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Condensed Matter > Statistical Mechanics
Title: Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $β$-ensemble and double confluent Heun equation
(Submitted on 5 Jul 2021 (v1), last revised 9 Jan 2023 (this version, v4))
Abstract: We consider the discrete defocusing nonlinear Schr\"odinger equation in its integrable version, which is called defocusing Ablowitz-Ladik lattice. We consider periodic boundary conditions with period $N$ and initial data sample according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular $\beta$-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.
Submission history
From: Guido Mazzuca [view email][v1] Mon, 5 Jul 2021 22:28:40 GMT (210kb,D)
[v2] Thu, 15 Jul 2021 17:44:39 GMT (209kb,D)
[v3] Wed, 13 Oct 2021 08:12:30 GMT (216kb,D)
[v4] Mon, 9 Jan 2023 10:34:07 GMT (216kb,D)
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