Related papers: The bispectral problem and polynomial solutions
We consider a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, we state localisation theorems for the pair-eigenvalues and discuss relations to a class of non-self-adjoint spectral…
We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive…
A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove…
In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions…
For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
In this paper, we investigate the unique solvability of a mixed boundary value problem for a fractional partial differential equation featuring a degenerate coefficient. By introducing a novel operator and applying the method of separation…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
We consider inverse problems for wave, heat and Schr\"odinger-type operators and corresponding spectral problems on domains of ${\bf R}^n$ and compact manifolds. Also, we study inverse problems where coefficients of partial differential…
Scattering problem for a self-adjoint integro-differential operator, which is the sum of the operator of second derivative and of a finite-dimensional self-adjoint operator, is studied. Jost solutions are found and it is shown that the…
A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators…
Inverse initial and inverse source problems of a time-fractional differential equation with Bessel operator are considered. Results on existence and uniqueness of solutions to these problems are presented. The solution method is based on…
We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary…
Using the concept of $\mathcal{D}$-operator and the classical discrete family of dual Hahn, we construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of higher order difference operators.
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We discuss approaches to computing eigenfunctions of the Ornstein--Uhlenbeck (OU) operator in more than two dimensions. While the spectrum of the OU operator and theoretical properties of its eigenfunctions have been well characterized in…
In this paper, we study the direct and inverse spectral problems for the Schrodinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic…