The Finite Spectrum Problem Of Sturm - Liouville Problem With Finite Transfer Condition

The Sturm-Liouville problems (SLP) are derived from many mathematical mod-els, such as solid heat conduction. In recent years, the Sturm-Liouville problems with transmission conditions are important research topics in mathematical physics. Boundary value problems with discontinuity conditions inside the interval often appear in applications, such problems are connected with discontinuous material properties, such as heat and mass transfer, varied assortment of physical transfer problems, vi-brating string problems, and diffraction problems. The study of the structure of the solution of the matching region leads to the consideration of an eigenvalue prob-lem for a second order differential operator with piecewise continuous coefficients and transmission conditions at interior points.As is well-known, the classical Sturm-Liouville theory states that the spectrum of a regular or singular, self-adjoint Sturm-Liouville problem (SLP) is unbounded and therefore infinite. This result is generally established under the assumption that the leading coefficient and the weight function are both positive. Atkinson in his book suggested that if the coefficients of SLP satisfy r=1/p,q,w∈L(J,C), the problem may have finite eigenvalues. But he did not elaborate with either an example or a theorem. In 2001, Kong et al.constructed a class of SLPs with exactly n eigenvalues for any positive integer n. Furthermore, Kong et al.gave these kinds of Sturm-Liouville problems with self-adjoint boundary conditions with matrix representations. This fur-ther illustrate the reality that the SLPs may have finite spectrum. Now, one question arises, what is it about the SLPs with finite transmission conditions? Whether they also have finite spectrum when SLPs with transmission condition? We will show, in this paper, that the answer is definite.In this paper, through investigating the finite spectrum of the SLPs with finite ransmission conditions, we obtain some new interesting results.The thesis is divided into two chapters.In the first chapter, we investigate the finite spectrum of the SLPs with two trans-mission conditions-(py’)’＋qy =λwy, y=y(t),t∈J=(a,c1)∪(c1,c2)∪(c2,b), -∞<a<b<+∞, r=1/p,q,w∈ L(J,C), where L(J,C) denotes the Hilbert space of all complex valued functions which are Lebesgue integrable on J. The boundary conditions are where M2(C) are complex valued 2×2 matrices, and the transmission conditions are C1Y(c1-) + D1Y(c1+) = 0,C1,D1∈M2(R), |C1| = ρ1 > 0,|D1|= θ1>0, C2Y(c2-) + D2Y(c2+) = 0,C2,D2∈M2(R), |C2| = ρ2 > 0,|D2| = θ2 > 0, where M2(R) are real valued 2×2 matrices.We construct a class of SLPs with two transmission conditions with exactly nl eigenvalues, and show that these eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere on the real line in the self-adjoint case.In the second chapter, we investigate the formula of the finite spectrum of the SLPs with finite transmission conditions.-(py’)’ + qy = λwy, y = y(t),t∈J= (a,c1)∪(c1,c2)∪…∪(ci-1,ci)∪(ci,b),-∞<a<b<+∞,1≤≤n,n∈N+,r=1/p,q,w∈L(J,C), where L(J,C) denotes the Hilbert space of all complex valued functions which are Lebesgue integrable on J, the boundary conditions are where M2(C) are complex valued 2×2 matrices, and the transmission conditions are CiY(ci-) + DiY(ci+) = 0,Ci,Di∈M2(R), |Ci|=ρi > 0, |Di|= θi > 0, where M2(R) are real valued 2×2 matrices.We construct a class of SLPs with n transmission conditions which have finite eigenvalues, and show that these eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere on the real line in the self-adjoint case.