Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

Nonlinear Stark--Wannier Equation

“…Concerning estimate (19) we consider the term u n , W u n when |n| ≤ N ; let where χ Ω0 is the characteristic function on Ω 0 . Then (the properties below concerning w 0 are given in Lemma 4.iii by [13], where w 0 (x) is the single well ground state defined in §3.2)…”

“…Finally, concerning the nonlinear term we recall the following result which follows by [13] (where we choose σ = 1, for the purpose of completeness the detailed proof is given in a separate appendix).…”

“…the map r 3 (54) and f n 2 = u n , ψ 3 − ψ 3 1 . By the proof of Lemma 3 in [13] we have that for any ρ ∈ (0, S 0 ) there exists C := C ρ > 0 such that the vector f 1 = {f n 1 } n∈Z can be estimated as follows…”

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

“…Recently, it has been proved that (1) admits a family of stationary solutions by reducing it to discrete nonlinear Schrödinger equations [11,17,23]. Concerning the reduction of the time-dependent equation to a discrete time-dependent nonlinear Schrödinger equation much less is known and rigorous results are only given under some conditions: for instance, in [4] the authors prove the validity of the reduction Date: October 10, 2019.…”

“…where β ∼ e −S0/h is an exponentially small positive constant in the semiclassical limit h ≪ 1 (in fact, S 0 > 0 is the Agmon distance between two adjacent wells, and for a precise estimate of the coupling parameter β we refer to (11)). Furthermore, ξ n = u n , W u n and C 1 = u n 4 L 4 where, roughly speaking (a precise definition for u n is given by [9,11,23]), {u n } n∈Z is an orthonormal base of vectors of the eigenspace associated to the first band of the Bloch operator such that u n ∼ ψ n as h goes to zero; where ψ n is the ground state with associated energy Λ 1 of the Schrödinger equation with a single well potential V n obtained by filling all the wells, but the n-th one, of the periodic potential V : −h 2 ∂ 2 ψn ∂x 2 + V n ψ n = Λ 1 ψ n . In fact, the linear operator −h 2 ∂ 2 ∂x 2 + V n has a single well potential and thus it has a not empty discrete spectum, we denote by Λ 1 the first eigenvalue (which is independent on the index n by construction).…”