arXiv subjects

Maxim Derevyagin

Publications and source records attributed to Maxim Derevyagin.

At least 19 recordsLinked to original sources

arXiv2026

A unified approach via Geronimus transformation to various types of orthogonal polynomials

We present a unified Geronimus-type construction that leads to various orthogonal polynomials including indefinite orthogonal polynomials, exceptional and, more generally, lacunary orthogonal polynomials, and multiple orthogonal polynomials. We use generalized Hermite polynomials to demonstrate the idea and obtain various results for the Geronimus-type transformation such as zero location, recurrence relations, and differential equations.

math.CA↗

arXiv2026

A remark on Chebyshev rational functions, multipoint Pad\'e approximants and Noise

Motivated by the recent interest in multipoint Pad\'e approximants in the physics community, we discuss Chebyshev rational functions and show how they give rise to multipoint Pad\'e approximants in exactly the same way that Chebyshev polynomials produce Pad\'e approximants. We present recurrence relations for Chebyshev rational functions, as well as the underlying continued fraction of Thiele type (also known as $R_{II}$ type). Finally, we provide numerical evidence illustrating the effects of noise on this interpolation scheme and show that a phenomenon similar to that recently observed by Costin, Dunne, and Meynig for Pad\'e approximants also occurs in the multipoint setting.

math.CA↗

arXiv2024

Jacobi matrices that realize perfect quantum state transfer and early state exclusion

In this paper we show how to construct 1D Hamiltonians, that is, Jacobi matrices, that realize perfect quantum state transfer and also have the property that the overlap of the time evolved state with the initial state is zero for some time before the transfer time. If the latter takes place we call it an early exclusion state. We also show that in some case early state exclusion is impossible. The proofs rely on properties of Krawtchouk and Chebyshev polynomials.

quant-ph↗

arXiv2023

DEK-Type orthogonal polynomials and a modification of the Christoffel formula

In this note we revisit one of the first known examples of exceptional orthogonal polynomials that was introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. This characterization also leads to a family of general orthogonal polynomials with missing degrees and this approach manifests its relation to biorthogonal polynomials introduced by Iserles and Nørsett, which are applicable to a whole range of problems in computational and applied analysis. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.

math-ph↗

arXiv2022

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to $R_{II}$-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.

math.CA↗

arXiv2021

Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality

We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are in fact functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem. At the end, we make some connections to other problems.

math-ph↗

arXiv2021

Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs

We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.

math-ph↗

arXiv2020

A note on an asymptotic formula for integrals of products of Jacobi polynomials

We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.

math.CA↗

arXiv2020

A Theorem of Joseph-Alfred Serret and its Relation to Perfect Quantum State Transfer

In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.

math.CA↗

arXiv2020

Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs

In this paper we study the spectral features, on fractal-like graphs, of Hamiltonians which exhibit the special property of perfect quantum state transfer: the transmission of quantum states without dissipation. The essential goal is to develop the theoretical framework for understanding the interplay between perfect quantum state transfer, spectral properties, and the geometry of the underlying graph, in order to design novel protocols for applications in quantum information science. We present a new lifting and gluing construction, and use this to prove results concerning an inductive spectral structure, applicable to a wide variety of fractal-like graphs. We illustrate this construction with explicit examples for several classes of diamond graphs.

math-ph↗

arXiv2020

Perfect quantum state transfer on diamond fractal graphs

In the quest for designing novel protocols for quantum information and quantum computation, an important goal is to achieve perfect quantum state transfer for systems beyond the well-known one dimensional cases, such as 1d spin chains. We use methods from fractal analysis and probability to find a new class of quantum spin chains on fractal-like graphs (known as diamond fractals) which support perfect quantum state transfer, and which have a wide range of different Hausdorff and spectral dimensions. The resulting systems are spin networks combining Dyson hierarchical model structure with transverse permutation symmetries of varying order.

math-ph↗

arXiv2017

Jacobi matrices generated by ratios of hypergeometric functions

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.

math.CA↗

arXiv2017

Asymptotics for polynomials orthogonal in an indefinite metric

We continue studying polynomials generated by the Szegő recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then translate them to the real line to obtain the Szegő asymptotics for the resulting polynomials. The latter polynomials give rise to a non-symmetric tridiagonal matrix but it is a finite-rank perturbation of a symmetric Jacobi matrix.

math.CA↗

arXiv2016

A note on Wall's modification of the Schur algorithm and linear pencils of Jacobi matrices

In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carathéodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szegő mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disc. For example, it will be shown that, when applying the Wall transformation, instead of OPRL, we get a sequence of orthogonal rational functions that satisfy three-term recurrence relation of the form $(H-λJ)u=0$, where $u$ is a semi-infinite vector, whose entries are the rational functions. Besides, $J$ and $H$ are Hermitian Jacobi matrices for which a version of the Denisov-Rakhmanov theorem holds true. Finally we will demonstrate how pseudo-Jacobi polynomials (aka Routh-Romanovski polynomials) fit into the picture.

math.CA↗

arXiv2016

On Szegő's theorem for a nonclassical case

In this paper we prove Szegő's Theorem for the case when a finite number of Verblunsky coefficients lie outside the closed unit disk. Although a form of this result was already proved by A.L. Sakhnovich, we use a very different method, which shows that the OPUC machinery can still be applied to deal with such nonclassical cases. The basic tool we use is Khrushchev's formula that in the classical case relates the absolutely continuous part of the measure and the $N$-th iterate of the Schur algorithm. It is noteworthy that Khrushchev's formula makes the proof short and extremely transparent. Also, we discuss Verblunsky's theorem for the case in question.

math.CA↗

arXiv2016

Multidimensional Toda Lattices: Continuous and Discrete Time

In this paper we present multidimensional analogues of both the continuous- and discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

math-ph↗

arXiv2015

Discrete integrable systems generated by Hermite-Padé approximants

We consider Hermite-Padé approximants in the framework of discrete integrable systems defined on the lattice $\mathbb{Z}^2$. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Padé approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.

math.CA↗

arXiv2015

Truncations of a class of pseudo-Hermitian operators

We consider the class of non-Hermitian operators represented by infinite tridiagonal matrices, selfadjoint in an indefinite inner product space with one negative square. We approximate them with their finite truncations. Both infinite and truncated matrices have eigenvalues of nonpositive type: either a single one on the real axis or a couple of complex conjugate ones. As a tool to evaluate the reliability of the use of truncations in numerical simulations, we give bounds for the rate of convergence of their eigenvalues of nonpositive type. Numerical examples illustrate our results.

math-ph↗