From the physical problem to the singular perturbation problem.131

In order to make clear the choice of a scale between both confinements, let us de-rive equations (4.1.1)-(4.1.5) from the Schrödinger-Poisson system written in physical variables. This system reads :

i~∂_tΨ=−~²

2m∆Ψ+eV₁(x)Ψ+V₂(z)Ψ+V(x,z)Ψ (4.1.7)

−∆V= e

₀ |Ψ|², (4.1.8)

wherem is the effective mass,ethe elementary charge of the electron and₀ denotes the electric permittivity of the material. Now, each dimensionless quantity used in (4.1.1)-(4.1.5) is the associated physical quantity normalized by a typical scale as follows :

t= t

t, x= x

x, z = z

z, V₁ = V1

V1

, V₂ = V2

V2

, V^ε= V

V ,Ψ^ε= Ψ

√N. (4.1.9) We now define two energy scales that will define the different scales of the three main effects that are the horizontal confinement, the vertical confinement and the self-consistent Poisson potential. The definition of these scales allows us to convey a physical meaning to the dimensionless equations (4.1.1)-(4.1.5) and aposteriori justify its expression. In that view, we define a strong energyE_conf, which will correspond to the energy of the vertical confinement (in z), and a transport energy Etransp, which will be the typical energy of the longitudinal transport (in x) and the one of the self-consistent effects.

We then define the small dimensionless parameter ε as the following quotient : ε:=

Etransp

E_conf 1/2

<<1. (4.1.10)

We now come to the scaling assumptions. We first set to the scale Econf the vertical confinement potential V₂ and the kinetic energy along z :

E_conf :=eV₂ = ~² 2mz².

Then, we set to the scale Etransp the horizontal confinement potential V1, the kinetic energy along the x= (x₁, x₂)directions, the self-consistent potential V^ε, and we choose a time-scale adapted to this energy :

E_transp :=eV₁ =eV = ~²

2mx² = ~ t.

Note that these choices of energy scales imply that ε is also the ratio between the transversal and longitudinal scales :

ε = z x.

Inserting the dimensionless quantities defined by (4.1.9) in the physical equations (4.1.7) leads to :

i∂_tΨ^ε =−∆_xΨ^ε− E_conf

E_transp∂_z²Ψ^ε+V₁(x)Ψ^ε+ E_conf

E_transpV₂(z)Ψ^ε+V^εΨ^ε.

Combined with the definition of the parameter ε given in (4.1.10), this provides us with equation (4.1.1).

The same arguments give

−ε²∆_xV^ε−∂_z²V^ε = e²x²

₀E_transpN ε² |Ψ^ε|².

Therefore, in order to avoid a trivial formal limit of the Poisson equation, we choose to work with high densities and set

N = ₀E_transp e²xz

1 ε.

This additional assumption allows us to derive the dimensionless equation (4.1.4) from the physical equations.

Remark 4.1.2. Consider now the case, studied in [3] and [21] where the authors study the transport of an electron gas that is strongly confined in the z direction.

In that case, we again choose two different energy scales : a strong energy that correspond to the vertical confinement E_conf and a transport energy E_transp that is the typical energy of both transport in the xdirections and self-consistent effects.

Therefore, starting with the adapted equation (4.1.7) : i~∂_tΨ=−~²

2m∆Ψ+V_c(z)Ψ+V(x,z)Ψ (4.1.11)

−∆V= e 0

|Ψ|². (4.1.12)

and defining the same dimensionless quantities (4.1.9) and the same quotient (4.1.10), let us define the following adapted energies.

E_conf :=eV_c= ~²

2mz², E_transp :=eV = ~² 2mx² = ~

t.

Combining with (4.1.11),(4.1.12), (4.1.9) and (4.1.10) still leads to the following dimensionless Schrödinger-Poisson system :

i∂_tψ^ε=−∆_xψ^ε+ 1

ε² −∂_z²+V_c(z)

ψ^ε+V^εψ^ε

−ε²∆_xV^ε−∂_z²V^ε = e²x²

₀E_transpN ε²|ψ^ε|².

Here comes the main difference between what is usually called the "hard wall potential case", where we add Dirichlet boundary condition in thez direction, and the same system with no Dirichlet boundary conditions, usually called "soft wall potential case". Indeed, in [3, 21, 13], the authors choose in the soft wall potential case to work with low densities and set

N = 0Etransp

e²xz .

The starting dimensionless soft wall potential 3D Schrödinger-Poisson confined system studied in these papers therefore reads

i∂_tψ^ε=−∆_xψ^ε+ 1

ε² −∂_z²+V_c(z)

ψ^ε+V^εψ^ε

−ε²∆_xV^ε−∂_z²V^ε =ε|ψ^ε|².

4.1.3 Heuristic approach of the asymptotic model

In this section, we focus on the formal asymptotic model for the Schrödinger-Poisson system with Dirichlet conditions (4.1.1)-(4.1.5). First of all, Equation (4.1.4) allows us to expect the formal limit of the three-dimensional Poisson potentialV^ε to be the solution W(|ψ^ε|²) of the following equation

−∂_z²W(t,·) =|ψ^ε(t,·)|², t ≥0,(x, z)∈Ω (4.1.13) W(t, x,0) =W(t, x,1) = 0, t >0, x∈R². (4.1.14)

In that case, W(|ψ^ε|²)is explicit and reads, for t >0and (x, z)∈Ω: W(|ψ^ε|²)(t, x, z) =

Z 1

0

[z(1−z⁰)−(z−z⁰)1z⁰≤z]|ψ^ε(t, x, z⁰)|²dz⁰. Let us define the following kernel, in order to simplify notations

∀z, z⁰ ∈(0,1), K(z, z⁰) :=z(1−z⁰)−(z−z⁰)1^z0≤z then, W(|ψ^ε|²) reads

W(|ψ^ε|²)(t, x, z) = Z 1

0

K(z, z⁰)|ψ^ε(t, x, z⁰)|²dz⁰. (4.1.15) The first step of our work will therefore be to make precise the functional frame-work for the convergence of (4.1.1)-(4.1.5) towards the following model in which the former Poisson nonlinearity is replaced by its formal asymptoticW(|ψ^ε|²)defined by (4.1.13)-(4.1.14) and state such a convergence result.

Consider the following model in which the Poisson equation (4.1.4)-(4.1.5) is replaced by its formal asymptotic (4.1.13)-(4.1.14). It will be referred to as the in-termediate model in the sequel.

i∂tψ^ε =Hxψ^ε+_ε¹2Hzψ^ε+W(|ψ^ε|²)ψ^ε, t >0,(x, z)∈Ω (4.1.16) ψ^ε(t, x,0) =ψ^ε(t, x,1) = 0, t >0, x∈R². (4.1.17) Here, Hx and Hz respectively denote the longitudinal Hamiltonian defined by :

Hx :=−∆x+V1(x) (4.1.18)

with domain

D(H_x) :={u∈H²(R²), V₁u∈L²(R²)} (4.1.19) and the transversal Hamiltonian defined by :

H_z :=−∂_z²+V₂(z)with homogeneous Dirichlet boundary conditions (4.1.20) and with domain

D(H_z) :=

u∈L²(0,1), ∂_z²u∈L²(0,1), u(0) =u(1) = 0 =H²∩H₀¹(0,1).

(4.1.21) Thanks to Assumption 4.1.1, V₂(z) is a smooth nonnegative function, and thus, the operatorH_zhas a discrete spectrum. In the sequel, the collection of its eigenvalues is denoted by E_p ≥ 0 and their associated eigenfunctions, chosen so as to form a Hilbertian basis ofD(H_z)are denoted byχ_p(z), aspruns in N. They satisfy, for any index p,

H_zχ_p = −∂_z²+V₂(z)

χ_p =E_pχ_p.

The second step consists in studying the asymptotics of this intermediate model as εgoes to zero. The probably most natural approach is to first project the Schrödinger equation (4.1.16) over the orthonormal basis (χ_p)p≥0. Its decomposition over(χ_p)p≥0

reads

ψ^ε(t, x, z) =X

p≥0

ψ_p^ε(t, x)χ_p(z) with ψ^ε_p(t, x) =hψ^ε(t, x,·)χ_pi where we used the notation

hfi:=

Z 1

0

f(z)dz.

Now, inserting this decomposition in the Schrödinger equation (4.1.16) and formally projecting over the (χ_p)p≥0 basis leads to the following infinite system of coupled, nonlinear Schrödinger equations filter out the time oscillations induced by the ^E_ε2^pψ^ε_p term. Therefore, letφ^ε_p be defined as the filtered ψ^ε_p :

φ^ε_p(t, x) = exp(itE_p/ε²)ψ_p^ε(t, x).

The φ^ε_p’s then satisfy the filtered system : i∂_tφ^ε_p =H_xφ^ε_p +X However, according to definition (4.1.15), we have :

W(|X Finally, combining (4.1.23) with (4.1.24) allows us to conclude under nice regularity assumptions, and provided the series at hand in (4.1.24) converge, that theφ^ε_p satisfy the following infinite system

Now that each ∂tφ^ε_p is of order O(1), notice that the infinite system of coupled nonlinear Schrödinger equation satisfied by theφ^ε_p’s (p∈N) is of form :

∂_tu^ε =Au^ε+B(t/ε², u^ε), (4.1.28) where the nonlinearityB happens to have some kind of periodicity in time due to the oscillatory e^{it(Ep+Eq−Er−Es)/ε2} factor. More precisely, as we will see in the following parts, the nonlinearity is almost periodic in time.

It now becomes quite tempting to average in time Equation (4.1.23) or, equiva-lently the toy model (4.1.28). Here, we use a key tool developed in [1], adapted from the well detailed work on the ODE’s in [23] and from Schochet’s work [24]. Assume that the function B(τ, u) entering in (4.1.28) possesses some ergodicity in time, i.e that one can define, in a functional framework we have to make precise later, the limit

B_av(u) = lim

T→+∞

1 T

Z T

0

B(τ, u)dτ.

Then, the toy-system (4.1.28) converges, asεgoes to zero towards the following limit system :

∂_tu=Au+B_av(u). (4.1.29)

For these reasons, and despite the differential system satisfied by the φ^ε_p’s is infinite, we can expect theφ^ε_p’s solving (4.1.23) to converge at least formally towards the solution of the following infinite averaged system :

i∂_tφ_p =H_xφ_p +X X X

(q,r,s)∈Λ_p

α_p,q,r,sφ_rφ_qφ_s, t >0, (x, z)∈Ω (4.1.30) φ_p(0, x) = hψ₀(x,·)χ_pi, x∈R² (4.1.31) where

∀p≥0, Λ_p :={(q, r, s)∈N³, E_p+E_q =E_r+E_s}. (4.1.32) This paper therefore aims at rigorously proving the convergence towards (4.1.30) in an appropriate framework.

Remark 4.1.3. Re-consider now the case where no Dirichlet conditions are imposed, referred to as the "soft wall potential case", in Remark (4.1.2) and where the Poisson equation is given upon the whole spaceR³ by :

−∆_x− 1 ε²∂_z²

V^ε= 1

ε|ψ^ε|², (x, z)∈R³. Its solution V^ε can be written with a convolution as

V^ε:= 1

4πp

|x|²+ε²z² ∗ |ψ^ε|².

The asymptotic of the Poisson potential is thus given by : V^ε(t, x, z)∼ 1

4π|x| ∗ |ψ^ε(t, x, z)|² = 1 4π|x| ∗x

Z

R

|ψ^ε(t,·, z⁰)dz⁰|²

(the reader can refer to [13] for more information). Therefore, in the "soft wall po-tential case", the asymptotic of the Poisson popo-tential does not depend on z. Note that, on the contrary, in our "hard wall potential case" the nonlinearity W(|ψ^ε|²) defined by (4.1.15) obviously depends on z. This dependence in z is crucial as it radically changes the nature of the analysis at hand in our work. Indeed, it induces fast oscillating in time terms that will not be dealt with as easily as previously.

Dans le document Fanny Delebecque préparée à l’unité de recherche UMR 6625 IRMAR (Page 131-137)