Derivation of the tight-binding approximation for time-dependent nonlinear Schr\"odinger equations

TL;DR

This paper rigorously derives a tight-binding approximation for the nonlinear time-dependent Schrödinger equation with periodic potential, providing precise estimates of the approximation error in the large potential limit.

Contribution

It establishes a mathematically rigorous connection between the continuous nonlinear Schrödinger equation and the discrete tight-binding model in a time-dependent setting.

Findings

Validates the tight-binding approximation with explicit error bounds

Shows the approximation holds in the large potential limit

Bridges continuous and discrete models for nonlinear quantum dynamics

Abstract

In this paper we consider the nonlinear one-dimensional time-dependent Schroedinger equation with a periodic potential and a local 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 Schroedinger equation of the tight-binding model.

Equations

$$\displaystyle\left\{\begin{array}[]{l}i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{1}{\epsilon}V\psi+\alpha_{1}{W}\psi+\alpha_{2}|\psi|^{2}\psi\,,\psi(\cdot,t)\in L^{2}({\mathbb{R}})\\ \psi(x,0)=\psi_{0}(x)\end{array}\right.$$

$$h=\hslash \sqrt{ϵ/2m},$$

$$\tau =\frac{\hslash }{h}t$$

$$F={\alpha }_1\frac{2m{h}^2}{{\hslash }^2}\text{mbox}and\eta ={\alpha }_2\frac{2m{h}^2}{{\hslash }^2},$$

$$ih\frac{\partial \psi }{\partial \tau }=-h^2\frac{{\partial }^2\psi }{\partial x^2}+V\psi +FW\psi +\eta |\psi {|}^2\psi$$

$$ih{\dot{g}}_n=-\beta (g_{n+1}+g_{n-1})+F{\xi }_n{g}_n+\eta C_1|g_n{|}^2{g}_n,n\in \mathbb{Z},$$

$$F\sim C_1\eta \sim \beta .$$

$$V(x)=V(x+a),\forall x\in \mathbb{R},$$

$$V(x)>V(x_0),\forall x\in \left[-\frac{1}{2}a,+\frac{1}{2}a\right)\setminus \{x_0\}.$$

$$ϵ\ll 1.$$

$$\underset{ϵ\to 0^{+}}{\lim }\frac{F}{\beta }\in \mathbb{R}\setminus \{0\}\text{mbox}and\underset{ϵ\to 0^{+}}{\lim }\frac{C_1\eta }{\beta }\in \mathbb{R}\setminus \{0\},$$

$$H_B:=-h^2\frac{d^2}{d{x}^2}+V.$$

$$\sigma (H_B)={\cup }_{\ell =1}^{\infty }[E_{\ell }^b,E_{\ell }^t],\text{mbox}where{E}_{\ell }^t\le E_{\ell +1}^b<E_{\ell +1}^t.$$

$$\frac{1}{C}h\le E_1^b\le Ch\text{mbox}and\frac{1}{C}h\le E_2^b-E_1^t\le Ch$$

$$\psi ={\psi }_1+{\psi }_{\perp }\text{mbox}where{\psi }_1=\Pi \psi \text{mbox}and{\psi }_{\perp }={\Pi }_{\perp }\psi .$$

$${\Vert \psi (\tau ,\cdot )-\sum _{n\in \mathbb{Z}}{g}_n(\tau )e^{i{\Lambda }_1\tau /h}{u}_n(\cdot )\Vert }_{L^2}\le C{h}^{1/2}.$$

$${\Vert \psi (\tau ,\cdot )-\sum _{n\in \mathbb{Z}}{g}_n(\tau )e^{i{\Lambda }_1\tau /h}{u}_n(\cdot )\Vert }_{L^2}\le C{e}^{-\zeta /h}$$

$$\underset{h\to 0^{+}}{\lim }\frac{f}{g}\in \mathbb{R}\setminus \{0\}.$$

$$\Vert f{\Vert }_{L^p}\le C\Vert \nabla f{\Vert }_{L^2}^{\delta }\Vert f{\Vert }_{L^2}^{1-\delta },\delta =\frac{1}{2}-\frac{1}{p}=\frac{p-2}{2p},p\in [2,+\infty ],$$

$$u(\tau )\le \alpha (\tau )+{\int }_0^{\tau }\delta (q)u(q)dq,\forall \tau \ge 0,$$

$$u(\tau )\le \alpha (\tau )e^{{\int }_0^{\tau }\delta (q)dq},\forall \tau \ge 0.$$

$$d_A(x,y)=\underset{\gamma \in \mathcal{C}}{\inf }{\int }_0^1\sqrt{{\left[V(\gamma (t))-E\right]}_{+}}|{\gamma }^{\prime }(t)|dt$$

$$H:=H_B+FW$$

$$\psi ,\psi \frac{\partial (V+FW)}{\partial x},\frac{\partial \psi }{\partial x}\in L^8([-T,T];L^4(\mathbb{R})),$$

$$\Vert \psi (\cdot ,\tau ){\Vert }_{L^2}=\Vert {\psi }_0(\cdot ){\Vert }_{L^2}$$

$$\mathcal{E}\left[\psi (\cdot ,\tau )\right]=\mathcal{E}\left[{\psi }_0(\cdot )\right]$$

$$\mathcal{E}(\psi )$$

$$\Vert \psi {\Vert }_{L^2}=\Vert {\psi }_0{\Vert }_{L^2}=1\text{mbox}and\Vert \nabla \psi {\Vert }_{L^2}\le C{h}^{-1/2},$$

$$\Vert {\psi }_1{\Vert }_{L^2}\le \Vert {\psi }_0{\Vert }_{L^2}=1\text{mbox}and\Vert \nabla {\psi }_1{\Vert }_{L^2}\le C{h}^{-1/2},$$

$$\Vert {\psi }_{\perp }{\Vert }_{L^2}\le \Vert {\psi }_0{\Vert }_{L^2}=1\text{mbox}and\Vert \nabla {\psi }_{\perp }{\Vert }_{L^2}\le C{h}^{-1/2};$$

$$\Vert {\psi }_0{\Vert }_{L^2}^2=\Vert \psi {\Vert }_{L^2}^2=\Vert {\psi }_{\perp }{\Vert }_{L^2}^2+\Vert {\psi }_1{\Vert }_{L^2}^2;$$

Full text

Ams classification (MSC 2010): 35Q55, 81Qxx, 81T25.

Keywords: Nonlinearity, PDEs, Tight-binding

This paper is partially supported by GNFM-INdAM

1. Introduction

Here we consider the nonlinear one-dimensional time-dependent Schrödinger equation with a cubic nonlinearity, a periodic potential $V$ and a perturbing potential $W$

[TABLE]

in the limit of large periodic potential, i.e. $0<\epsilon\ll 1$; $\alpha_{1}$ represents the strength of the perturbing potential $W$ and $\alpha_{2}$ represents the strength of the nonlinearity term.  Equation (3) is the so called Gross-Pitaevskii equation for Bose-Einstein condensates where $\hbar$ is the Planck’s constant and $m$ is the mass of the single atom.  Such a model describes, for instance, one-dimensional Bose-Einstein condensates in an optical lattice and under the effect of an external field with potential $\alpha_{1}W$; in particular, when such a perturbing potential is a Stark-type potential, that is it is locally linear, then recently has been shown the existence of Bloch oscillations for the wavefunction condensate and a precise measurement of the gravity acceleration has given [10, 19].

In the physical literature a standard way to study equation (3) consists in reducing it to a discrete Schrödinger equation taking into account only nearest neighbor interactions, the so called tight-binding model [3].  The validity of such an approximation is, as far as we know, not yet rigorously proved in a general setting.

Recently, it has been proved that (3) 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 to discrete nonlinear Schrödinger equations for large times when $V$ is multiple-well trapped potential; while, in [18] a similar result for periodic potentials $V$ satisfying a sequence of specific technical conditions (see Theorem 2.5 [16] for a resume) is obtained.  We must also recall the papers [1, 2, 5] where applications of the orbital functions in a similar context is developed; in particular, in [2] the authors prove the validity of the reduction to discrete nonlinear Schrödinger equations of the Gross-Pitaevskii equation with a periodic linear potential and a sign-varying nonlinearity coefficient.  In [5] the authors consider the case of a two-dimensional lattice; in particular, they show that tight-binding approximation is justified for simple and honeycomb lattices provided that the initial wavefunction is exponentially small.

In this paper we are able to show that the reduction of (3) to the time-dependent discrete nonlinear Schrödinger equations properly works with a precise estimate of the error, and that we don’t need of special technical assumptions on the shape of the initial wavefunction and/or on the periodic potential; in fact, we have only to assume that the initial wavefunction is prepared on one band of the Bloch operator, let us say for argument’s sake the first one.

By introducing the new semiclassical parameter

[TABLE]

the new time variable

[TABLE]

and the effective perturbation and nonlinearity strengths

[TABLE]

then the above equation (3) takes the semiclassical form

[TABLE]

with $h\ll 1$.

In the tight-binding approximation solutions to (5) are approximated by solutions to the time-dependent discrete nonlinear Schrödinger equation

[TABLE]

where $\beta\sim e^{-S_{0}/h}$ is an exponentially small positive constant in the semiclassical limit $h\ll 1$ (in fact, $S_{0}>0$ is the Agmon distance between two adjacent wells, and for a precise estimate of the coupling parameter $\beta$ we refer to (17)).  Furthermore, $\xi_{n}=\langle u_{n},Wu_{n}\rangle$ and $C_{1}=\|u_{n}\|_{L^{4}}^{4}$ where, roughly speaking (a precise definition for $u_{n}$ is given by [9, 11, 23]), $\{u_{n}\}_{n\in{\mathbb{Z}}}$ is an orthonormal base of vectors of the eigenspace associated to the first band of the Bloch operator such that $u_{n}\sim\psi_{n}$ as $h$ goes to zero; where $\psi_{n}$ is the ground state with associated energy $\Lambda_{1}$ 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 $V$: $-h^{2}\frac{\partial^{2}\psi_{n}}{\partial x^{2}}+V_{n}\psi_{n}=\Lambda_{1}\psi_{n}$.  In fact, the linear operator $-h^{2}\frac{\partial^{2}}{\partial x^{2}}+V_{n}$ has a single well potential and thus it has a not empty discrete spectum, we denote by $\Lambda_{1}$ the first eigenvalue (which is independent on the index $n$ by construction).

We must underline that usually the tight-binding approximation is constructed by making use of the Wannier’s functions instead of the vectors $u_{n}$ [3, 16].  Indeed, the decomposition by means of the Wannier’s functions turns out to be more natural and it works for any range of $h$; on the other hand, the use of a suitable base $\{u_{n}\}_{n\in{\mathbb{Z}}}$ in the semiclassical regime of $h\ll 1$ has the great advantage that the vectors $u_{n}$ are explicitly constructed by means of the semiclassical approximation.  In fact, Wannier’s functions may be approximated by such vectors $u_{n}$ as pointed out by [14].

The analysis of the discrete nonlinear Schrödinger equations (6) depends on the relative value of the perturbative parameters $F$ and $\eta$ with respect to the coupling parameter $\beta$.  In this paper we consider two situations.

In the first case, named model 1 corresponding to Hypothesis 3a), we assume that $\alpha_{1}$ and $\alpha_{2}$ are fixed and independent of $\epsilon$.  In such a case we have that $\beta\ll|F|$ and $\beta\ll C_{1}|\eta|$ and then the analysis of (6) is basically reduced to the analysis of a system on infinitely many decoupled equations.  Indeed, the perturbative terms with strength $F$ and $\eta$ dominate the coupling term with strength $\beta$ between the adjacent wells.  In fact, this model has some interesting features; for instance, when $W$ represents a Stark-type perturbation then the analysis of the stationary solutions exhibits the existence of a cascade of bifurcations [22, 23].  On other hand, due to the fact that the perturbation is large, when compared with the coupling term, the validity of the tight-binding approximation is justified only for time intervals rather small.

In the second case, named model 2 corresponding to Hypothesis 3b), we assume that both $\alpha_{1}$ and $\alpha_{2}$ go to zero when $\epsilon$ goes to zero.  In particular, we assume that

[TABLE]

That is the perturbative terms are of the same order of the coupling term.  In such a case the validity of the tight-binding approximation holds true for times of the order of the inverse of the coupling parameter $\beta$, that is the time interval is exponentially large.

We must remark that one could consider, in principle, other limits for $\alpha_{1}$ and $\alpha_{2}$ when $h$ goes to zero and Theorem 4 is very general and it holds true under different assumptions concerning $\alpha_{1}$ and $\alpha_{2}$ provided that $F={\mathcal{O}}(h^{2})$ and $\eta={\mathcal{O}}(h^{2})$.  In fact, Hyp. 3a) and Hyp. 3b) represents, in some sense, two opposite situations concerning the choice of the parameters.

In §2 we state the assumptions on equation (5) and we state our main results in Corollaries 1 and 2, they follow from a more technical Theorem 4 we state and prove in §5.  In §3 we prove a priori estimate of the wavefunction $\psi$ and of its gradient $\nabla\psi$.  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 $\ell^{1}$, while in the present paper estimates in the norm $\ell^{2}$ 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 $\alpha_{1}$ and $\alpha_{2}$ are fixed, i.e. model 1, and in §5.2 we consider the case where $\alpha_{1}$ and $\alpha_{2}$ goes to zero as $\epsilon$ goes to zero in a suitable way, i.e. model 2.

2. Description of the model and main results

2.1. Assumptions

Here, we consider the nonlinear Schrödinger equation (3) where the following assumptions hold true.

Hypothesis 1. $V(x)$* is a smooth, real-valued, periodic and non negative function with period $a$, i.e.*

[TABLE]

and with minimum point $x_{0}\in\left[-\frac{1}{2}a,+\frac{1}{2}a\right)$ such that

[TABLE]

For argument’s sake we assume that $V(x_{0})=0$ and $x_{0}=0$.

Remark 1**.**

We could, in principle, adapt our treatment to a more general case where $V(x)$ has more than one absolute minimum point in the interval $\left[-\frac{1}{2}a,+\frac{1}{2}a\right)$.

Hypothesis 2. The perturbation ${W}(x)$ is a smooth real-valued function.  We assume that $W\in L^{\infty}({\mathbb{R}})$.

Concerning the parameters $m$, $\hbar$, $\alpha_{1}$, $\alpha_{2}$ and $\epsilon$ we make the following assumption

Hypothesis 3. We assume the limit of large periodic potential, i.e. $\epsilon$ is a real and positive parameter small enough

[TABLE]

Concerning the other parameters we assume that:

• a)

The parameters $m$, $\hbar$, $\alpha_{1}$, $\alpha_{2}$ are real-valued and independent of $\epsilon$;

or

• b)

The parameters $m$, $\hbar$ are real-valued and independent of $\epsilon$ while the parameters $\alpha_{1}$, $\alpha_{2}$ are real-valued and they go to zero as $\epsilon$ goes to zero, in particular we assume that

[TABLE]

where $F$ and $\eta$ are defined in (4), and where the parameters $\beta$ and $C_{1}$ depend on $\epsilon$ (by means of $h$) and they are defined by (17) and (18).

*For argument’s sake we assume in both cases that $\alpha_{1}\geq 0$; hence $F\geq 0$. *

Remark 2**.**

In both cases we have that $0\leq F\leq Ch^{2}$ and $|\eta|\leq Ch^{2}$ for some positive constant $C$.  In the case b), in particular, $F$ and $\eta$ are exponentially small when $h$ goes to zero.

Let $H_{B}$ be the Bloch operator formally defined on $L^{2}({\mathbb{R}},dx)$ as

[TABLE]

It is well known that this operator admits self-adjoint extension on the domain $H^{2}({\mathbb{R}})$, still denoted by $H_{B}$, and its spectrum is given by bands:

[TABLE]

The intervals $(E^{t}_{\ell},E^{b}_{\ell+1})$ are named gaps; a gap may be empty, that is $E^{b}_{\ell+1}=E^{t}_{\ell}$, or not.  It is well known that in the case of one-dimensional crystals all the gaps are empty if, and only if, the periodic potential is a constant function.  Because we assume that the periodic potential is not a constant function then one gap, at least, is not empty (for a review of Bloch operator we refer to [20]).  In particular, when $h$ is small enough then the following asymptotic behaviors [24, 25]

[TABLE]

hold true for some $C>1$; hence, the first gap between $E_{1}^{t}$ and $E_{2}^{b}$ is not empty in the semiclassical limit.

Let $\Pi$ the projection operator associated to the first band $[E_{1}^{b},E_{1}^{t}]$ of $H_{B}$ and let $\Pi_{\perp}=\mathds{1}-\Pi$.  Let

[TABLE]

We assume that

Hypothesis 4. $\Pi_{\perp}\psi_{0}=0$, where $\psi_{0}(x)=\psi(x,0)$; that is the wave function $\psi$ is initially prepared on the first band.  Through the paper we assume, for argument’s sake, that $\psi_{0}$ is normalized, i.e. $\|\psi_{0}\|_{L^{2}}=1$.

2.2. Main results

Here, we state our main results; they are a consequence of a rather technical Theorem 4 we postpone to §5.  Let ${\mathbf{g}}\in C({\mathbb{R}},\ell^{2}({\mathbb{Z}}))$ be the solution to the tight-binding model, that is the discrete nonlinear Schrödinger equation (6); let $\psi(\tau,x)\in C({\mathbb{R}},H^{1}({\mathbb{R}}))$ be the solution to the nonlinear Schrödinger equation (5) with initial condition $\psi_{0}(x)=\sum g_{n}(0)u_{n}(x)$.

Corollary 1**.**

Under the assumption Hypothesis 3a) we have that for any $\tau\in[0,Ch^{-1/2}]$ then

[TABLE]

Corollary 2**.**

Under the assumption Hypothesis 3b) we have that for any $\tau\in[0,C\beta^{-1}h]$, where $\beta^{-1}$ is exponentially large as $h$ goes to zero, then

[TABLE]

for some $C>0$ and $\zeta>0$ independent of $h$.

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 $\|u_{0}\|_{H^{1}}\sim h^{-1/2}$ would appear instead of the term $\|u_{0}\|_{L^{2}}=1$ and therefore the estimate of the error becames meaningless.  On the other hand, this argument is not critical in the case of the extension of Corollary 2 to the $H^{1}$-norm because the term $\|u_{0}\|_{H^{1}}$ is controlled by means of the exponentially small term $e^{-\zeta/h}$.  In fact, we expect that Corollary 2 still hold true with the $H^{1}$-norm even if we don’t dwell here with the detailed proof.

2.3. Notation and some functional inequalities

Hereafter, we denote by $\|\cdot\|_{L^{p}}$, $p\in[+1,+\infty]$, the usual norm of the Banach space $L^{p}({\mathbb{R}},dx)$; we denote by $\|\cdot\|_{\ell^{p}}$, $p\in[+1,+\infty]$, the usual norm of the Banach space $\ell^{p}({\mathbb{Z}})$.

Hereafter, we omit the dependence on $\tau$ in the wavefunctions $\psi$ and in the vectors ${\mathbf{c}}$ when this fact does not cause misunderstanding.

By $C$ we denote a generic positive constant independent of $h$ whose value may change from line to line.

If $f$ and $g$ are two given quantities depending on the semiclassical parameter $h$, then by $f\sim g$ we mean that

[TABLE]

Furthermore, we recall some well known results for reader’s convenience:

One-dimensional Gagliardo-Nirenberg inequality by §B.5 [16]:

[TABLE]

• -

Gronwall’s Lemma by Theorem 1.3.1 [15]: let $u(\tau)$ be a non negative and continuous function such that

[TABLE]

where $\alpha(\tau)$ and $\delta(\tau)$ are non negative and monotone not decreasing functions, then

[TABLE]

• -

Agmon distance: let $E$ be a given energy and $V(x)$ be a potential function, let $[z]_{+}=z$ if $z\geq 0$ and $[z]_{+}=0$ if $z<0$; then the Agmon distance $d_{A}(x,y)$ between two points $x,y\in{\mathbb{R}}^{d}$ is induced by the Agmon metric $[V(x)-E]_{+}dx^{2}$ where $dx^{2}$ is the standard metric on $L^{2}({\mathbb{R}}^{d})$:

[TABLE]

where ${\mathcal{C}}$ is the set of piecewise paths $\gamma$ in ${\mathbb{R}}^{d}$ connecting $\gamma(0)=x$ and $\gamma(1)=y$ (see [12] for a resume).  In particular, in dimension $d=1$ and for energy $E=V_{min}$ we denote by $S_{0}=\int_{x_{n}}^{x_{n+1}}\sqrt{V(x)-V_{min}}dx$ the Agmon distance between the bottoms $x_{n}$ and $x_{n+1}$ of two adjacent wells; by the periodicity of $V(x)$ then $S_{0}$ does not depend on the index $n$.

3. Preliminary results

We recall here some results by [6, 7, 8] concerning the solution to the time-dependent nonlinear Schrödinger equation with initial wavefunction $\psi_{0}$.  The linear operator ${H}$, formally defined as

[TABLE]

on the Hilbert space $L^{2}({\mathbb{R}},dx)$, admits a self-adjoint extension on the domain $H^{2}({\mathbb{R}})$, still denoted by ${H}$.  In order to discuss the local and global existence of solutions to (5) we apply Theorem 4.2 by [8]: if $\psi_{0}\in H^{1}({\mathbb{R}})$ there is a unique solution $\psi\in C([-T,T],H^{1}({\mathbb{R}}))$ to (5) with initial datum $\psi_{0}$, such that

[TABLE]

for some $T>0$ depending on $\|\psi_{0}\|_{H^{1}}$.

In fact (see [7]), this solution is global in time for any $\eta\in{\mathbb{R}}$ (because in the case of one-dimensional nonlinear Schrödinger equations the cubic nonlinearity in sub-critical) and (5) enjoys the conservation of the mass

[TABLE]

and of the energy

[TABLE]

where

[TABLE]

Here, we prove some useful preliminary a priori estimates.

Theorem 1**.**

The following a priori estimates hold true for any $\tau\in{\mathbb{R}}$:

[TABLE]

for some positive constant $C$.

Proof.

From the conservation of the norm we have that

[TABLE]

hence

[TABLE]

From the conservation of the energy we may obtain a priori estimate of the gradient of the wavefunction.  Let

[TABLE]

where $\langle H_{B}\psi_{0},\psi_{0}\rangle\sim h$ since $\psi_{0}$ is restricted to the eigenspace associated to the first band.  Recalling that $V\geq 0$ then we have that

[TABLE]

which implies $\|\nabla\psi_{0}\|_{L^{2}}\leq Ch^{-1/2}$.  From this fact, using the fact that $W$ is a bounded potential and by the Gagliardo-Nirenberg inequality we have that

[TABLE]

Hence, ${\mathcal{E}}(\psi_{0})\sim h$ since $F\leq Ch^{2}$ and $|\eta|\leq Ch^{2}$ (see Remark 2).  Thus, the conservation of the energy implies the following inequality:

[TABLE]

since $V_{min}=0$ and by the conservation of the norm.  Let us set

[TABLE]

then $|\Gamma|\leq C$ and $\Lambda\sim h^{-1}$ as $h$ goes to zero.  The previous inequality becomes

[TABLE]

Again, the Gagliardo-Nirenberg inequality implies that

[TABLE]

and thus we get

[TABLE]

from which it follows that

[TABLE]

for some positive constant $C$.

Since $\Pi H_{B}=H_{B}\Pi$, we have that

[TABLE]

since $V_{min}\geq 0$.  Then,

[TABLE]

hence,

[TABLE]

Similarly we get

[TABLE]

and thus the proof of the Theorem is so completed. ∎

Corollary 3**.**

We have the following estimates:

[TABLE]

Proof.

They immediately follow from the one-dimensional Gagliardo-Nirenberg inequality (where $p=+\infty$ and $\delta=\frac{1}{2}$) and from the previous result. ∎

4. Construction of the discrete time-dependent nonlinear Schrödinger equation

By the Carlsson’s construction [9] resumed and expanded by [11] (see also §3 [23] for a short review of the main results) we may write $\psi_{1}$ by means of a linear combination of a suitable orthonormal base $\{u_{n}\}_{n\in{\mathbb{Z}}}$ of the space $\Pi\left[L^{2}({\mathbb{R}})\right]$, that is

[TABLE]

where $u_{n}\in H^{1}({\mathbb{R}})$ and ${\mathbf{c}}=\{c_{n}\}_{n\in{\mathbb{Z}}}\in\ell^{2}({\mathbb{Z}})$ and where we omit, for simplicity’s sake, the dependence on $\tau$ in the wavefunctions $\psi$, $\psi_{1}$, $\psi_{\perp}$ as well as in the vector ${\mathbf{c}}$.

By inserting (9) and (10) in equation (5) then it takes the form (where $\dot{}=\frac{\partial}{\partial\tau}$)

[TABLE]

where ${\mathbf{c}}\in\ell^{2}$ and $\psi_{\perp}$ are such that for any $\tau\in{\mathbb{R}}$

[TABLE]

By mean of the gauge choice $\psi(x,\tau)\to e^{i\Lambda_{1}\tau/h}\psi(x,\tau)$, and then $\psi_{\perp}(x,\tau)\to e^{i\Lambda_{1}\tau/h}\psi_{\perp}(x,\tau)$ and $c_{n}(\tau)\to e^{i\Lambda_{1}\tau/h}c_{n}(\tau)$, (13) takes the form

[TABLE]

where $\Lambda_{1}$ is the energy associated to the ground state of the Schrödinger operator $-h^{2}\frac{\partial^{2}}{\partial x^{2}}+V_{n}$, with single well potential $V_{n}$ obtained by filling all the wells of the periodic potential $V$, but the $n$-th one; since $V_{n}(x)=V_{m}(x-x_{n}+x_{m})$ by construction (see [11, 23] for details) then the spectrum of this linear operator is independent on the index $n$ and the eigenvetor $\psi_{n}$ associated to the ground state $\Lambda_{1}$ is such that $\psi_{m}(x)=\psi_{n}(x-x_{m}+x_{n})$ .

We have that

[TABLE]

where $\Lambda_{1}$ and $\beta$ are independent of the index $n$ and $\beta$ is such that for any $0<\rho<S_{0}$ there is $C:=C_{\rho}$ such that

[TABLE]

the remainder term $r_{1,n}$ is defined as

[TABLE]

where $\tilde{D}_{n,m}$ satisfies Lemma 1 in [23].  Furthermore,

[TABLE]

where we set

[TABLE]

Finally

[TABLE]

where we set

[TABLE]

and where by Lemma 1.vi [23] it follows that

[TABLE]

Therefore, (16) may be written

[TABLE]

where we set

[TABLE]

Tight-binding approximation (6) is obtained by putting $\psi_{\perp}\equiv 0$ and by neglecting the coupling term $r_{n}$ in (21).

We have the following estimates.

Lemma 1**.**

[TABLE]

for some positive constants $C$ and $\zeta$ independent of $h$.

Proof.

Such an estimate directly comes from Lemma 1 by [23]. ∎

Lemma 2**.**

For any $0<\rho<S_{0}$ there is a positive constant $C:=C_{\rho}$ such that

[TABLE]

Proof.

We set

[TABLE]

then $r_{2,n}=\sum_{m\in{\mathbb{Z}}}W_{n,m}c_{m}$.  By Example 2.3 §III.2 [13] it follows that

[TABLE]

where $M^{\prime}$ and $M^{\prime\prime}$ are such that $\sum_{m\in{\mathbb{Z}}}|W_{n,m}|\leq M^{\prime}$ and $\sum_{n\in{\mathbb{Z}}}|W_{n,m}|\leq M^{\prime\prime}$ for any $n\in{\mathbb{Z}}$; then $M^{\prime}=M^{\prime\prime}$ because $|W_{n,m}|=|W_{m,n}|$.  Since $W$ is a bounded operator and by Lemma 1.iv [23] it immediately follows that $M^{\prime}=Ce^{(S_{0}-\rho)/h}$ for any $0<\rho<S_{0}$ and for some positive constant $C:=C_{\rho}$.  Hence, Lemma 2 is so proved. ∎

Lemma 3**.**

[TABLE]

Proof.

Since $r_{3,n}=\langle u_{n},W\psi_{\perp}\rangle_{L^{2}}$ where $\{u_{n}\}_{n\in{\mathbb{Z}}}$ is an orthonormal base of the space $\Pi\left[L^{2}({\mathbb{R}})\right]$; then, from the Parseval’s identity it follows that

[TABLE]

because $\Pi W\Pi_{\perp}$ is a bounded potential. ∎

For what concerns the vector ${\mathbf{r}}_{4}$ let

[TABLE]

where we set

[TABLE]

and

[TABLE]

where $\sum_{j,m,\ell\in{\mathbb{Z}}}^{\star}$ means that at least one of three indexes $j$, $\ell$ and $m$ is different from the index $n$.

Lemma 4**.**

Let ${\mathbf{B}}=\{B_{n}\}_{n\in{\mathbb{Z}}}$, then for any $0<\rho<S_{0}$ there is a positive constant $C$ such that

[TABLE]

Proof.

For argument’s sake let us assume that $m$ is the index different from the index $n$ in the sum (23); then we have to check the term

[TABLE]

where

[TABLE]

Let $0<\rho<S_{0}$ be fixed; from Lemma 1.iv [23] it follows that for any $\rho^{\prime},\rho^{\prime\prime}>0$ such that $\rho^{\prime}+\rho^{\prime\prime}<\rho$ then there exists a positive constant $C>0$, independent of the indexes $n$ and $m$ and of the semiclassical parameter $h$, such that

[TABLE]

Now, observing that $|c_{m}|\leq 1$ since $\|{\mathbf{c}}\|_{\ell^{2}}\leq 1$, then

[TABLE]

where we make use of the estimate (24) and where $\rho^{\prime},\rho^{\prime\prime}>0$ are such that $\rho^{\prime}+\rho^{\prime\prime}<\rho$.  Hence,

[TABLE]

for some positive constant $C$.  For what concerns the term $B_{2,n}$ we have that

[TABLE]

since $\|u_{\ell}\|_{L^{\infty}}=\|u_{0}\|_{L^{\infty}}\leq Ch^{-1/4}$ and $\left\|\sum_{m\in{\mathbb{Z}}}|u_{m}|\right\|_{L^{\infty}}\leq Ch^{-1/2}$ (see Lemma 1 [23]), from which it follows that

[TABLE]

where $\rho^{\prime},\rho^{\prime\prime}>0$ are such that $\rho^{\prime}+\rho^{\prime\prime}<\rho<S_{0}$.  Finally,

[TABLE]

[TABLE]

since $|c_{m}|\leq 1$.  Hence,

[TABLE]

From these estimates it follows that

[TABLE]

and Lemma 4 is so proved. ∎

Now we deal with the vector ${\mathbf{A}}$ with elements

[TABLE]

where

[TABLE]

Lemma 5**.**

[TABLE]

Proof.

Indeed, since $\{u_{n}\}_{n\in{\mathbb{Z}}}$ is an orthonormal base of $\Pi\left[L^{2}({\mathbb{R}})\right]$ from the Parseval’s identity it follows that

[TABLE]

from Corollary 3. ∎

In conclusion we have proved the following Lemma;

Lemma 6**.**

[TABLE]

5. Validity of the tight-binding approximation

First of all we need of the following estimate:

Lemma 7**.**

Let us set

[TABLE]

and let ${\mathbf{c}}$ and $\psi_{\perp}$ be the solutions to (21); then

[TABLE]

Proof.

Indeed, from (21) it immediately follows that

[TABLE]

from which the estimate (25) follows since $\|{{\mathbf{c}}}\|_{\ell^{2}}\leq 1$ and $|c_{n}|\leq 1$, $|\xi_{n}|\leq C$ because $W$ is a bounded potential and $C_{1}\sim h^{-1/2}$. ∎

Hereafter, we denote by $\omega$ a quantity, whose value may change from line to line, such that

[TABLE]

for some $\zeta>0$ and some $C>0$ independent of $h$.

Theorem 2**.**

[TABLE]

Proof.

Indeed, collecting the results from Lemmata 2,3 and 6 and from Remark 2 we have that

[TABLE]

from which the statement immediately follows. ∎

Since $\psi_{\perp}(x,0)=\Pi_{\perp}\psi_{0}=0$, then the second differential equation of the system (21) may be written as an integral equation of the Duhamel’s form

[TABLE]

Theorem 3**.**

We have the following estimate

[TABLE]

Proof.

Let

[TABLE]

then the previous equation (27) becomes

[TABLE]

where

[TABLE]

are such that

Lemma 8**.**

The following estimates hold true:

[TABLE]

and

[TABLE]

Proof.

In order to prove the estimates (29) and (30) we remark that $e^{-i(H_{B}-\Lambda_{1})(\tau-s)/h}$ is an unitary operator; hence,

[TABLE]

Now,

[TABLE]

from Theorem 1 and Corollary 3; hence, (30) follows.  In order to prove (29) we make use of an integration by parts:

[TABLE]

From this fact and since $\|[H_{B}-\Lambda_{1}]^{-1}\Pi_{\perp}\|=\left[\mbox{dist}(\Lambda_{1},E_{2}^{b})\right]^{-1}\sim h^{-1}$ then it follows that

[TABLE]

since

[TABLE]

and

[TABLE]

by Lemma 7, Theorem 2, and from the draft estimate $\|\psi_{\perp}\|_{L^{2}}\leq 1$. ∎

Hence, we have the following integral inequality

[TABLE]

and then the Gronwall’s Lemma implies that

[TABLE]

Theorem 3 is so proved. ∎

Now, we deal with the first differential equation of the system (21)

[TABLE]

where

[TABLE]

We compare it with the equation

[TABLE]

which represents the tight-binding approximation of (5), up to a phase factor $e^{-i\Lambda_{1}\tau/h}$ depending on time.

The Cauchy problem (34) is globally well-posed, that is there exists a unique solution ${\mathbf{g}}\in C({\mathbb{R}},\ell^{2}({\mathbb{Z}}))$ that depends continuously on the the initial data (see, e.g. Theorem 1.3 [16]).  We must underline that we have the following a priori estimate $\|{\mathbf{c}}\|_{\ell^{2}}\leq 1$ and the conservation of the norm of ${\mathbf{g}}$

[TABLE]

indeed, an immediate calculus gives that

[TABLE]

because $\beta$, $F$, $\eta$, $C_{1}$ and $\xi_{n}$ are real-valued.

Then, it follows that the vector ${\mathbf{c}}-{\mathbf{g}}$ satisfies to the following integral equation

[TABLE]

from which

[TABLE]

Lemma 9**.**

${\mathbf{G}}$* is a Lipschitz function such that*

[TABLE]

Proof.

Indeed

[TABLE]

since $|c_{n}|^{2}-|g_{n}|^{2}=c_{n}\left[\bar{c}_{n}-\bar{g}_{n}\right]+\bar{g}_{n}\left[c_{n}-g_{n}\right]$ and (18). ∎

By Theorems 2 and 3, it turns out that the vector ${\mathbf{r}}$ is norm bounded by

[TABLE]

for some positive constant $C$ independent of $h$ and where

[TABLE]

Then, we get the integral inequality

[TABLE]

where

[TABLE]

and

[TABLE]

By the Gronwall’s Lemma we finally get the estimate

[TABLE]

Therefore, we have proved that

Lemma 10**.**

Let $a$, $b$ and $c$ dafined by (37-39), then Equazioni Differenziali della Fisica Matematica

[TABLE]

for some positive constant $C$ independent of $h$.

In conclusion

Theorem 4**.**

Let ${\mathbf{g}}\in C({\mathbb{R}},\ell^{2}({\mathbb{Z}}))$ be the solution to the discrete nonlinear Schrödinger equation (34); let $\psi(\tau,x)\in C({\mathbb{R}},H^{1}({\mathbb{R}}))$ be the solution to the nonlinear Schrödinger equation (5) with initial condition $\psi_{0}(x)=\sum g_{n}(0)u_{n}(x)$; let $a$, $b$ and $c$ defined by Lemma 10; let $\lambda$ be defined by Lemma 7.  Then, for some positive constant $C$ independent of $h$ it follows that

[TABLE]

Proof.

Indeed, recalling that we made use of the gauge choice $\psi\to e^{i\Lambda_{1}\tau/h}\psi$, we have that

[TABLE]

where

[TABLE]

because $\{u_{n}\}$ is an orthonormal set of vectors. ∎

5.1. Proof of Corollary 1

Here we assume, according with Hypothesis 3a), that the real-valued parameters $\alpha_{1}$ and $\alpha_{2}$ are fixed; in such a case we have that

[TABLE]

Therefore:

[TABLE]

Then the estimate (43) makes sense for times of order $\tau\in[0,Ch^{-\gamma}]$ for some fixed $\gamma\leq\frac{1}{2}$.  In such an interval we have that

[TABLE]

In particular, for $\gamma=\frac{1}{2}$ Corollary 1 follows.

5.2. Proof of Corollary 2

Here we assume, according with Hypothesis 3b), that the real-valued parameters $\alpha_{1}$ and $\alpha_{2}$ are not fixed, but both go to zero when $\epsilon$ goes to zero; in particular we have that

[TABLE]

In such a case we have that

[TABLE]

The estimate (43) makes sense for times of order $\tau\in[0,\beta^{-1}h]$.  In such an interval we have that

[TABLE]

and

[TABLE]

for some $\zeta>0$.  Hence, Corollary 2 is proved.