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

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.

In the physical literature a standard way to study equation (1) consists in reducing it to a discrete Schrödinger equation taking into account only nearest neighbor interactions, the so called tight-binding model [1]. 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 [7,13,17]. 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 [2] proved the validity of the reduction to discrete nonlinear Schrödinger equations for large times when V is multiple-well trapped potential; while [14] were able to obtain a similar result for a periodic potential V by assuming a specific technical condition on the initial wavefunction.
In this paper we are able to show that the reduction of (1) 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 Date: February 26, 2019. This paper is partially supported by GNFM-INdAM. 1 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, let us say the first one for argument's sake, of the Bloch operator.
By introducing the new semiclassical parameter then the above equation (1) takes the semiclassical form with h ≪ 1.
In the tight-binding approximation solutions to (3) 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 is an exponentially small positive constant in the semiclassical limit h ≪ 1. 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 [6,7,17]), {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 of the Schrödinger equation with a single well potential obtained by filling all the wells, but the n-th one, of the periodic potential V .
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 [1,12]. In fact, 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 [10].
The analysis of the discrete nonlinear Schrödinger equations (4) 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 ǫ. In such a case we have that β ≪ |F | and β ≪ C 1 |η| and then the analysis of (4) 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 [16,17]. 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. One may extend the validity of such an approximation to larger intervals of time by assuming some further conditions of the initial wavefunction or on the potential V as done by [2,14].
In the second case, named model 2 corresponding to Hypothesis 3b), we assume that both α 1 and α 2 go to zero when ǫ 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.
In §2 we state the assumptions on equation (3). In §3 we prove a priori estimate of the wavefunction ψ and of its gradient ∇ψ. In §4 we formally construct the discrete nonlinear Schrödinger equations; in this Section we make use of some ideas already developed by [7,17] and we refer to these papers as much as possible. We must underline that in [7,17] 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 §4 we finally prove the validity of the tight-binding approximation with a precise estimate of the error, the method used is based on an idea already used by [15] 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. model 1, and in §5.2 we consider the case where α 1 and α 2 goes to zero as ǫ goes to zero in a suitable way, i.e. model 2.
2. Description of the model 2.1. Assumptions. Here, we consider the nonlinear Schrödinger equation (3) where F and η are defined in (2) and 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.
and with minimum point x 0 ∈ − 1 2 a, + 1 2 a such that 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 . Hypothesis 2. The perturbation W (x) is a smooth real-valued function. We assume that: W ∈ L ∞ (R) and there are C > 0 and s ≥ 1 Concerning the parameters m, , α 1 , α 2 and ǫ we assume that Hypothesis 3. We assume the limit of large periodic potential, i.e. ǫ is a real and positive parameter small enough ǫ ≪ 1 .

Concerning the other parameters we assume that:
a) The parameters m, , α 1 , α 2 are real-valued and independent of ǫ; or b) The parameters m, are real-valued and independent of ǫ while the parameters α 1 , α 2 are real-valued and they go to zero as ǫ goes to zero, in particular we assume that where the parameters β and C 1 depend on ǫ (by means of h) and they are defined by (11) and (12).

Remark 2.
In both cases we have that |F | ≤ Ch 2 and |η| ≤ Ch 2 . 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, still denoted by H B , and its spectrum is given by bands: are named gaps; a gap may be empty, that is E b ℓ+1 = E t ℓ , 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. In particular, when h is small enough then the following asymptotic behaviors [18,19] hold true for some C > 0; hence, the first gap between E t 1 and E b 2 is not empty in the semiclassical limit.
Let Π the projection operator associated to the first band We assume that Hypothesis 4. Π ⊥ ψ 0 = 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.

2.2.
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 · ℓ p , p ∈ [+1, +∞], the usual norm of the Banach space ℓ p (Z). In particular; if ψ = ψ(x, τ ) is a wavefunction then by ψ L p we mean Hereafter, we omit the dependence on τ in the wavefunctions ψ and in the vector 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 ∼ g we mean that Let d A (x, y) be the Agmon distance between two points x, y ∈ R (for a definition of the Agmon distance see [8]) and let S 0 := d A (x n , x n+1 ), n ∈ Z and x n := x 0 +na, be the Agmon distance between the bottom points x n and x n+1 of two adjacent wells of the periodic potential V ; by periodicity S 0 does not depend on the index n.

Preliminary results
We recall here some results by [3,4,5] concerning the solution to the timedependent nonlinear Schrödinger equation with initial wavefunction ψ 0 . The linear operator H, formally defined as , admits a self-adjoint extension, still denoted by H. In order to discuss the local and global existence of solutions to (3) we apply Theorem 4.2 by [5]: for some T > 0 depending on ψ 0 H 1 .
In fact (see [4]), this solution is global in time for any η ∈ R (because 1 < 2/d, where d = 1 is the spatial dimension) and (3) enjoys the conservation of the mass ψ(·, τ ) L 2 = ψ 0 (·) L 2 and of the energy 3.1. A priori estimates. We have that 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 . From this fact, by Remark 2, using the fact that W is a bounded potential and by the Gagliardo-Nirenberg inequality we have that 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 as h goes to zero, and Λ ∼ h −1 since |F | ≤ Ch 2 and E(ψ 0 ) ∼ h. The previous inequality becomes Again, the Gagliardo-Nirenberg inequality implies that and thus we get Similarly we get ∇ψ ⊥ L 2 ≤ Ch −1/2 , 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 and from the previous result.

Construction of the discrete time-dependent nonlinear Schrödinger equation
By the Carlsson's construction [6] resumed and expanded by [7] (see also §3 [17] 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 ∈ ℓ 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.
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 [17] it follows that Therefore, (10) 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 (4) is obtained by putting ψ ⊥ ≡ 0 and by neglecting the coupling term r n in (13).
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 [17].
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 [9] it follows that where M ′ and M ′′ are such that m∈Z |W n,m | ≤ M ′ and m∈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 [17] 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. Let s be defined by Hypothesis 2; then, the following estimate holds true: Proof. Since n∈N |a n | 2 ≤ n∈N |a n | 2 then Now, we are going to estimate the term n∈Z |W | |u n | L 2 ; to this end let us set A = [−R, +R] and B = R \ A, where R = 1 2 + N a and N > 0 is a large enough positive number which will be defined later. Then since u n L 2 = 1 and Lemma 1.iv [17]. For what concerns the other term let χ A be the characteristic function on the set A, we have that since W 2 ∈ L 2 because of Hypothesis 2 and because for any ρ ′ > 0 and ρ ′′ > 0 there is a positive constant C such that as one can see by means of the same arguments used in the proof of Lemma 2 [7], and where the Agmon distance between x n / ∈ A and A is given by Then, we have obtained that If we set N = ⌈h −α ⌉ + 1 (where ⌈x⌉ is the integer part of a real number x) for α > 0 such that −α = −1 − α(1 − 2s), that is α = 1/2s, then we finally gets By collecting all these results then Lemma 3 follows.

Remark 3.
We remark that if W has compact support then Indeed, it immediately follows by means of a simple fact: where N is a fixed positive and integer number such that the set A = − N + 1 2 a, + N + 1 2 a contains the support set of W .
For what concerns the vector r 4 let r 4,n = u n , |ψ| 2 ψ − C 1 |c n | 2 c n = A n + B n where we set where ⋆ j,m,ℓ∈Z means that at least one of three indexes j, ℓ and m is different from the index n. Observing that |c m | ≤ 1 since c ℓ 2 ≤ 1, then where we make use of the following property (see Lemma 1.iv by [17]): for any ρ ′ , ρ ′′ ∈ (0, S 0 ) there is a positive constant C > 0 independent of the indexes n and m such that For what concerns the term B 2,n we have that Lemma 1 [17]), from which it follows that Finally, since |c m | ≤ 1. Hence, From these estimates it follows that and Lemma 4 is so proved.

Now we deal with the vector A with elements
Proof. Indeed, from Theorem 1 and Corollary 1.
In conclusion we have proved the following Lemma;

Validity of the tight-binding approximation
First of all we need of the following estimate: and let c and ψ ⊥ be the solutions to (13); then Proof. Indeed, from (13) it immediately follows that from which the estimate (15) follows since c ℓ 2 ≤ 1 and |c n | ≤ 1.
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.
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 (13) may be written as an integral equation of the Duhamel's form Theorem 3. We have the following estimate Proof. Let then the previous equation (17) becomes is such that Lemma 8. The following estimates hold true: and Proof. In order to prove the estimates (19) and (20) we remark that e −i(HB −Λ1)(τ −s)/h is an unitary operator; hence, from Theorem 1 and Corollary 1; hence, (20) follows. In order to prove (19) we make use of an integration by parts:

From this fact and since [H
by Lemma 7 and Theorem 2.
We compare it with the equation which represents the tight-binding approximation of (3), up to a phase factor e −iΛ1τ /h depending on time. We must underline that we have the following a priori estimate c ℓ 2 ≤ 1 and the conservation of the norm of g g ℓ 2 = g(0) ℓ 2 = c(0) ℓ 2 = 1 ; indeed, an immediate calculus gives that Hence, Lemma 9. G is a Lipschitz function such that Proof. Indeed since |c n | 2 − |g n | 2 = c n [c n −ḡ n ] +ḡ n [c n − g n ] and (12).
By Theorems 2 and 3, it turns out that the vector r is norm bounded by for some positive constant d independent of h and where Then, we get the integral inequality By the Gronwall's Lemma we finally get the estimate Therefore, we have proved that for some positive constants d and C independent of h.

Model 2.
Here we assume, according with Hypothesis 3b), that the realvalued parameters α 1 and α 2 are not fixed, but both go to zero when ǫ goes to zero; in particular we have that In such a case we have that The estimate (26) makes sense for times of order τ ∈ [0, β −1 h]. In such an interval we have that ψ ⊥ L 2 ≤ Ce −(S0−ρ)/h and c − g ℓ 2 ≤ Ce −ζ/h for some ζ > 0.
In conclusion we have that