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  • Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation
Title: Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation
Abstract We consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz–Ladik lattice. We consider periodic boundary conditions with period N and initial data sampled 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 $$\beta $$ β -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.  more » « less
Award ID(s):
1928930
PAR ID:
10420205
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Communications in Mathematical Physics
Volume:
399
Issue:
3
ISSN:
0010-3616
Page Range / eLocation ID:
1689 to 1729
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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