Preprint: N. Demni and Z Mouayn 01-2025

[Zouhair Mouayn]

Dear All,

Comments about Preprint https://arxiv.org/abs/2501.17291 are welcome.
Best regards
Zouhair Mouayn

[Submitted on 28 Jan 2025]

Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

Nizar Demni, Zouhaïr Mouayn

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 us to identify them with the 2D-Hermite polynomials associated with 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.

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