of discontinuity x=m and x=n where ab>0, α1β1>0,α2β2>0, sinφacosφa>0, sinφbcosφb<0.