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An extended anyon Fock space and noncommutative Meixnertype orthogonal polynomials in infinite dimensions
Russian Mathematical Surveys, Volume: 70, Issue: 5, Start page: 857
Swansea University Authors: Eugene Lytvynov , Irina Rodionova
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DOI (Published version): 10.1070/RM2015v070n05ABEH004965
Abstract
Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R...
Published in:  Russian Mathematical Surveys 

Published: 
2015

URI:  https://cronfa.swan.ac.uk/Record/cronfa22143 
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Abstract: 
Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operatorvalued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$space generated by the algebra of polynomials in variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixnertype class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon L\'evy white noise of the Meixner type is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. Furthermore, we get the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x \eta\partial_x^\dag\partial_x\partial_x+\partial_x$.Here $\partial_x$ and $\partial_x^\dag$ are annihilation and creation operators at point $x$. 

College: 
College of Science 
Issue: 
5 
Start Page: 
857 