Sturm-liouville Operators With Discontinuity At Interior Points

In this paper we study Sturm–Liouville operator with discontinuity at interior points.In linear operators,differential operators are unbounded linear operators and have many important applications.Many problems in mathematics,physics and others can be transformed into the problems of obtaining the eigenvalues,eigenvalue functions of a differential operator and the eigenfunction expansion of a function.And many practical prob-lems such as lamination heat transfer,vibrating string with nodes,and the differential operator with potential function?which is a generalized function?can be transformed into the problem of differential operator with discontinuity at interior points.Billiards system receives much attention and can be studied using the spectral theory of differential operator,i.e.,it is considered as a class of relevant differential operators with an infinite number of discontinuity points.The collisions of mass point are described by additional transmission conditions at discontinuity points.Therefore,the study of the interior discontinuous differential operators has attract-ed wide attention of many mathematic researchers in this field.In this paper,the Sturm–Liouville operator with an infinite number of discon-tinuous points in the interior and the indefinite Sturm–Liouville operator with discontinuity in the interior are studied.Our research focuses on the deficiency indices,the characterization of self-adjoint extension,the discreteness of the spectrum of Sturm–Liouville operator with an infinite number of discontinuous points in the interior and also focus on the spec-tral analysis of left definite Sturm–Liouville operator with discontinuity in the interior and the existence and the number of non-real eigenvalue of the indefinite Sturm–Liouville operator with discontinuity in the interior.In the first part of this paper,the Sturm–Liouville operator with an infinite number of discontinuous points in the interior is studied.In this part,we first discusses the characterization of self-adjoint extension of Sturm-Lioulle operator with an infinite number of discontinuous points in the interior.We notice that the symmetric differential operator with an infinite number of deficiency indices needs an infinite number of function-s to characterize the domain of the self-adjoint extension and that these functions satisfy“maximum selection”condition.New inner product and Hilbert space are defined using the additional transmission conditions at discontinuity points.Then,the problems are discussed in the new Hilbert space.We introduce new concepts i.e.,maximum operator_maxand min-imum operator_minwhich are relevant to the transmission conditions.We prove that_minin the new Hilbert space is closed symmetric operator with finite deficiency indices and that_minand_maxare conjugates of each other.Hence the infinite deficiency indices problem can ingeniously reduce to the finite deficiency indices problem and consequently the restriction,i.e.,“maximum selection”condition can be removed.The characteriza-tions of all self-adjoint extension of_minare given directly and completely.Further,construction of the self-adjoint extension of the minimum operator_minis discussed.Based on the above discussion,the deficiency indices,the Friedrichs extension and the discreteness of the spectrum of Sturm–Liouville operator with an infinite number of discontinuous points in the interior are studied.This problem is considered in the new space which relates the transmission condition and the range of the deficiency indices is obtained.And we obtain a sufficient condition for the deficiency indices of the operator of this kind to be?1,1?,i.e.,the condition that the coefficient function,,discontinuous points and coefficient matrix for transmission condition satisfy.We also discuss the existence of lower bound for the op-erator of this kind and further characterize the Friedrichs extension.In the end of this part,we obtain a sufficient condition for the spectrum of the operator of this kind to be discrete,using the decomposition of operator.In the second part of this paper,the indefinite Sturm–Liouville op-erator with discontinuity in the interior is studied.Firstly,we study the spectral analysis of left definite Sturm–Liouville operator with discontinu-ity in the interior by using the spectral theory of linear operator in indefinite metric space and characteristic curve,and prove that the spectrum of the left definite Sturm–Liouville operator with discontinuity in the interior is real,discrete and has no finite accumulation point,no upper and lower bounds.Secondly,we discuss the problem of the existence and the number of non-real eigenvalue of the indefinite Sturm–Liouville operator with dis-continuity in the interior,and obtain several sufficient conditions for judg-ing the existence and the number of non-real eigenvalue of the indefinite Sturm–Liouville operator with discontinuity in the interior.Finally,under the separable boundary condition,we study the indefinite Sturm–Liouville operator with discontinuity in the interior and prove the analyticity of its characteristic curve and then study the number of non-real eigenvalue of the operator of this kind and use two examples to illustrate our results.This paper is organized into six chapters.Chapter 1 is introduction part.Chapter 2 introduces the relevant fundamental concepts and im-portant lemmas.Chapter 3 defines a new Hilbert space relevant to the discontinuity of differential operator in the interior,and investigates the problem of the characterization of self-adjoint extension of Sturm–Liouville operator with an infinite number of discontinuous points in the interior.Chapter 4 discusses the deficiency indices,the Friedrichs extension and the discreteness of the spectrum of Sturm–Liouville operator with an in-finite number of discontinuous points in the interior.Chapter 5 studies the spectral analysis of indefinite regular Sturm–Liouville operator with discontinuity in the interior.Chapter 6 discusses the non-real eigenvalue problem of the indefinite Sturm–Liouville operator with discontinuity in the interior and with separable boundary condition..