Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

TL;DR

This paper introduces polyanalytic Hermite polynomials linked to the elliptic Ginibre model, expanding the mathematical tools for analyzing eigenvalues in non-Hermitian random matrix theory and their connections to quantum physics.

Contribution

It derives new polyanalytic Hermite polynomials for the elliptic Ginibre model using Bogoliubov transformations, linking them to 2D-Hermite polynomials and quantum states.

Findings

Polynomials share orthogonality with holomorphic Hermite polynomials.

Explicit kernel formulas for Landau levels are obtained.

Connections to two-photon coherent states and SU(1,1) group are established.

Abstract

Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we appeal to squeezed creation and annihilation operators arising from the Bogoliubov transformation of creation and annihilation operators on the Bargmann-Fock space. The obtained polynomials are then expressed as linear combinations of products of Hermite polynomials and share the same orthogonality relation with holomorphic Hermite polynomials. Moreover, this expression allows to identify them with the 2D-Hermite polynomials associated to a unimodular complex symmetric 2x2 matrix. Afterwards, we derive, for any Landau level, a closed formula for the kernel of the isometry mapping the basis of (rescaled) holomorphic Hermite polynomials to the corresponding complex Hermite polynomials. This kernel is also interpreted in terms of the two-photon coherent states and the metaplectic representation of the SU(1,1) group.