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

In this paper, we consider the nonlinear one-dimensional time-dependent 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.


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 i ∂ψ ∂t = − 2 2m ∂ 2 ψ ∂x 2 + 1 V ψ + α 1 W ψ + α 2 |ψ| 2 ψ, ψ(·, t) ∈ L 2 (R) ψ(x, 0) = ψ 0 (x) (1) in the limit of large periodic potential, i.e., 0 < 1; α 1 represents the strength of the perturbing potential W and α 2 represents the strength of the nonlinearity term. Equation (1) is the so-called Gross-Pitaevskii equation for Bose-Einstein condensates where 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 α 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 Eq. (1) 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 (1) admits a family of stationary solutions by reducing it to discrete nonlinear Schrödinger equations [11,18,23]. Concerning the reduction of the time-dependent equation to a discrete timedependent 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 [17] 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 (1) to the timedependent discrete nonlinear Schrödinger equations properly works with a precise estimate of the error, and that we do not 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 the new time variable τ = h t and the effective perturbation and nonlinearity strengths then Eq. (1) takes the semiclassical form with h 1.
Vol. 21 (2020) Tight-Binding Approximation for NLS 629 In the tight-binding approximation, solutions to (4) are approximated by solutions to the time-dependent discrete nonlinear Schrödinger equation ihġ n = −β(g n+1 + g n−1 ) + F ξ n g n + ηC 1 |g n | 2 g n , n ∈ Z, 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 (13)). 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 nth one, of the periodic potential V : 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 spectrum, we denote by Λ 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∈Z in the semiclassical regime of h 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 Eq. (5) depends on the relative value of the perturbative parameters F and η with respect to the coupling parameter β. In this paper, we consider two situations.
In the first case, named model 1 corresponding to Hypothesis (3a), we assume that α 1 and α 2 are fixed and independent of . In such a case, we have that β |F | and β C 1 |η| and then the analysis of (5) is basically reduced to the analysis of a system on infinitely many decoupled equations. Indeed, the perturbative terms with strength F and η dominate the coupling term with strength β 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 α 1 and α 2 go to zero when goes to zero. In particular, we assume that 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 β, that is the time interval is exponentially large. We must remark that one could consider, in principle, other limits for α 1 and α 2 when h goes to zero and Theorem 6 is very general and it holds true under different assumptions concerning α 1 and α 2 provided that F = O(h 2 ) and η = O(h 2 ). In fact, Hyp. (3a) and Hyp. (3b) represents, in some sense, two opposite situations concerning the choice of the parameters.
In Sect. 2, we state the assumptions on Eq. (4) and we state our main results in Theorems 1 and 2, they follow from a more technical Theorem 6 we state and prove in Sect. 5. In Sect. 3, we prove a priori estimate of the wavefunction ψ and of its gradient ∇ψ. In Sect. 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 1 , while in the present paper estimates in the norm 2 are necessary and thus most of the material of Sect. 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 Sect. 5, we finally prove the validity of the tightbinding approximation with a precise estimate of the error in Theorem 6, 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 Sect. 5.1, we consider the case where α 1 and α 2 are fixed, i.e., model 1, and in Sect. 5.2, we consider the case where α 1 and α 2 goes to zero as goes to zero in a suitable way, i.e., model 2.

Assumptions
Here, we consider the nonlinear Schrödinger Eq. (1) where the following assumptions hold true.
is a smooth, real-valued, periodic and non-negative function with period a, i.e., and with minimum point 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 − 1 2 a, + 1 2 a .

Vol. 21 (2020)
Tight-Binding Approximation for NLS 631 Concerning the parameters m, , α 1 , α 2 and we make the following assumption Hypothesis 3. We assume the semiclassical limit of large periodic potential, i.e., is a real and positive parameter small enough

1.
Concerning the other parameters we assume that: (a) The parameters m, , α 1 , α 2 are real-valued and independent of ; or (b) The parameters m, are real-valued and independent of while the parameters α 1 , α 2 are real-valued and they go to zero as goes to zero; in particular, we assume that there exist > 0 and a positive constant C such that for any 0 < < then where F and η are defined in (3), and where the parameters β and C 1 depend on (by means of h) and they are defined by (13) and (14).

Remark 2.
In both cases, we have that 0 ≤ F ≤ Ch 2 and |η| ≤ Ch 2 for some positive constant C. In the case (b), in particular, F and η are exponentially small when h goes to zero.
Let H B be the Bloch operator formally defined on L 2 (R, dx) as It is well known that this operator admits self-adjoint extension on the domain H 2 (R), still denoted by H B , and its spectrum is given by bands: 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] hold true for some C > 1; hence, the first gap between E t 1 and E b 2 is not empty in the semiclassical limit.
We assume that 0); that is the wave function ψ is initially prepared on the first band. Through the paper we assume, for argument's sake, that ψ 0 is normalized, i.e., ψ 0 L 2 = 1.

Main Results
Here, we state our main results; they are a consequence of a rather technical Theorem 6 we postpone to Sect. 5. Let g ∈ C(R, 2 (Z)) be the solution to the tight-binding model, that is the discrete nonlinear Schrödinger Eq. (5); let ψ(τ, x) ∈ C(R, H 1 (R)) be the solution to the nonlinear Schrödinger Eq. (4) with initial condition ψ 0 (x) = g n (0)u n (x), let us recall (2) and, finally, let Λ 1 be the first eigenvalue of the single-well operator (6).

Theorem 1. Under the assumption Hypothesis (3a), we have that there exist
> 0 and a positive constant C independent of such that for any 0 < < then

Theorem 2. Under the assumption Hypothesis (3b), we have that there exists
> 0 and two positive constants C and ζ independent of such that for any 0 < < then Remark 3. In [17], 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 Theorem 1 to the H 1 -norm, it is clear that one has to pay a price; indeed, in the proof of Theorem 6, 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 became meaningless.
On the other hand, this argument is not critical in the case of the extension of Theorem 2 to the H 1 -norm because the term u 0 H 1 is controlled by means of the exponentially small term e −ζ/h . In fact, we expect that Theorem 2 still hold true with the H 1 -norm even if we do not dwell here with the detailed proof.

Notation and Some Functional Inequalities
Hereafter, we denote by · L p , p ∈ [+1, +∞], the usual norm of the Banach space L p (R, dx); we denote by · p , p ∈ [+1, +∞], the usual norm of the Banach space p (Z). Hereafter, we omit the dependence on τ in the wavefunctions ψ and in the vectors c when this fact does not cause misunderstanding.
By C, we denote a generic positive constant independent of h [and then, because of (2), by ] 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 ∼ g we mean that Furthermore, we recall some well known results for reader's convenience: -One-dimensional Gagliardo-Nirenberg inequality by §B.5 [16]: -Gronwall's Lemma by Theorem 1.3.1 [15]: let u(τ ) be a non-negative and continuous function such that where α(τ ) and δ(τ ) are non-negative and monotone not decreasing functions, then -Agmon distance let E be a given energy and V (x) be a potential function, let [z] + = z if z ≥ 0 and [z] + = 0 if z < 0; then the Agmon distance d A (x, y) between two points x, y ∈ R d is induced by the Agmon metric [V (x) − E] + dx 2 where dx 2 is the standard metric on L 2 (R d ): where C is the set of piecewise paths γ in R d connecting γ(0) = x and γ(1) = y (see [12] for a resume). In particular, in dimension d = 1 and for energy E = V min we denote by S 0 = xn+1 xn 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.

Preliminary Results
We recall here some results by [6][7][8] concerning the solution to the timedependent nonlinear Schrödinger equation with initial wavefunction ψ 0 . The linear operator H, formally defined as Ann. Henri Poincaré on the Hilbert space L 2 (R, dx), admits a self-adjoint extension on the domain H 2 (R), still denoted by H. In order to discuss the local and global existence of solutions to (4), we apply Theorem 4.2 by [8]: if ψ 0 ∈ H 1 (R) there is a unique solution ψ ∈ C([−T, T ], H 1 (R)) to (4) with initial datum ψ 0 , such that for some T > 0 depending on ψ 0 H 1 .
In fact (see [7]), this solution is global in time for any η ∈ R (because in the case of one-dimensional nonlinear Schrödinger equations the cubic nonlinearity in sub-critical) and (4) enjoys the conservation of the mass ψ(·, τ) L 2 = ψ 0 (·) L 2 and of the energy Here, we prove some useful preliminary a priori estimates.

Theorem 3.
The following a priori estimates hold true for any τ ∈ R: for some positive constant C.
Proof. From the conservation of the norm, we have that From the conservation of the energy, we may obtain a priori estimate of the gradient of the wavefunction. Let where H B ψ 0 , ψ 0 ∼ h since ψ 0 is restricted to the eigenspace associated to the first band. Recalling that V ≥ 0 then we have that implies ∇ψ 0 L 2 ≤ Ch −1/2 . From this fact, using the fact that W is a bounded potential and by the Gagliardo-Nirenberg inequality we have that Hence, E(ψ 0 ) ∼ h since F ≤ Ch 2 and |η| ≤ Ch 2 (see Remark 2). Thus, the conservation of the energy implies the following inequality: since V min = 0 and by the conservation of the norm. Let us set then |Γ| ≤ C and Λ ∼ h −1 as h goes to zero. The previous inequality becomes

Again, the Gagliardo-Nirenberg inequality implies that
hence, Similarly we get and thus, the proof of the Theorem is so completed.

Corollary 1.
We have the following estimates: Proof. They immediately follow from the one-dimensional Gagliardo-Nirenberg inequality (where p = +∞ and δ = 1 2 ) and from the previous result.

Construction of the Discrete Time-Dependent Nonlinear Schrödinger Equation
By the Carlsson's construction [9] resumed and expanded in Appendix A by [11] (see also §3 [23] for a short review of the main results) we may write ψ 1 by means of a linear combination of a suitable orthonormal base {u n } n∈Z of the space Π L 2 (R) , that is where u n ∈ H 1 (R) and c = {c n } n∈Z ∈ 2 (Z) and where we omit, for simplicity's sake, the dependence on τ in the wavefunctions ψ, ψ 1 , ψ ⊥ as well as in the vector c. By inserting (9) and (10) in Eq. (4), it takes the form (where˙= ∂ ∂τ ) where c ∈ 2 and ψ ⊥ are such that for any τ ∈ R By mean of the gauge choice ψ(x, τ ) → e iΛ1τ/h ψ(x, τ ), and then ψ ⊥ (x, τ ) → e iΛ1τ/h ψ ⊥ (x, τ ) and c n (τ ) → e iΛ1τ/h c n (τ ), (11) takes the form (12) where Λ 1 is the energy associated to the ground state of the Schrödinger operator −h 2 ∂ 2 ∂x 2 + V n , with single-well potential V n obtained by filling all the wells of the periodic potential V , but the nth 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 ψ n associated to the ground state Λ 1 is such that ψ m (x) = ψ n (x − x m + x n ) .
We have that where Λ 1 and β are independent of the index n and β is such that for any 0 < ρ < S 0 there is C := C ρ such that 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 Therefore, (12) may be written where we set r n = r 1,n + F r 2,n + F r 3,n + ηr 4,n .
Tight-binding approximation (5) is obtained by putting ψ ⊥ ≡ 0 and by neglecting the coupling term r n in (15). We have the following estimates. Lemma 1.
for some positive constants C and ζ independent of h.
Proof. Such an estimate directly comes from Lemma 1 by [23].

Lemma 2.
For any 0 < ρ < S 0 , there is a positive constant C := C ρ such that Proof. We set then r 2,n = m∈Z W n,m c m . By Example 2.3 §III.2 [13], it follows that where M and M are such that m∈Z |W n,m | ≤ M and n∈Z |W n,m | ≤ M for any n ∈ Z; then M = M because |W n,m | = |W m,n |. Since W is a bounded operator and by Lemma 1.iv [23] it immediately follows that M = Ce (S0−ρ)/h for any 0 < ρ < S 0 and for some positive constant C := C ρ . Hence, Lemma 2 is so proved.

Lemma 3.
Proof. Since r 3,n = u n , W ψ ⊥ L 2 where {u n } n∈Z is an orthonormal base of the space Π L 2 (R) ; then, from the Parseval's identity it follows that For what concerns the vector r 4 , let where j,m, ∈Z means that at least one of three indexes j, and m is different from the index n. Let 0 < ρ < S 0 be fixed; from Lemma 1.iv [23], it follows that for any ρ , ρ > 0 such that ρ + ρ < ρ then there exists a positive constant C > 0, independent of the indexes n and m and of the semiclassical parameter h, such that Vol. 21 (2020)

Tight-Binding Approximation for NLS 639
Now, observing that |c m | ≤ 1 since c 2 ≤ 1, then where we make use of estimate (17) and where ρ , ρ > 0 are such that ρ +ρ < ρ. Hence, for some positive constant C. For what concerns the term B 2,n , we have that Lemma 1 [23]), from which it follows that since |c m | ≤ 1. Hence, From these estimates, it follows that and Lemma 4 is so proved.

Lemma 5.
Vol. 21 (2020) Tight-Binding Approximation for NLS 641 Proof. Indeed, since {u n } n∈Z is an orthonormal base of Π L 2 (R) from the Parseval's identity it follows that from Corollary 1.
In conclusion, we have proved the following Lemma; Lemma 6.

Validity of the Tight-Binding Approximation
First of all we need of the following estimate: and let c and ψ ⊥ be the solutions to (15); then Proof. Indeed, from (15) it immediately follows that from which estimate (18) follows since c 2 ≤ 1 and |c n | ≤ 1, |ξ n | ≤ C because W is a bounded potential and Hereafter, we denote by ω a quantity, whose value may change from line to line, such that |ω| ≤ Ce −(S0+ζ)/h for some ζ > 0 and some C > 0 independent of h. Theorem 4. Proof. Indeed, collecting the results from Lemmata 2, 3 and 6 and from Remark 2, we have that from which the statement immediately follows.
Since ψ ⊥ (x, 0) = Π ⊥ ψ 0 = 0, then the second differential equation of the system (15) may be written as an integral equation of the Duhamel's form Theorem 5. We have the following estimate Proof. Let are such that Lemma 8. The following estimates hold true: and Proof. In order to prove estimates (22) and (23) Tight-Binding Approximation for NLS 643 from Theorem 3 and Corollary 1; hence, (23) follows. In order to prove (22), we make use of an integration by parts: From this fact and since [ by Lemma 7, Theorem 4, and from the draft estimate ψ ⊥ L 2 ≤ 1.
Hence, we have the following integral inequality and then the Gronwall's Lemma implies that Theorem 5 is so proved. Now, we deal with the first differential equation of the system (15) ihċ n = G n (c) + r n , where G n (c) = −β(c n+1 + c n−1 ) + F ξ n c n + ηC 1 |c n | 2 c n .
for some positive constant C independent of h.

Proof of Theorem 1
Here, we assume, according with Hypothesis (3a), that the real-valued parameters α 1 and α 2 are fixed; in such a case we have that Therefore: In particular, for γ = 1 2 then Theorem 1 follows.

Proof of Theorem 2
Here, we assume, according with Hypothesis (3b), that the real-valued parameters α 1 and α 2 are not fixed, but both go to zero when goes to zero; in particular we have that Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.