Namig J. Guliyev
Institute of Mathematics and Mechanics,
Azerbaijan National Academy of Sciences,
9 B. Vahabzadeh str., AZ1141, Baku, Azerbaijan.
njguliyev@gmail.com
Recent publications
Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin. The Quarterly Journal of Mathematics 74 (2023), no. 3, 889–910. arXiv:2001.00061
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 Schrödinger 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.
We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand–Levitan integral equation for boundary coefficients only.
A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter. Analysis and Mathematical Physics 10 (2020), no. 1, Paper No. 2, 8 pp. arXiv:1905.07952
We establish a criterion for a set of eigenfunctions of the one-dimensional Schrödinger operator with distributional potentials and boundary conditions containing the eigenvalue parameter to be a Riesz basis.
On extensions of symmetric operators. Operators and Matrices 14 (2020), no. 1, 71–75. arXiv:1807.11865
We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices (1,1).
Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter. Journal of Mathematical Physics 60 (2019), no. 6, 063501, 23 pp. FEATURED ARTICLE arXiv:1806.10459
We study various direct and inverse spectral problems for the one-dimensional Schrödinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.
Corrections to the published version
page 8: the title of Subsection III.A should read "Transformation of Herglotz–Nevanlinna functions"
page 18: in the first sentence of the proof of Theorem 5.1, all three appearances of the index n−J should read n+J
On two-spectra inverse problems. Proceedings of the American Mathematical Society 148 (2020), no. 10, 4491–4502. arXiv:1803.02567
We consider a two-spectra inverse problem for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this problem.
Essentially isospectral transformations and their applications. Annali di Matematica Pura ed Applicata 199 (2020), no. 4, 1621–1648. arXiv:1708.07497
We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.