The Initial Boundary Value Problem of the Generalized Wigner System

The Initial Boundary Value Problem of the

Generalized Wigner System

Bin Li, and Jie-qiong Shen

Abstract—This paper is concerned over the initial boundary value problem of the generalized Wigner system, which models quantum mechanical (charged) particles transport under the influence of a generalized Hartree potential field. Existence and uniqueness of the mild solution for one and three-dimensional problem are established on weighted-L2 space by semigroup theory. The main difficulties in establishing mild solutions are to derive a priori estimates on the generalized Hartree potential field.

Index Terms—mild solution, semigroup theory, generalized Wigner system, initial boundary value problem, Hartree poten-tial field.

I. INTRODUCTION

I

i¯hψt(t, x) =− ¯

h2

2 ∆ψ(t, x) +V(t, x)ψ(t, x), x∈R n ₍₁₎ coupled the generalized Hartree potential

V(t, x) = λ

|x|α∗xρ(t, x) (2) has been studied by many researchers in the quantum me-chanics, see [1], [2] and the references therein for more de-tails. This model describes the time-evolution of a complex-valued wave function ψ(t, x), under the influence of a generalized Hartree potential (see [3], [4], [19] for a broader introduction). Where _|xλ_|α denotes a given real-valued

inter-action potential kernel, the symbol∗is a convolution,¯his a Planck constant, and λ∈R.

The macroscopic density ρ = ρ(t, x) is now defined by the zeroth order moment in the kinetic variable v, i.e., by the physical observables from both the wave functionψand the Winger functionw, namely,

ρ(t, x) =|ψ|2=

Z

Rn

w(t, x, v)dv. (3) The Wigner transform ofψ(t, x)is

w(t, x, v) = 1

(2π)n

Z

Rn

ψ(t, x−¯h

2y)ψ(t, x+ ¯

h

2y) exp(iv·y)dy (4) where ψ denotes the complex conjugate of ψ. A straight-forward calculation by applying the Wigner transform (4) to the Hartree equation (1) shows that w(t, x, v) satisfies the so-called Wigner-like equation, see e.g. [5], [6],

wt+ (v· ∇x)w−Θ¯h[V]w= 0, (5)

Manuscript received April 13, 2015; revised April 21, 2015.

Bin Li is with the Department of Mathematics, Jincheng College of Sichuan University, Chengdu, 611731, China e-mail: endure2008@163.com. Jie-qiong Shen is with the Department of Mathematics, Jincheng Col-lege of Sichuan University, Chengdu, 611731, China e-mail: jieqiong-shen123@163.com.

where the Wigner functionw=w(t, x, v)is a probabilistic quasi-distribution function of particles at timet≥0, located atx∈Rn with velocity v ∈Rn. The operator Θ¯h[V]w in the equation (5) is a pseudo-differential operator, as in [7], [13], [14], formally defined by

Θ¯h[V]w(t, x, v) =

i

(2π)n_¯h

Z

Rn Z

Rn

δ[V](t, x, η)w(t, x, v′)ei(v−v

′ )η

dv′dη, (6)

δ[V](t, x, η) =V µ

t, x+¯hη 2

¶

−V µ

t, x−¯hη

2

¶ .

It is clear that the equations (5)-(6) coupled eqution (2) contain the Wigner-Poisson equation (e.g., n = 3, α = 1 and n = 1, α = −1). Over the past years, there have been many mathematical studies of mild or classical solution for the WP system, which models the charge transport in a semiconductor device under the Poisson potential. For instance, it has been studied in the whole spaceR3

x×R3v(see [8] and the references therein), in a bounded spatial domain with periodic [9], or absorbing [10], or time-dependent inflow [13], [14], boundary conditions, and on a discrete lattice [11], [12]. Some different aspects regarding the derivation of the model have been reviewed in [20], [21].

The present paper is devoted to investigating the equations (5)-(6) coupled equation (2) and establishing certain mathe-matical results on the existence and uniqueness of the local mild solution, with the initial boundary conditions, which is very difficult: first, the function V(t, x) does not satisfy ∆xV =ρ as in WP problem (see [14] and so on); and the second is to derive priori estimates on the nonlinear operators Θ¯h[V]w. Therefore, the mathematical analysis must be done on the some new methods such as splitting the singular kernel

1

|x|α with ½

α∈(0,1], n= 3;

α∈[−1,−1

2), n= 1.

On the other hand, the natural choice of the functional setting for the study of the WP problem is the Hilbert space

L2(Rn

x×Rnv), see [15], [5]. However, it can be immediately observed that the densityρ(t, x), given by (3), is not well-defined for any w(t, x, v) belonging to this space. In other words, the nonlinear termΘ¯h[V]wis not defined point-wise int on the state space of the Wigner function. Therefore, in Section II we introduce two Hilbert spaces

X=L2([0, l]3×R_v3,(1 +|v|2)2dxdv)

and

X1=L2([0, l]×Rv,(1 +|v|2)dxdv),

see also [17], [9], [14], such that, the existence of the density

With the aforementioned notations, the main results of this paper can be described as the following theorems:

Theorem 1 Let 0 < α ≤ 1, for every w0 ∈ X, the equations (5)-(6) coupled with the equation (2), with initial boundary value conditions

w(t,0, x2, x3, v) =w(t, v, l, x2, x3),

w(t, x1,0, x3, v) =w(t, x1, l, x3, v),

w(t, x1, x2,0, v) =w(t, x1, x2, l, v),

w(t= 0, x, v) =w0(x, v), 1< l <∞,

has a unique mild solutionw∈C([0, tmax), X).

Theorem 2 Let −1 ≤ α < −1₂, for every w0 ∈ X1, the equations (5)-(6) coupled the equation (2), with initial boundary value conditions

w(t,0, v) =w(t, l, v), w(t= 0, x, v) =w0(x, v)

has a unique mild solutionw∈C([0, tmax), X1).

Remark 1 This result extends the previous result by Li, Shen [16] and generalizes the part previous result by Manzini [13].

Remark 2 It is straightforward to extend our results to the high dimensional case (n >3) with0< α≤n−2.

Our paper is structured as follows: In section II we introduce two weighted spaces X and X1 for the Wigner functionwthat allows to define the nonlinear termΘ¯h[V]w in one and three dimensions. In section III, we obtain a local-in-time mild solution on the weightedL2_{spaceX, X}

1 via the the Lumer-Phillips theorem [18] in one and three-dimensional, respectively.

II. THE FUNCTIONAL SETTING&PRELIMINARIES

In this section we shall discuss the functional analytic pre-liminaries for studying the nonlinear Wigner problem (5)-(6) coupled equation (2). Or more accurately, we shall introduce two appropriate state spaces for the Wigner functionwwhich allows to control the nonlinear term Θh¯[V]w, which will be considered as a perturbation of the generator A defined in (23), respectively. This is one of the key ingredients for proving the theorems.

In the ensuing, we show indeed that the nonlinear operator Θ¯h[V]wis respectively (local) bounded in the weighted L2 spaces, in symbols:

X :=L2(Ix×R3v,(1 +|v|2)2dxdv), Ix= [0, l]3x, (7)

X1:=L2(E×Rv,(1 +|v|2)dxdv), E= [0, l]x, (8) endowed with the following scalar products

hf, gi_X :=

Z

I

Z

R3

v

f(x, v)·g(x, v)(1 +v2)2_{dvdx, (9)}

hf, gi_X1 :=

Z

I

Z

Rv

f(x, v)·g(x, v)(1 +v2)dvdx, (10)

for f, g ∈ X, X1. In our calculations, we shall use the following equivalent norms:

||f||2_X :=||f||2_L2+

3

X

i=1

||v2_if||2_L2, (11)

||f||2X1 :=||f||

2

L2+||vf||2_L2. (12)

The following proposition motivates our choice of the space

X, X1 for the analysis.

Lemma 1 For n= 3, let w∈ X and ρ(t, x) defined in (3), for allx∈I, thenρbelongs to L2(I)and satisfies

||ρ||L2_(I)≤C||w||_X, (13)

ρalso belongs toL1_{(I)and satisfies}

||ρ||L1

(I)≤C||w||X. (14) Moreover, for every p ∈ [1,2], ρ belongs to Lp_{(I) and} satisfies

||ρ||Lp_(I)≤C||w||_X. (15)

Proof: The first assertion follows directly by using Cauchy-Schwartz inequality inv-integral, see also [13], [14]. On the other hand, by H¨older inequality, we have

kρkL1_(I)≤

Z

I

¯ ¯ ¯ ¯ ¯ Z

R3

v

w(t, x, v)dv ¯ ¯ ¯ ¯ ¯dx≤ ·Z

I 12dx

¸1 2

 Z l

0

¯ ¯ ¯ ¯ ¯ Z

R3

v

w(t, x, v)dv ¯ ¯ ¯ ¯ ¯

2

dx  

1 2

≤CkwkX.

Using the interpolation inequality, we get

kρkLp_(I) ≤ kρkθ_L2_(I)kρk1−θ

L1_(I)≤Ckwk_X. Lemma 2 Forn= 1, let w ∈X1 andρ(t, x)defined in (3), for allx∈E, thenρbelongs toL2_{(I)and satisfies}

||ρ||L2

(E)≤C||w||X1, (16)

ρalso belongs toL1(E)and satisfies

||ρ||L1_(E)≤C||w||_X

1. (17)

Moreover, for every p ∈ [1,2], ρ belongs to Lp_{(E) and} satisfies

||ρ||Lp_(E)≤C||w||_X1. (18) Proof:See the proof of the Lemma 1.

Next, we consider the Lipschitz properties of the pseudo-differential operator Θ¯h[V]w defined by (6). But by the definition of it, thewhave to be0-extended to R3_x andRx. We will show indeed that this operator is well defined from the spaceX, X1 to themselves. Moreover, we can state the following results:

Lemma 3 Forn= 3, let 0< α≤1, for all w∈X, the operatorΘh¯[V]wmapsX into itself and there existsC >0 such that

||Θ¯h[V]w||X≤C||w||2X. (19)

Proof: Indeed, the operator Θh¯[V]w can be rewritten in a more compact form as,

Fv→η(Θ¯h[V]w)(x, η) =

i

¯

hδV(x, η)F

−1

η→vw(x, η), (20) where the symbol∗v is the partial convolution with respect to the variable v, Fv→η is the Fourier transformation with respect to the variablev andF−1

η→v its inverse:

Fv→η[f(x,·)](η) =

Z

Rn

f(x, v)eiv·ηdv,

Fη−→1v[g(x,·)](v) = 1 (2π)n

Z

Rn

for suitable functionsf andg. Then one has

kΘ¯h[V]wkL2 ≤CkδV(x, η)F−1

η→vwkL2≤

CkVkL∞kF−1

η→vwkL2 ≤CkVk

L∞kwk L2.

Let k(·) = _|·|1α, k1 = k(·)||·|≤1 and k2 = k(·)||·|>1, so,

k(·) = k1+k2 with k1 ∈ Lp(R3) for all p ∈ [1,_α3) and

k2∈Lq(R3)for allq∈(3_α,+∞]. On the other hand, since

V = 1

|·|α∗ρ, using Ho¨lder’s inequality we have

kk1∗ρk_L∞_(B)≤Ckk1k_L2(B)kρk_L2(B)≤

CkρkL2_(B)≤Ckρk

L2_(I),

whereBis the three dimensional unit ball. Likewise, outside

B we get

kk2∗ρkL∞_(R3_\B)≤Ckk₂k_L∞_(R3_\B)kρk

L1_(I\B)≤

CkρkL1_(I\B)≤Ckρk_L1_(I).

By Lemma 1, we can get

kΘ¯h[V]wkL2 ≤CkVk_L∞kwk

L2 ≤Ckwk2

X.

On the other hand, by [22], we have

v_i2Θ¯h[V]w= 1 4Θ¯h

£

∂_i2V¤w+ Θ¯h[V]v2iw+ Ω¯h[∂iV]w

with the pseudo-differential operator Ω¯h[ϕ]w=

i

(2π)n_¯h

Z

Rn Z

Rn

κ[ϕ]w(t, x, v′)ei(v−v′)ηdv′dη, κ[ϕ] =ϕ

µ

t, x+¯hη 2

¶

+ϕ µ

t, x−¯hη

2

¶ .

In the sequel we use the abreviation∂i=∂xi, and get

kv_i2Θ¯h[V]wkL2≤ 1

4kΘ¯h

£

∂2_iV¤wkL2+kΩ_¯h[∂_iV]wk_L2+kΘ_¯h[V]v2

iwkL2.

The first two terms can be estimated as follows:

kΘ¯h

£

∂2_iV¤wkL2≤Ckδ(∂2

iV)Fv→ηwkL2≤

Ck∂_i2VkL2_(R3₎kF_v→ηwk

L2_(I;L∞_(R3

η))≤ Ck|x|−2−α∗xρkL2_(R3₎k(1 +|v|2)wk_L2≤

C(k 1

|x|2+α ∗xρkL2_(B)+k 1

|x|2+α ∗xρkL2_(R3_\B))kwk_X≤

C(kρkL2

(I)+kρkL1

(I))kwkX≤Ckwk2X by applying H¨older’s inequality, _2+α3 < 2 with 0 < α ≤

1 and the Sobolev imbedding Fv→ηw ∈ W2,2(R3η) ֒→

L∞(R3_η).

kΩ¯h[∂iV]wkL2≤Ckδ(∂_iV)∂_η

iFv→ηwkL2≤ Ck∂iVkL4

(R3

)k∂ηiFv→ηwkL2(I;L4(R3η))≤ Ck|x|−1−α_∗

xρkL4_(R3₎k(1 +|v_i|2)wk

L2≤

C(k 1

|x|1+α ∗xρkL4(B)+k 1

|x|1+α ∗xρkL4(R3\B))kwkX≤

Ck 1

|x|1+αkL75_(B)kρk_L 28

15_(B)kwkX+

Ck 1

|x|1+αkL4_(R3_\B)kρk

L1_(I\B)kwk_X≤

C(kρk

L

28

15_(B)+kρkL

1_(I\B))kwk_X≤Ckwk2

X

by the Sobolev imbedding ∂ηiFv→ηw ∈ W

1,2_(R3 η) ֒→

L4_(R3

η), 1+α3 > 7 5 and

3

1+α < 4 with 0 < α ≤ 1, and Lemma 1. We also get

kΘ¯h[V]vi2wkL2 ≤CkV ∂2

ηiFv→ηwkL

2 ≤

CkVkL∞ (R3

)k∂2ηiFv→ηwkL

2 ≤

C(k 1

|x|α ∗xρkL∞(B)+k 1

|x|α∗xρkL∞(R3\B))kwkX ≤

Ck 1

|x|αkL2(B)kρkL2(B)kwkX+

Ck 1

|x|αkL∞(R3\B)kρkL1(I\B)kwkX ≤

C(kρkL2_(B)+kρk

L1_(I\B))kwk_X ≤Ckwk2

X

by applying Ho¨lder’s inequality and Lemma 1. This con-cludes the proof of result.

Lemma 4 Forn= 1, let−1≤α <−1

2, for allw∈X1, the operator Θh¯[V]w mapsX1 into itself and there exists

C >0 such that

||Θ¯h[V]w||X1 ≤C||w||

2 X1.

Proof:Indeed, the solution of Poisson equationV andVe, its extension with value zero outside [0,l], are essentially bounded. Moreover, we can get

kVekL∞_(R)≤Ckρk L1_(E),

where E = [0, l], see also [14]. Moreover, using Lemmas 2-3, we have

kΘ¯h[V]wkL2 ≤

CkVkL∞kwk

L2 ≤Ckρk

L1_(E)kwk

L2≤Ckwk2

L2.

On the other hand, by [23], we get

vΘ¯h[V]w= Θh¯[V]vw−Ω¯h[Vx]w

with the pseudo-differential operator Ω¯h[Vx]w=

i

2π¯h Z

R

Z

R

κ[Vx]w(t, x, v′)ei(v−v ′

)η

dv′dη, κ[Vx] =Vx

µ

t, x+¯hη 2

¶

+Vx

µ

t, x−¯hη

2

¶ .

Hence,

kvΘ¯h[V]wkL2≤ kΩ_¯h[V_x]wk_L2+kΘ_h¯[V]vwk_L2.

Setting r=−αthen 1₂ < r≤1, and the first term can be estimated as follows:

kΩ¯h[Vx]wkL2 ≤Ck 1

|x|1−r ∗ρkL∞(E)kwkL2≤

C(kρkL2_(B)+kρk_L1_(E\B))kwk_L2 ≤Ckwk2

L2.

For the last term, we have

kΘ¯h[V]vwkL2≤CkVk

L∞_(E)kvwk L2≤

CkρkL1_(E)kvwk_L2 ≤Ckwk

L2kvwk

L2.

The combination of these estimates yields the result.

Lemma 5 Forn= 3, let 0< α≤1, for all w∈X, the operatorΘ¯h[V]w is of classC∞inX, and satisfies

kΘ¯h[V1]w1−Θ¯h[V2]w2kX≤

Proof: For allwi∈X, i= 1,2, setting Π = Θ¯h[V1]w1−Θ¯h[V2]w2, Π1= Θ¯h[V1]w1−Θ¯h[V1]w2, Π2= Θ¯h[V1]w2−Θ¯h[V2]w2,

we have

kΠkX ≤ kΠ1kX+kΠ2kX

with

kΠ1kX =kΘ¯h[V1] (w1−w2)kX ≤

kΘ¯h[V1] (w1−w2)kL2+

3

X

i=1

kv2_iΘ¯h[V1] (w1−w2)kL2 ≤

CkδV[w1]Fv→η(w1−w2)kL2+

C

3

X

i=1

kδ∂_i2V[w1])Fv→η(w1−w2)kL2+

C

3

X

i=1

kδ(∂iV[w1])∂ηiFv→η(w1−w2)kL2+

C

3

X

i=1

kδV[w1]∂η2iFv→η(w1−w2)kL

2 ≤

CkV[w1]kL∞kw

1−w2kL2+

C

3

X

i=1

k∂2_iV[w1]kL2kF[w₁−w₂]k

L2_(I;L∞_(R3

η))+

C

3

X

i=1

k∂iV[w1]kL4k∂_η

iF[w1−w2]kL2(I;L4(R3η))+

C

3

X

i=1

kV[w1]kL∞k∂2

ηiF[w1−w2]kL

2 ≤

Ckw1kL2kw₁−w₂k_X; kΠ2kX =kΘ¯h[V1−V2]w2kX ≤

kΘ¯h[V1−V2]w2kL2+

3

X

i=1

kv_i2Θ¯h[V1−V2]w2kL2 ≤

CkδV[w1−w2]Fv→ηw2kL2+

C

3

X

i=1

kδ∂2_iV[w1−w2])Fv→ηw2kL2+

C

3

X

i=1

kδ(∂iV[w1−w2])∂ηiFv→ηw2kL2+

C

3

X

i=1

kδV[w1−w2]∂η2iFv→ηw2kL

2 ≤

CkV[w1−w2]kL∞kw₂k L2+

C

3

X

i=1

k∂_i2V[w1−w2]kL2kFw₂k

L2

(I;L∞ (R3

η))+

C

3

X

i=1

k∂iV[w1−w2]kL4k∂_η

iFw2kL2(I;L4(R3η))+

C

3

X

i=1

kV[w1−w2]kL∞k∂2

ηiFv→ηw2kL

2 ≤

Ckw2kXkw1−w2kL2,

and the assertion is proved.

Lemma 6 Forn= 1, let−1≤α <−1

2, for allw∈X1, the operatorΘ¯h[V]wis of classC∞ inX1, and satisfies

kΘ¯h[V1]w1−Θ¯h[V2]w2kX1 ≤

C(kw1kX1+kw2kX1)kw1−w2kX1.

Proof: For allwi∈X, i= 1,2, by Lemmas 4-5, we have

kΘh¯[V1]w1−Θ¯h[V2]w2kX1 ≤ kΘ¯h[V1]w1−Θ¯h[V1]w2kX1+ kΘh¯[V1]w2−Θ¯h[V2]w2kX1

with

kΘ¯h[V1] (w1−w2)kX1 ≤ kΘ¯h[V1] (w1−w2)kL2+kvΘ_¯h[V₁] (w₁−w₂)k

L2 ≤

CkδV[w1]Fv→η(w1−w2)kL2+

CkΩ¯h[Vx[w1]] (w1−w2)kL2+

CkΘ¯h[V[w1]]v(w1−w2)kL2 ≤

CkV[w1]kL∞kw₁−w₂k L2+

CkVx[w1]kL∞kw

1−w2kL2+

CkV[w1]kL∞kv(w₁−w₂)k L2 ≤

Ckw1kL2kw₁−w₂k_X 1; kΘ¯h[V1−V2]w2kX1 ≤ kΘh¯[V1−V2]w2kL2+kvΘ_¯h[V₁−V₂]w₂k_L2 ≤

CkδV[w1−w2]Fv→ηw2kL2+

CkΩh¯[Vx[w1−w2]]w2kL2+

CkΘ¯h[V[w1−w2]]vw2kL2 ≤

CkV[w1−w2]kL∞kw₂k L2+

CkVx[w1−w2]kL∞kw₂k L2+

CkV w1−w2]kL∞kvw₂k L2 ≤

Ckw2kX1kw1−w2kL2,

and the assertion is proved.

III. PROOF OF THE MAIN RESULTS

In this section, we will prove the main results of this paper by semigroup theory. Let we rewrite the Wigner equation as

wt=Aw+ Θ¯h[V]w, t >0, (21)

w(t= 0) =w0, (22) where linear operatorA:D(A)→X or A:D1(A)→X1 by

Af=−v· ∇xw (23)

and theirs domain

D(A) ={w∈X|v· ∇xw∈X,

w(0, x2, x3) =w(l, x2, x3),

w(x1,0, x3) =w(x1, l, x3),

w(x1, x2,0) =w(x1, x2, l), l >1};

D1(A) ={w∈X1|v· ∇xw∈X1,

w(t,0, v) =w(t, l, v), l >1}.

considerΘ¯h[V]was a bounded perturbation of the generator

A. On the other hand, since Θ¯h[V]w is locally Lipschitz continuous (see Lemmas 3, 5 for detail), Theorem 6.1.4 of [18] shows that the problem (21)-(22) coupled equation (2) has a unique mild solution for every w0∈X on some time interval[0, tmax), wheretmaxdenotes the maximal existence time of the mild solution. Moreover, iftmax<∞, then

lim t→tmax

kwkX =∞.

This concludes the proof of result.

Proof of Theorem 2: In fact, we can get the assertion by repeating the analogous strategies in proof of Theorem 1.

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