Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schrödinger Equations

Abstract

In this paper we consider the nonlinear one-dimensional timedependent Schrödinger equation with a periodic potential and a bounded perturbation. In the limit of large periodic potential the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schrödinger equation of the tight-binding model. Ams classification (MSC 2010): 35Q55, 81Qxx, 81T25.

Mean Field Derivation of DNLS from the Bose–Hubbard Model

“…Thus, equality (37) can be considered as the version, in the Fock-Bargmann space, of the Quantum Wick Theorem showed in [25] that works in the Fock space and with the related bosonic creation and annihilation operators of quantum field theory.…”

“…However, it seems that a direct mean field derivation of the DNLS (1) together with quantitative, explicit estimates is missing. In some works (see for example [1,33,37] and references therein) the DNLS is obtained directly from the NLS equation, in the framework of the tight-binding approximation. However, combining these two kinds of results, a growth exponential in time of the mean field estimate for DNLS follows (essentially due to the Grönwall lemma).…”

A weak KAM approach to the periodic stationary Hartree equation

“…Thus, the asymptotics for N → +∞ furnishes an integral with increasing dimension, with normalization factor 1/(2π) n . Second, we introduce the stationary Hartree equation related to many body operator (1.1), (see for example [18,[24][25][26][27][28][29][30][31]39,40] and references therein) here for…”