Related papers: Linear spectral transformations for multivariate o…
We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…
We estabish an analog of the Cauchy-Poincare separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borg-type result) for a normal matrix. Using this result we…
In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner…
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with…
Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn).
We illustrate the basic notions of {\em additional non-isospectral symmetries} and their interplay with the discrete {\em \DB transformations} of integrable systems at the instance of {\em constrained Kadomtsev-Petviashvili} (\cKP)…
This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{\alpha }e^{-x}dx+N\delta (x-c), \] for $x\in[0,\infty)$, $\alpha>-1$, a free…
Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We study general linear perturbations of a class of 4d real-dimensional hyperkahler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic…
Crum's theorem and its modification a la Krein-Adler are formulated for the discrete quantum mechanics with real shifts, whose eigenfunctions consist of orthogonal polynomials of a discrete variable. The modification produces the associated…
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary…
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis…
An alternative expression for the Christoffel--Darboux formula for multiple orthogonal polynomials of mixed type is derived from the $LU$ factorization of the moment matrix of a given measure and two sets of weights. We use the action of…
1. The 2-Toda lattice and its generic symmetries 2. A Larger class of symmetries for special initial conditions 3. Borel decomposition of Moment matrices, tau-functions and string-orthogonal polynomials 4. From string-orthogonal Polynomials…
In this paper we analyze the quantum homological invariants (the Poincar\'e polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the…
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…