Inverse Spectral Problem Of Regular Sturm-Liouville Operator

Driven by other disciplines and many engineering and technical fields, the s-tudy of the inverse Sturm-Liouville problem has aroused great interest and high attention to domestic and overseas scholars. So far, it has become one of the fastest development and growth fields in applied mathematics. The inverse spectral and inverse nodal problems are two basic and important subjects in the inverse problem study, they have extensive and direct applications in the earth physics, the quantum physics, the meteorology and other fields, they are also the effective way of solving nonlinear evolution equation in mathematical physics.In this present paper, the inverse spectral and inverse nodal problem are s-tudied first for classical Sturm-Liouville operator then for Sturm-Liouville operator with spectral parameter in the discontinuity conditions, the uniqueness theorems are proved and the reconstruction algorithms are provided by the corresponding spectrum data, respectively. The mains works are given as follows:In the first chapter, we first give a summary of the physical backgrounds of the Sturm-Liouville operators, and elaborate the research advances of the inverse Sturm-Liouville problems, then introduce the main work of this paper.In the second chapter, the inverse two spectra theorem of classical Sturm-Liouville operators is considered. Our main goal here is to consider a variation of the classical inverse two-spectra problem in that we shall replace the second spectrum by adding a interface conditions. Using the obtained spectrum and the original spectrum, we prove the uniqueness theorem. This urges us to generalize the obtained result to a more general circumstance that the two spectra are both ob-tained by changing the parameters of the interface conditions. We first analysis the relationship between the new two spectra, then consider the uniqueness problems. We observe that, when the first parameters are the same but the second parame-ters are different, they satisfy the classical interlacing property and the uniqueness theorem is valid; however, when the first parameters are not the same, they are no longer alternate and the uniqueness result does not remain valid in general, there are at most finite numbers of potentials corresponding to two spectra. We further provide additional spectral information to attack the uniqueness problem.In the third chapter, the uniqueness problem of inverse Sturm-Liouville prob-lems with the potential known on an interior subinterval is considered. We prove that the potential on the entire interval and boundary conditions are uniquely de-termined in terms of the known eigenvalues and interior spectral data. Moreover, we also concern with the situation where the potential is smooth at some given points, in this case, some eigenvalues and interior spectral data can be missing. The obtained results are generalizations of the Hochstadt-Lieberman theorem and Gesztesy-Simon theorem.In the fourth chapter, the inverse nodal problem for the classical Sturm-Liouville operator with dense nodal subset on an interior subinterval is considered. We give a comprehensive discussion on the overdetermined phenomenon of the inverse nodal problem. We first treat the case of 1/2 belonging to this known subinterval, prove that the unknown coefficients are uniquely determined by a twin dense subnodal set on this subinterval. Note that here the interval length can be arbitrarily small. In particular, if the subinterval is symmetrical about 1/2, then the known nodal information is optimized for recovering the unknown coefficients uniquely, which may be closest to the non-overdetermined situation in our opinion. We also consider the case of 1/2 not belonging to this known subinterval, by virtue of additional spectral information we prove the uniqueness theorem. Thus, the open question raised by Yang in 2001 is solved effectively by means of this additional spectral information.In the fifth chapter, the direct and inverse Sturm-Liouville problem are con-sidered under the circumstance of the discontinuity conditions involved spectral parameter at finite interior points inside a finite interval. We first give a self-adjoint operator-theoretic formulation for this problem, analyzing the spectrum characteris-tic, obtain the expansion theorem by reference to the self-adjointness of the operator; then get the asymptotic behavior of the solutions and eigenvalues; further we provide several uniqueness results for this inverse spectral problem in terms of its spectral characteristics, such as Weyl function, the discrete spectral data, two spectra, re-spectively; finally we provide the constructive solutions by the method of spectral mappings.In the sixth chapter, we are concerned with the inverse nodal problem for the Sturm-Liouville problem that the spectral parameter appears in the interior point conditions. Since the asymptotic estimates of eigenvalues and nodal points of this problem are different from the classical ones, thus many well-known methods are no longer applicable, in this case, we apply the method established in chapter four, prove the uniqueness theorem by means of a bilaterally dense subset of the nodal points. Further, we establish the algorithm for finding piecewise constant approximation to the unknown potential.