Mathematics

On Recovering Differential Operators on a Closed Set from Spectra

The Sturm – Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set.

Pieri Formulae and Specialisation of Super Jacobi Polynomials

We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra osp(2m + 1, 2n) can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties.

An Inverse Spectral Problem for Sturm – Liouville Operators with Singular Potentials on Graphs with a Cycle

This paper is devoted to the solution of inverse spectral problems for Sturm – Liouville operators with singular potentials from class W2−1 on graphs with a cycle. We consider the lengths of the edges of investigated graphs as commensurable quantities. For the spectral characteristics, we take the spectra of specific boundary value problems and special signs, how it is done in the case of classical Sturm – Liouville operators on graphs with a cycle. From the spectra, we recover the characteristic functions using Hadamard’s theorem.