Second Order Singular Sturm - Liouville Spectral Problem With Transition

Recently, more and more researchers are interested in the discontinuous Sturm-Liouville problem with inner singular points since these problems are of widely applications in engineer and mechanics.(see[1-30]for details). Various physics applications of this kind of problem are found, such as mass and heat transfer problems. The Sturm-Liouville problem with one inner discontinuous point is widely considered, see [1,5,9,12,16,22,30],and the Sturm-Liouville problem with two inner discontinuous points is also researched in papers [7, 9, 11, 14, 16, 28]. Based on these papers, some researchers considered the Sturm-Liouville problem with a weight function, see [8, 10, 17]. As we know, the singular Sturm-Liouville problems are of great importance and have drawn more and more researchers’ attention. However, little is known on the singular Sturm-Liouville problems with transmission conditions except that a singular Sturm-Liouville problem with eigenparameter dependent boundary conditions and transmission conditions at one inner discontinuous point is researched in paper [11]. The two-order Sturm-Liouville problem with limit circle endpoints on a ?nite interval is considered in this paper.By modifying the inner product space and using the classical analysis technique,we de?ne a new self-adjoint operator A such that the eigenvalues of such a problem is coincided with those of A. We construct its fundamental solutions, get the asymptotic formulas for its eigenvalues, discuss some properties of its spectrum, and obtain its Green Function and the resolvent operator.The thesis is divided into two chapters according to contents.Chapter 1 based on [11], we research the singular Sturm-Liouville problem with two inter discontinuous points and one limit endpoint. where b is a limit circle point, u1, u2 are linearly independent real-valued solutions of equation-(p(x)u′(x))′+ q(x)u(x) = 0 and [u1, u2](b) ?= 0, where [y, z](x) = p(yz′- y′z)is the sesquilinear form; and the transmission conditions at the points of discontinuity x = ξ1and x = ξ2: where p(x) = p21 for x ∈ [a, ξ1), p(x) = p22 for x ∈(ξ1, ξ2) and p(x) = p23 for x ∈(ξ2, b);ω(x) = ω21for x ∈ [a, ξ1), ω(x) = ω22for x ∈(ξ1, ξ2) and ω(x) = ω23for x ∈(ξ2, b); λ is a complex eigenparameter; function q(x) is real and continuous on [a, ξ1)∪(ξ1, ξ2)∪(ξ2, b)and has a ?nite limit q(±ξi) = lim x�'(ξi±0)q(x)(i = 1, 2); wi, pi(i = 1, 2, 3), αj, θj, ρj(j =1, 2, 3, 4) and m1, m2 are non-zero real numbers.Chapter 2 based on [8]-[14], particularly [8] and [11], using the same method, we consider the singular Sturm-Liouville problem with two inter discontinuous points and two limit endpoints on a ?nite interval.where a and b are both limit circle points, y1 and y2are linearly independent real-valued solutions of equation-(p(x)y′(x))′+ q(x)y(x) = 0 and [y1, y2](a) ?= 0, [y1, y2](b) ?= 0,where [y, z](x) = p(yz′- y′z) is the sesquilinear form; and the transmission conditions at the points of discontinuity x = ξ1and x = ξ2:where p(x) = p21 for x ∈(a, ξ1), p(x) = p22 for x ∈(ξ1, ξ2) and p(x) = p23 for x ∈(ξ2, b); ω(x) = ω21for x ∈(a, ξ1), ω(x) = ω22for x ∈(ξ1, ξ2) and ω(x) = ω23for x ∈(ξ2, b); λ is a complex eigenparameter; function q(x) is real and continuous on(a, ξ1)∪(ξ1, ξ2)∪(ξ2, b) and has a ?nite limit q(±ξi) = lim x�'(ξi±0)q(x)(i = 1, 2); wi, pi(i =1, 2, 3), θj, ρj(j = 1, 2, 3, 4), m1, m2, n1, n2 are non-zero real numbers.