arXiv subjects

Eugene Lytvynov

Publications and source records attributed to Eugene Lytvynov.

At least 19 recordsLinked to original sources

arXiv2026

Umbral calculus over a vector space

Let $V$ be a vector space over $\mathbb F=\mathbb R$ or $\mathbb C$. We develop a basis-free umbral calculus over $V$. We define the vector space of polynomials over $V$, and polynomial sequences in it. We discuss shift-invariant operators acting in polynomials over $V$. We define polynomial sequences of binomial type and Sheffer sequences over $V$. We provide equivalent characterizations of these polynomial sequences. We prove two recurrence formulas for Sheffer sequences. With each Sheffer sequence, we associate a linear operator acting in polynomials over $V$, which we call a Sheffer operator. We prove that the set of Sheffer operators is a group for the usual product of linear operators, which is isomorphic to the Riordan group of pairs of formal tensor power series in a variable from $V$. Under the assumption that $V$ is an algebra, we lift every Sheffer sequence over $\mathbb F$ to a Sheffer sequence over $V$. We provide examples of such lifting.

math.CO↗

arXiv2026

Generalized Segal-Bargmann transform for Poisson distribution revisited

For $α>0$ and $σ> 0$, we consider the following probability distribution on $α\mathbb N_0$: $π_{α,σ} = \exp \big(- \fracσ{α^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\fracσ{α^2}\big)^n δ_{αn}$, where $δ_y$ denotes the Dirac measure with mass at $y$. For $α=1$, $π_{1,σ}$ is the Poisson distribution with parameter $σ$. Furthermore, the centered probability distribution $\tilde π_{α,σ} = \exp \big(- \fracσ{α^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\fracσ{α^2}\big)^n δ_{αn-σ/α}$ weakly converges to $μ_σ$ as $α\to0$. Here $μ_σ$ is the Gaussian distribution with mean zero and variance $σ$. Let $(c_n)_{n=0}^\infty$ be the monic polynomial sequence that is orthogonal with respect to the measure $μ_{α,σ}$. In particular, for $α=1$, $(c_n)_{n=0}^\infty$ is a sequence of Charlier polynomials. Let $\mathbb F_σ(\mathbb C)$ denote the Bargmann space of all entire functions $f(z)=\sum_{n=0}^\infty f_nz^n$ with $f_n \in \mathbb C$ satisfying $ \sum_{n=0}^{\infty} {| f_n |}^2 \, n! \, σ^n < \infty$. The generalized Segal--Bargmann transform associated with the measure $π_{α,σ}$ is a unitary operator $\mathcal S:L^2(α\mathbb N_0,π_{α,σ})\to \mathbb F_σ(\mathbb C)$ that satisfies $(\mathcal Sc_n)(z)=z^n$ for $n\in\mathbb N_0$. We present some new results related to the operator $\mathcal S$. In particular, we observe how the study of $\mathcal S$ naturally leads to the normal ordering in the Weyl algebra.

math-ph↗

arXiv2025

The projection spectral theorem, quasi-free states and point processes

In this review paper, we demonstrate that several classes of point processes in a locally compact Polish space $X$ appear as the joint spectral measure of a rigorously defined particle density of a representation of the canonical anticommutation relations (CAR) or the canonical commutation relations (CCR) in a Fock space. For these representations of the CAR/CCR, the vacuum state on the corresponding $*$-algebra is quasi-free. The classes of point process that arise in such a way include determinantal and permanental point processes.

math-ph↗

arXiv2025

Lie structures of the group of Sheffer operators

Let $Φ$ be an (LB)-space over $\mathbb F=\mathbb R$ or $\mathbb C$, and let $Φ'$ be the dual space of~$Φ$. We study the set $\mathbb S(Φ)$ of Sheffer operators acting in polynomials on $Φ'$. We prove that $\mathbb S(Φ)$ is a group for the usual product of operators. We equip $\mathbb S(Φ)$ with a natural topology which makes $\mathbb S(Φ)$ into an infinite-dimensional manifold with a global parametrization. We show that $\mathbb S(Φ)$ is an infinite-dimensional, regular Lie group, and provide an explicit description of the Lie algebra of $\mathbb S(Φ)$, including an explicit form of the Lie bracket on it. Our main results are new even in the one-dimensional case, $Φ=\mathbb{F}$. Furthermore, our results lead to improved understanding of the Lie algebra of the Riordan group, cf.\ Cheon, Luzón, Morón, Prieto-Martinez, {\it Adv. Math.} 319 (2017) 522--566.

math.FA↗

arXiv2025

Segal-Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials

Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal--Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal--Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct and study a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences.

math-ph↗

arXiv2024

Finite difference calculus in the continuum

The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations, find explicit form for certain finite difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.

math-ph↗

arXiv2024

Analysis on the cone of discrete Radon measures

We study analysis on the cone of discrete Radon measures over a locally compact Polish space $X$. We discuss probability measures on the cone and the corresponding correlation measures and correlation functions on the sub-cone of finite discrete Radon measures over $X$. For this, we consider on the cone an analogue of the harmonic analysis on the configuration space developed in [12]. We also study elements of the difference calculus on the cone: we introduce discrete birth-and-death gradients and study the corresponding Dirichlet forms; finally, we discuss a system of polynomial functions on the cone which satisfy the binomial identity.

math-ph↗

arXiv2023

Particle-hole transformation in the continuum and determinantal point processes

Let $X$ be an underlying space with a reference measure $σ$. Let $K$ be an integral operator in $L^2(X,σ)$ with integral kernel $K(x,y)$. A point process $μ$ on $X$ is called determinantal with the correlation operator $K$ if the correlation functions of $μ$ are given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{det}[K(x_i,x_j)]_{i,j=1,\dots,n}$. It is known that each determinantal point process with a self-adjoint correlation operator $K$ is the joint spectral measure of the particle density $ρ(x)=\mathcal A^+(x)\mathcal A^-(x)$ ($x\in X$), where the operator-valued distributions $\mathcal A^+(x)$, $\mathcal A^-(x)$ come from a gauge-invariant quasi-free representation of the canonical anticommutation relations (CAR). If the space $X$ is discrete and divided into two disjoint parts, $X_1$ and $X_2$, by exchanging particles and holes on the $X_2$ part of the space, one obtains from a determinantal point process with a self-adjoint correlation operator $K$ the determinantal point process with the $J$-self-adjoint correlation operator $\widehat K=KP_1+(1-K)P_2$. Here $P_i$ is the orthogonal projection of $L^2(X,σ)$ onto $L^2(X_i,σ)$. In the case where the space $X$ is continuous, the exchange of particles and holes makes no sense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-free representation of the CAR. This transformation acts identically on the $X_1$ part of the space and exchanges the creation operators $\mathcal A^+(x)$ and the annihilation operators $\mathcal A^-(x)$ for $x\in X_2$. This leads to a quasi-free representation of the CAR, which is not anymore gauge-invariant. We prove that the joint spectral measure of the corresponding particle density is the determinantal point process with the correlation operator $\widehat K$.

math-ph↗

arXiv2023

Quasi-free states on a class of algebras of multicomponent commutation relations

Multicomponent commutations relations (MCR) describe plektons, i.e., multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix $Q(x_1,x_2)$ that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several conditions, including the functional Yang--Baxter equation. The aim of the paper is to give an appropriate definition of a quasi-free state on an MCR algebra, and construct such states on a class of MCR algebras. We observe a significant difference between the classical setting for bosons and fermions and the setting of MCR algebras. We show that the developed theory is applicable to systems that contain quasiparticles of opposite type. An example of such a system is a two-component system in which two quasiparticles, under exchange, change their respective types to the opposite ones ($1\mapsto 2$, $2\mapsto1$). Fusion of quasiparticles means intuitively putting several quasiparticles in an infinitely small box and identifying the statistical behaviour of the box. By carrying out fusion of an odd number of particles from the two-component system as described above, we obtain further examples of quantum systems to which the developed theory is applicable.

math-ph↗

arXiv2021

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations -- discrete Radon measures $γ=\sum_iδ_{x_i}$ on $X$, where $δ_{x_i}$ is the Dirac measure with mass at $x_i$.The spatial falling factorials $(γ)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}δ_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\niω\mapsto (ω)_n\in M^{(n)}(X)$, where $M^{(n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space $\mathcal {CF}^{(n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$, are linear operators, acting between spaces $\mathcal {CF}^{(n)}(X)$ and $\mathcal {CF}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(ω)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*ω^{\otimes k}$ and $ω^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(ω)_k$, respectively. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

math.CO↗

arXiv2021

Hafnian point processes and quasi-free states on the CCR algebra

Let $X$ be a locally compact Polish space and $σ$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $σ$ is the Lebesgue measure). Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb K(y,x)$. We say that a point process $μ$ in $X$ is hafnian with correlation kernel $\mathbb K(x,y)$ if, for each $n\in\mathbb N$, the $n$th correlation function of $μ$ (with respect to $σ^{\otimes n}$) exists and is given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\,$. Here $\operatorname{haf}(C)$ denotes the hafnian of a symmetric matrix $C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process $Π_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying $\int_Δ\mathbb E(|G(x)|^2)σ(dx)<\infty$ for each compact $Δ\subset X$. Then the Cox process $Π_R$ with $R(x)=|G(x)|^2$ is a hafnian point process. The main result of the paper is that each such process $Π_R$ is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCR), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.

math.PR↗

arXiv2019

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bożejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

math-ph↗

arXiv2019

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $α\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^α_{\mathrm{min}}(\mathbb C)$, the Fréchet topological space of entire functions of order at most $α$ and minimal type (when the order is equal to $α>0$). In particular, every function $f\in \mathcal E^α_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^α_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $Φ$ and its dual space $Φ'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $Φ'$. In particular, for $Φ=Φ'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $Φ'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^α_{\mathrm{min}}(Φ')$ when $α>1$. The latter result is new even in the one-dimensional case.

math.FA↗

arXiv2018

Construction of solutions to parabolic and hyperbolic initial-boundary value problems

We show that infinitely differentiable solutions to parabolic and hyperbolic equations, whose right-hand sides are analytical in time, are also analytical in time at each fixed point of the space. These solutions are given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\infty$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show that exact regular and weak solutions to the initial-boundary value problems for parabolic and hyperbolic equations can be determined as the sum of a function that satisfies the boundary conditions and the limit of the infinitely differentiable solutions for smooth approximations of the data of the corresponding problem with zero boundary conditions. These solutions are represented in the form of the Taylor expansion with respect to $t$. The suggested me

math.AP↗

arXiv2018

Quasi-invariance of completely random measure

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $μ$ be a completely random discrete measure on $X$, i.e., $μ$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_+)$ consisting of functions on $X$ that take values in $\mathbb R_+=(0,\infty)$ and are equal to 1 outside a compact set. Each element $θ\in C_0(X\to\mathbb R_+)$ maps $\mathbb K(X)$ onto itself; more precisely, $θ$ sends a discrete Radon measure $\sum_i s_iδ_{x_i}$ to $\sum_i θ(s_i)s_iδ_{x_i}$. Thus, elements of $C_0(X\to\mathbb R_+)$ transform the weights of discrete Radon measures. We study conditions under which the measure $μ$ is quasi-invariant under the action of the current group $C_0(X\to\mathbb R_+)$ and consider several classes of examples. We further assume that $X=\mathbb R^d$ and consider the group of local diffeomorphisms $\operatorname{Diff}_0(X)$. Elements of this group also map $\mathbb K(X)$ onto itself. More precisely, a diffeomorphism $φ\in \operatorname{Diff}_0(X)$ sends a discrete Radon measure $\sum_i s_iδ_{x_i}$ to $\sum_i s_iδ_{φ(x_i)}$. Thus, diffeomorphisms from $\operatorname{Diff}_0(X)$ transform the atoms of discrete Radon measures. We study quasi-invariance of $μ$ under the action of $\operatorname{Diff}_0(X)$. We finally consider the semidirect product $\mathfrak G:=\operatorname{Diff}_0(X)\times C_0(X\to \mathbb R_+)$ and study conditions of quasi-invariance and partial quasi-invariance of $μ$ under the action of $\mathfrak G$.

math.PR↗

arXiv2018

An infinite dimensional umbral calculus

The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.

math.FA↗

arXiv2017

Fock representations of $Q$-deformed commutation relations

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^†_t=Q(s,t)\partial_t^†\partial_s+δ(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish space), the function $Q:T^2\to \mathbb C$ satisfies $|Q(s,t)|\le1$ and $Q(s,t)=\overline{Q(t,s)}$, and $$\int_{T^2}h(s)g(t)δ(s,t)\,σ(ds)σ(dt):=\int_T h(t)g(t)\,σ(dt),$$ $σ$ being a fixed reference measure on $T$. In the case where $|Q(s,t)|\equiv 1$, the $Q$-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev (1995). These generalized statistics contain anyon statistics as a special case (with $T=\mathbb R^2$ and a special choice of the function $Q$). The related $Q$-deformed Fock space $\mathcal F(\mathcal H)$ over $\mathcal H:=L^2(T\to\mathbb C,σ)$ is constructed. An explicit form of the orthogonal projection of $\mathcal H^{\otimes n}$ onto the $n$-particle space $\mathcal F_n(\mathcal H)$ is derived. A scalar product in $\mathcal F_n(\mathcal H)$ is given by an operator $\mathcal P_n\ge0$ in $\mathcal H^{\otimes n}$ which is strictly positive on $\mathcal F_n(\mathcal H)$. We realize the smeared operators $\partial_t^†$ and $\partial_t$ as creation and annihilation operators in $\mathcal F(\mathcal H)$, respectively. Additional $Q$-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form $\partial^†_s\partial^†_t=Q(t,s)\partial^†_t\partial^†_s$, $\partial_s\partial_t=Q(t,s)\partial_t\partial_s$, valid for those $s,t\in T$ for which $|Q(s,t)|=1$.

math-ph↗

arXiv2016

Wick calculus for noncommutative white noise corresponding to $q$-deformed commutation relations

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to $q$-deformed commutation relations with $q\in(-1,1)$. We construct a Gel'fand triple centered at the $q$-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a Våge-type inequality for the Wick product on the dual space.

math.PR↗