Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin

We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter with "a negative number of poles". More precisely, we treat in a unified manner one-dimensional Schr\"{o}dinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz--Nevanlinna function of the eigenvalue parameter at each endpoint, and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter dependent boundary conditions to those with inverse square singularities, and vice versa.Comment: 20 pages, 3 TikZ figures, submitte

Darboux transformations on Sturm-Liouville Eigenvalue Problems with Eigenparameter Dependent Transmission Conditions

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg. In fulfillment of the requirements for the degree of Master of Science, 2017Sturm-Liouville eigenvalue problems arise prominently in mathematical physics. An innumerous amount of complexities have been encountered in solving these problems and a myriad of techniques have been explored over the century. In this work, we investigate one such technique, namely the Darboux-Crum trans formation. This transformation transforms an existing problem into one that is readily solvable or displays properties that are better understood. In particular, we focus our attention on the e↵ect the Darboux-Crum transformation has on the eigenparameter dependence of the transmission condition of our Sturm-Liouville eigenvalue problem.XL201

Local solvability and stability for the inverse Sturm-Liouville problem with polynomials in the boundary conditions

In this paper, we for the first time prove local solvability and stability of the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients

Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions

In this paper, we for the first time get constructive solution for the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction

Eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary and interface conditions

In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given

Forward and inverse spectral theory of Sturm-Liouville operators with transmission conditions

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, School of Mathematics, 2017.ForwardandinversespectralproblemsconcerningSturm-Liouvilleoperatorswithoutdiscontinuitieshavebeenstudiedextensively. Bycomparison,therehasbeenlimitedworktacklingthecase where the eigenfunctions have discontinuities at interior points, a case which appears naturally in physical applications. We refer to such discontinuity conditions as transmission conditions. We consider Sturm-Liouville problems with transmission conditions rationally dependent on the spectral parameter. We show that our problem admits geometrically double eigenvalues, necessitating a new analysis. We develop the forward theory associated with this problem and also consider a related inverse problem. In particular, we prove a uniqueness result analogous to that of H. Hochstadt on the determination of the potential from two sequences of eigenvalues. In addition, we consider the problem of extending Sturm’s oscillation theorem, regarding the number of zeroes of eigenfunctions, from the classical setting to discontinuous problems with general constant coefficient transmission conditionsGR201