Nonlinear Stark--Wannier Equation

Abstract

In this paper we consider stationary solutions to the nonlinear one-dimensional Schrödinger equation with a periodic potential and a Starktype perturbation. In the limit of large periodic potential the Stark-Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases.Ams classification (MSC 2010): 35Q55, 81Qxx, 81T25.

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

“…Remark 3. In [18] the estimate of the error was given in the energy norm, and even in [23] we used the H 1 -norm. If one wants to extend the result of Corollary 1 to the H 1 -norm it is clear that one has to pay a price; indeed, in the proof of Theorem 4 the term u 0 H 1 ∼ h −1/2 would appear instead of the term u 0 L 2 = 1 and therefore the estimate of the error becames meaningless.…”

“…the remainder term r 1,n is defined as Finally u n , |ψ| 2 ψ = C 1 |c n | 2 c n + r 4,n , C 1 = u n 4 L 4 , where we set r 4,n = u n , |ψ| 2 ψ − C 1 |c n | 2 c n and where by Lemma 1.vi [23] it follows that…”

“…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).…”

“…In §4 we formally construct the discrete nonlinear Schrödinger equations; in this Section we make use of some ideas already developed by [11,23] and we refer to these papers as much as possible. We must underline that in [11,23] the estimate of the remainder terms is given in the norm ℓ 1 , while in the present paper estimates in the norm ℓ 2 are necessary and thus most of the material of Section 3, and in particular Lemmata 2, 3, 4, 5 and 6, is original and it cannot be simply derived from the papers quoted above. In §5 we finally prove the validity of the tight-binding approximation with a precise estimate of the error in Theorem 4, the method used is based on an idea already applied by [21] for a double-well model and now applied to a periodic potential; in particular, in §5.1 we consider the case where α 1 and α 2 are fixed, i.e.…”