2025/01/28 by Nizar Demni, Demni, Nizar, Zouhaı̈r Mouayn +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2501.17291
openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.