Semiclassical analysis for a Schr\"odinger operator with a U(2) artificial gauge: the periodic case

0.19 MB·

by  Abderemane Morame

Semiclassical analysis for a Schr¨odinger operator
with a U(2) artificial gauge: the periodic case
4
1
0
2 A. Morame1 and F. Truc2

n
u June 16, 2014
J

4
2

]
h 1 Universit´e de Nantes, Facult´e des Sciences, Dpt. Math´ematiques,
p
UMR 6629 du CNRS, B.P. 99208, 44322 Nantes Cedex 3, (FRANCE),
-
h
E.Mail: [email protected]
t
a 2 Universit´e de Grenoble I, Institut Fourier,
m
UMR 5582 CNRS-UJF, B.P. 74,
[
38402 St Martin d’H`eres Cedex, (France),

2
E.Mail: [email protected]
v
7
4 Abstract
4
6 WeconsideraSchro¨dingeroperatorwithaHermitian2x2matrix-valued
.
1 potential which is lattice periodic and can be diagonalized smoothly on
0 the whole Rn. In the case of potential taking its minimum only on the
4
lattice, weprovethatthewell-known semiclassical asymptotic offirstband
1
: spectrum for a scalar potential remains valid for our model.
v
i
X
Keywords : semiclassical asymptotic, spectrum, eigenvalues, Schrodinger,
r
a periodic potential, BKW method, width of the first band, magnetic field.
AMS MSC 2000 : 35J10, 35P15, 47A10, 81Q10, 81Q20.
Contents
1 Introduction 2
2 Preliminary: the artificial gauge model 3
3 Proof of Theorem 2.1 5
4 Asymptotic of the first band 7
5 B.K.W. method for the Dirichlet ground state 11
1
6 Sharp asymptotic for the width of the first band 14
1 Introduction
Schro¨dinger operators with periodic matrix-valued potentials appear in many
models in physics. Such models have been used recently to describe the motion
of an atom in optical fields ([Co], [Co-Da], [Da-al]), see also [Ca-Yu]. The aim
of this paper is to investigate their spectral properties using semiclassical anal-
ysis. We focus on the first spectral band and assume that the potential has a
non degenerate minimum. The Schro¨dinger operators with a non-Abelian gauge
potential are Hamiltonian operators on L2(Rn;Cm) of the following form :
Hh = h2 n (D I −A )2 +V +hQ+h2R = Ph(x,hD) . (1.1)
k=1 xk k
P
The classical symbol of Ph(x,hD), Ph(x,ξ), for (x,ξ) ∈ Rn ×Rn, is given by
n
Ph(x,ξ) = (ξ I −hA (x))2 +ih2∂ A (x) +V(x)+hQ(x)+h2R(x) ,
k k xk k
Xk=1(cid:8) (cid:9)
(1.2)
I is the identity m×m matrix, V, Q, R and the A are hermitian m×m matrix
k
with smooth coefficients and Γ periodic:
A = (a (x)) , V = (v (x)) ,
k k,ij 1 i,j m ij 1 i,j m
≤ ≤ ≤ ≤
Q = (q (x)) , R = (r (x)) , 
ij 1 i,j m ij 1 i,j m
a , v , q ,≤r ≤∈ C (Rn;C), ≤ ≤ 
k,ij ij ij ij ∞  (1.3)
a = a , v = v , q = q , r = r 
k,ji k,ij ji ij ji ij ji ij 
a (x−γ) = a (x), v (x−γ) = v (x),
k,ij k,ij ij ij
q (x−γ) = q (x) and r (x−γ) = r (x) ∀γ ∈ Γ ; 
ij ij ij ij 




n
Γ is a lattice of Rn, Γ = { m β ; m ∈ Z},
k k k
X
k=1
β ,β ,...,β ∈ Rn form a basis, det(β ,β ,...,β ) 6= 0.
1 2 n 1 2 n
We use the notation D = (D ,...,D ) where D = −i∂ , k = 1...n,
x1 xn xk xk
so D2 = −∆ is the Laplacian operator on L2(Rn).
The dual basis {β⋆,...,β⋆} of the reciprocal lattice Γ⋆, is the basis of Rn
1 n
defined by the relations
n
β⋆.β = 2πδ : Γ⋆ = { m β⋆; m ∈ Z}.
j k jk k k k
X
k=1
The fundamental cell, the Wigner-Seitz cell,
n
1 1
Wn = { x β ; x ∈]− , [},
k k k
2 2
X
k=1
2
will be identified with the n-dimensional torus Tn = Rn/Γ and the dual cell, the
Brillouin zone, is defined by
n
1 1
Bn = { θ β⋆;θ ∈]− ,− [} .
k k k 2 2
X
k=1
We will identify L2(Tn;Cm) with Γ periodic functions of L2 (Rn;Cm) provided
loc
with the norm of L2(Wn;Cm). In the same way the Sobolev space Wk(Tn;Cm),
with k ∈ N, may be identified with Γ periodic functions of Wk (Rn;Cm) provided
loc
with the norm of Wk(Wn;Cm).
By Floquet theory, (see [Ea] or [Re-Si] ), we have
Hh = ⊕Hh,θdθ ,
ZBn
with Hh,θ the partial differential operator P (x,h(D −θ)) on L2(Tn;Cm).
h
The ellipticity of P (x,h(D −θ)) implies that the spectrum of Hh,θ is discrete
h
sp(Hh,θ) = {λh,θ; j ∈ N⋆}, λh,θ ≤ λh,θ ≤ ... ≤ λh,θ ≤ λh,θ ≤ ... (1.4)
j 1 2 j j+1
each λh,θ is an eigenvalue of finite multiplicity and each eigenvalue is repeated
j
according to its multiplicity.
1
(When m = 1 and V = Q = R = A = 0, ( eiω.x) is the Hilbert basis
k |Tn| ω∈Γ⋆
of L2(Tn) which is composed of eigenfunctionps of h2(D−θ)2).
The Floquet theory guarantees that
∞
sp(Hh) = sp(Hh,θ) = bh, (1.5)
j
θ[Bn j[=1
∈
where bh denotes the j-th band bh = {λh,θ,θ ∈ Bn}.
j j j
In the sequel h will be a non negative small constant, h will be in ]0,h [, and
0 0
any non negative constant which doesn’t depend on h will invariably be denoted
by C.
2 Preliminary: the artificial gauge model
We will be interested in the model of artificial gauge considered in [Co], [Co-Da]
and [Da-al]
m = 2, V = vI +W, A = Q = R = 0, ∀k,
k (2.1)
W = w.σ, with w = (w ,w ,w ), v and the w are in C (Rn;R), (cid:27)
1 2 3 j ∞
we denote σ = (σ ,σ ,σ ), where the σ are the Pauli matrices.
1 2 3 j
3
Let us remark that
V = vI +W, W = w.σ, W2 = |w|2I . (2.2)
In the sequel we will assume that
|w| > 0
(2.3)
v(x)−|w(x)| has a unique non degenerate minimum on Tn. (cid:27)
Due to the invariance of the Laplacian by translation and by the action of
O(n), we can assume, up to a composition by a translation of the potentials, that
v(γ)−|w(γ)| < v(x)−|w(x)|, ∀x ∈ Rn \Γ and ∀γ ∈ Γ,
n  (2.4)
v(x)−|w(x)| = E + τ2x2 +O(|x|3), as |x| → 0, 
0 k k 
X
k=1


(τ > 0, ∀k).
k
There exists U ∈ U(2), ( a unitary 2×2 matrix), such that
v −|w| 0
U⋆VU = V = . (2.5)
(cid:20) 0 v +|w| (cid:21)
e
As |w| never vanishes, U = U(x) can be chosen smooth and Γ periodic:
u u
U = 11 12 ∈ C (Tn;U(2));
∞
(cid:18) u u (cid:19)
21 22
1
for example u = (w −|w|+eiθ(w −iw )),
11 3 1 2
2 |w|(|w|−Re((w +iw )e iθ))
1 2 −
p 1
u = (w +iw −eiθ(w +|w|)),
21 1 2 3
2 |w|(|w|−Re(w +iw )e iθ))
1 2 −
p w2 +w2 π
u = u , u = −u and θ = χ( 2 3) ,
12 21 22 11 |w|2 2
where χ(t) is a smooth function on the real line, 0 ≤ χ(t) ≤ 1,
χ(t) = 1 when |t| ≤ 1/4 and χ(t) = 0 when |t| ≥ 1/2. So
U = (α,β,ρ).σ+iδσ , with (α,β,ρ,δ) ∈ C (Tn;S3); (2.6)
0 ∞
σ is the 2×2 identity matrix and S3 is the unit sphere of R4.
0
When w + iw 6= 0 or when w < 0, one can choose U such that δ = 0 by
1 2 3
1 w w
1 2
taking (α,β,ρ) = (− ,− , |w|−w ).
3
2|w| |w|−w |w|−w
3 3 p
Firstly let us exppand thepformula of thpe operator
n
Hh = U⋆HhU = h2D2I +U⋆VU −2ih2 (U⋆∂ U)D −h2U⋆∂2 U
xk xk xk
Xk=1(cid:2) (cid:3)
e
4
which can be rewritten as
n
Hh = U⋆HhU = h2 (D I −A )2 +V +h2R, (2.7)
xk k
X
k=1
e e
where A = iU⋆∂ U :
k xk
A = [(∂ α,∂ β,∂ ρ)∧(α,β,ρ)+(δ∂ α−α∂ δ,δ∂ β−β∂ δ,δ∂ ρ−ρ∂ δ)].σ,
k xk xk xk xk xk xk xk xk xk
(2.8)
and
n
R = (U⋆∂ U)2 +(∂ U⋆).(∂ U) . (2.9)
xk xk xk
Xk=1(cid:8) (cid:9)
So we can assume that Hh is of the form (1.1) with m = 2, Q = 0, A and R
k
given by (2.8) and (2.9), with U defined by (2.6), and V = V a diagonal matrix
given by (2.5).
e
Theorem 2.1 Under the above assumptions, the first bands bh, j = 1,2,..., of
j
Hh are concentrated around the value hµ + E j = 1,2,..., in the sense that,
j 0
there exist N > 1 and h > 0 such that
0 0
distance(hµ +E ,bh) ≤ Ch2, ∀j < N and ∀h, 0 < h < h ,
j 0 j 0 0
n
where µj = (2jk +1)τk, jk ∈ N, the (µℓ)ℓ N⋆ is the increasing sequence of the
∈
X
k=1
n
eigenvalues of the harmonic oscillator −∆+ τ2x2.
k k
X
k=1
3 Proof of Theorem 2.1
Proof. According to the above discussion, we can assume that
Ph(x,hD) Ph(x,hD)
Hh = Ph(x,hD), with Ph(x,hD) = 11 12 , (3.1)
(cid:18) Ph(x,hD) Ph(x,hD) (cid:19)
21 22
with
Ph(x,hD) = h2(D −a (x))2 + v(x)−|w(x)| + h2r (x)
11 .,11 11
Ph(x,hD) = h2(D +a (x))2 + v(x)+|w(x)| + h2r (x) 
22 .,11 22
Ph(x,hD) = −h2a (x).(D +a (x))−h2a (x).(D −a (x)) 
12 .,12 .,11 .,12 .,11  (3.2)
+ih2div(a (x)) + h2r (x) 
.,12 12 
Ph(x,hD) = −h2a (x)(D −a (x))−h2a (x).(D +a (x))
21 .,21 .,11 .,21 .,11
+ih2div(a (x)) + h2r (x). 
.,21 21 



5
(D = (D ,D ,...,D ) and a (x) = (a (x),a (x),...,a (x)).)
x1 x2 xn .,ij 1,ij 2,ij n,ij
(We used that a = −a ).
.,22 .,11
Let us denote by Hh,θ and Hh,θ the operators associated with
11 22
Ph(x,h(D −θ)) and Ph(x,h(D −θ)) on L2(Tn;C).
11 22
Then, if c = min|w(x)| and c = maxkR(x)k,
0 1
sp(Hh,θ) ⊂ [E −h2c ,+∞[ and sp(Hh,θ) ⊂ [E −h2c +2c ,+∞[.
11 0 1 22 0 1 0
To prove the theorem it is then enough to prove the proposition below.
Proposition 3.1 Let us consider a constant c, 0 < c < c . Then there exists
0
C > 0 such that, for any Eh ∈]−∞,E +2c[, we have
0 0
Eh ∈ sp(Hh,θ) ⇒ distance(Eh,sp(Hh,θ) ≤ C h2
11 0 (3.3)
Eh ∈ sp(Hh,θ) ⇒ distance(Eh,sp(Hh,θ)) ≤ C h2. (cid:27)
11 0
Proof. For such Eh, (Hh,θ −Eh) 1 exists and, thanks to semiclassical pseudod-
22 −
ifferential calculus of [Ro] (see also [Di-Sj] ), for h > 0 small, if 0 < h < h then
0 0
k(Hh,θ −Eh) 1k +kh(D −θ)(Hh,θ −Eh) 1k
22 − L2(Tn) 22 − L2(Tn)
+k(Hh,θ−Eh) 1h(D−θ)k +kh(D−θ)(Hh,θ−Eh) 1h(D−θ)k ≤ C,
22 − L2(Tn) 22 − L2(Tn)
and then
kPh(x,h(D −θ))(Hh,θ −Eh) 1Ph(x,h(D −θ))k ≤ h2C.
12 22 − 21 L2(Tn)
So if Eh ∈ sp(Hh,θ), then uh = (uh,uh) 6= (0,0) is an eigenfunction of Hh,θ
1 2
associated with Eh iff
Hh,θuh +Ph(x,h(D −θ))uh = Ehuh
11 1 12 2 1 (3.4)
uh = −(Hh,θ −EhI) 1Ph(x,h(D −θ))uh. (cid:27)
2 22 − 21 1
In fact Eh ∈]−∞,E +c[ will be an eigenvalue of Hh,θ iff there exists
0
uh in the Sobolev space W2(Tn;C), kuhk 6= 0, such that
1 1 L2(Tn)
Hh,θuh −Ph(x,h(D −θ))(Hh,θ −EhI) 1Ph(x,h(D −θ))uh = Ehuh,
11 1 12 22 − 21 1 1
then we get the first part of Proposition 3.1.
If Eh is an eigenvalue of Hh,θ satisfying the assumption of Proposition 3.1,
11
and uh an associated eigenfunction, then with
1
uh = (uh,−(Hh,θ −Eh) 1Ph(x,h(D −θ))uh), one has
1 22 − 21 1
k(Hh,θ −EhI)uhk
L2(Tn;C2)
= kPh(x,h(D −θ))(Hh,θ −Eh) 1Ph(x,h(D −θ))uhk
12 22 − 21 1 L2(Tn;C)
≤ h2Ckuhk ,
L2(Tn;C2)
we get the second part of Proposition 3.1.
Theorem 2.1 follows from Proposition 3.1 and [Si-1], [Si-2], [He-Sj-1] and
[He-Sj-2] results, (see also [He]), which guarantee that the sequence of eigenvalues
of H1h1,θ, (λj(H1h1,θ))j N⋆ satisfies ∀N0 > 1, ∃h0 > 0, C0 > 0 s.t. ∀h, 0 < h < h0
and ∀j ≤ N , |λ (∈Hh,θ)−(hµh +E )| ≤ C h2(cid:3)
0 j 11 j 0 0
6
4 Asymptotic of the first band
For any real Lipschitz Γ periodic function φ, and for any u ∈ W2(Tn;C2), we
have the identity
Re < Ph(x,h(D −θ))u|e2φ/hu > =
L2(Tn;C2)
n
(cid:0) (cid:1) 
h2k((Dxk −θk)I −Ak)eφ/huk2L2(Tn;C2)  (4.1)
X 
k=1
+ < (V −|∇φ|2I +h2R)u|e2φ/hu > .
L2(Tn;C2) 



e
This identity enables us to apply the method used in [He-Sj-1], (see also [He] and
[Ou]). We define the Agmon [Ag] distance on Rn
1
d(y,x) = inf v(γ(t))−|w(γ(t))|−E |γ˙(t)|dt , (4.2)
0
γ Z
0 p
the inf is taken among paths such that γ(0) = y and γ(1) = x.
For common properties of the Agmon distance, one can see for example [Hi-Si].
We will use that, for any fixed y ∈ Rn, the function d(y,x) is a Lipschitz
function on Rn and |∇ d(y,x)|2 ≤ v(x)−|w(x)|−E almost everywhere on Rn.
x 0
Using that the zeros of v(x)−w(x)−E are the elements of Γ and are non
0
degenerate, we get that the real function d (x) = d(0,x) satisfies, (see [He-Sj-1]),
0
|∇d (x)|2 = v(x)−|w(x)|−E in a neighbourhood of 0.
0 0
We summarize the properties of the Agmon distance we will need:
i) ∃R > 0 s.t. d (x) ∈ C (B (R ))
0 0 ∞ 0 0
ii) |∇d (x)|2 = v(x)−|w(x)|−E , ∀x ∈ B (R ) 
0 0 0 0 (4.3)
iii) |∇d (x)|2 ≤ v(x)−|w(x)|−E 
0 0 
iv) |∇d (x)|2 ≤ v(x)−|w(x)|−E
Γ 0



where d (x) = d(0,x), B (r) = {x ∈ Rn; d (x) < r}
0 0 0
and d (x) = d(Γ,x) = mind(ω,x).
Γ
ω Γ
The least Agmon dis∈tance in Γ is
S = inf d (β ) = inf d(ω,ρ) . (4.4)
0 0 k
1 k n ρ=ω, (ω,ρ) Γ2
≤ ≤ 6 ∈
The Agmon distance on Tn, dTn(.,.), is defined by its Γ-periodic extension on
(Rn)2
Tn
d (y,x) = mind(y,x+ω).
ω Γ
∈
Then
S
0 = sup{r > 0 s.t. {x ∈ Tn; dTn(x ,x) < r} is simply connected} , (4.5)
0
2
r
7
where x is the single point in Tn such that v(x )−|w(x )| = E . The Γ-periodic
0 0 0 0
function on Rn, d (x) is the one corresponding to the extension of dTn(x ,x).
Γ 0
If λh,θ is an eigenvalue of Hh,θ and if uh,θ is an associated eigenfunction, then
by (4.1) one gets as in the scalar case considered in [He-Sj-1]) and [He-Sj-2],
n
h2k((D −θ )I −A )eφ/huh,θk2
xk k k L2(Tn;C2) 
Xk=1  (4.6)

+ < [V −|∇φ|2I +h2R−λh,θI]+uh,θ|e2φ/huh,θ >L2(Tn;C2) 
=< [V −|∇φ|2I +h2R−λh,θI] uh,θ|e2φ/huh,θ > ,
e L2(Tn;C2) 
− 


so, when φ(xe) = dTn(x ,x), necessarily λh,θ −E +O(h2) > 0,
0 0
and if h/C < λh,θ − E < hC, then uh,θ is localized in energy near x , for any
0 0
η ∈]0,1[, ∃C > 0 such that
η
n
h2k((D −θ )I −A )eηdTn(x0,x)/huh,θk2
xk k k L2(Tn;C2) 
X
k=1  (4.7)
+(1−η2) < (v −|w|−E0)uh,θ|e2ηdTn(x0,x)/huh,θ >L2(Tn;C2) 
≤ hC |uh,θ(x)|2dx.
ηR{x∈Tn; dTn(x0,x)<√hC} 

Let Ω ⊂ Tn an open and simply connected set with smooth boundary satis-
fying, for some η, 0 < η < S /2,
0
S −η
{x ∈ Tn; dTn(x ,x) < 0 } ⊂ Ω ⊂ {x ∈ Tn; dTn(x ,x) < S /2} (4.8)
0 0 0
2
Let Hh be the selfadjoint operator on L2(Ω;C2) associated with Ph(x,hD) with
Ω
Dirichletboundarycondition. Wedenoteby(λj(HΩh))j N⋆ theincreasingsequence
of eigenvalues of Hh. Using the method of [He-Sj-1], w∈e get easily the following
Ω
results
Theorem 4.1 For any η, 0 < η < S /2, there exist h > 0 and N > 1 such
0 0 0
that, if 0 < h < h and j ≤ N , then
0 0
∀θ ∈ Bn, 0 < λ (Hh) − λh,θ ≤ Ce (S0 η)/(2h); (4.9)
j Ω j − −
so the length of the band bh satisfies |bh| ≤ Ce (S0 η)/(2h).
j j − −
For the first band, we have the following improvement
|bh| ≤ Ce (S0 η)/h. (4.10)
1 − −
Sketch of the proof.
AsΩ issimply connected andtheoneformθdxisclosed, thereexists a smooth
realfunctionψθ(x)onΩsuchthate−iψθ(x)Ph(x,hD)eiψθ(x) = Ph(x,h(D−θ)) :the
Dirichlet operators on Ω, Hh and Hh,θ associated to Ph(x,hD) and Ph(x,h(D−
Ω Ω
θ)) are gauge equivalent, so they have the same spectrum.
8
Therefore the min-max principle says that
0 < λ (Hh,θ)−λh,θ = λ (Hh) − λh,θ.
j Ω j j Ω j
But the exponential decay of the eigenfunction ϕh,θ(x) associated with λh,θ, given
j j
by (4.7) implies that
k(Ph(x,h(D −θ))−λh,θ)χϕh,θ(x)k ≤ Ce (S0 η+ǫ)/(2h),
j j L2(Ω;C2) − −
for some ǫ > 0, and for a smooth cut-off function χ supported in Ω and χ(x) = 1
if dTn(x ,x) ≤ (S −η +ǫ)/2.
0 0
So distance(λh,θ,sp(Hh)) ≤ Ce (S0 η+ǫ)/(2h).
j Ω − −
This achieves the proof of (4.9).
Let us denote Eh,θ, (respectively Eh,Ω), the first eigenvalue λh,θ, (respectively
1
λ (Hh)),andϕh,θ(x),(respectively ϕh,Ω(x)),theassociatednormalizedeigenfunc-
1 Ω
tions. Let χ be a cut-off function satisfying the same properties as before. Then
Ph(x,h(D−θ))(e−iψθ(x)χ(x)ϕh,Ω(x)) = λ1(HΩh)e−iψθ(x)χ(x)ϕh,Ω(x)+ e−iψθ(x)r0h(x)
with, thanks to the same identity as (4.7) for Dirichlet problem on Ω,
kr0hkL2(Tn;C2) ≤ Ce−S02−hη.
The same argument used in [He-Sj-1], (see also [He]), gives this estimate
|Eh,θ −Eh,Ω− < rh|χϕh,Ω > | ≤ Ce (S0 η)/h.
0 L2(Tn;C2) − −
As τ =< rh|χϕh,Ω > does not depend on θ, so
h 0 L2(Tn;C2)
∀θ ∈ Bn , |Eh,θ −Eh,Ω −τ | ≤ Ce (S0 η)/h
h − −
(cid:3)
this estimate ends the proof of (4.10)
As for the tunnel effect in [He-Sj-1] and [Si-2], we have an accuracy estimate
for the first band, like the scalar case in [Si-3] and in [Ou] (see also [He]).
Theorem 4.2 There exists h > 0 such that, if 0 < h < h then
0 0
|bh| ≤ Ce S0/h.
1 −
Sketch of the proof. Instead of comparing Hh,θ with an operator defined in a
subset of Tn, we have to work on the universal cover Rn of Tn.
We take Ω ⊂ Rn an open and simply connected set with smooth boundary
satisfying, for some η , 0 < η < η < S /2,
0 0 1 0
B ((S +η )/2) ⊂ Ω ⊂ B ((S +η )/2). (4.11)
0 0 0 0 0 1
So Ω contains the Wigner set Wn, more precisely
Wn ⊂ Ω ⊂ 2Wn and Ω ∩ Γ = {0}.
9
We let also denote Hh the Dirichlet operator on L2(Ω;C2) associated with
Ω
Ph(x,hD), and Eh its first eigenvalue. The associated eigenfunction is also de-
Ω
noted by ϕh,Ω(x).
In the same way as to get (4.7), we have
n
h2k(D I −A )ed0(x)/hϕh,Ωk2 ≤ hC |ϕh,Ω(x)|2dx, (4.12)
xk k L2(Ω;C2) Z
Xk=1 B0(√hC)
then the Poincar´e estimate gives
e2d0(x)/h|ϕh,Ω(x)|2dx ≤ h 1C |ϕh,Ω(x)|2dx. (4.13)
−
Z Z
Ω B0(√hC)
Let χ a smooth cut-off function satisfying
χ(x) = 1 if d (x) ≤ (S +η )/2 and χ(x) = 0 if x ∈/ Ω.
0 0 0
Then the function
ϕh,θ(x) = eiθ(x ω)χ(x−ω)ϕh,Ω(x−ω)
−
X
ω Γ
∈
is Γ-periodic and satisfies
(Ph(x,h(D −θ))−Eh)ϕh,θ(x) = rh,θ and
Ω (4.14)
krh,θkL2(Wn;C2) ≤ Ce−(S0+η0)/(2h)kϕh,θkL2(Wn;C2) (cid:27)
and
< rh,θ|ϕh,θ > =
L2(Wn;C2)
eiθ(ρ ω) ([Ph(x,hD);χ]ϕh,Ω)(x−ω).(χϕh,Ω)(x−ρ)dx
−
Z
X Wn
ω,ρ Γ0
∈
with Γ = {0,±β ,...,±β } and
0 1 n
n
[Ph(x,hD);χ] = −2h2i ∂ χ(D I −A ) − h2∆χI.
xk xk k
X
k=1
So
1
< rh,θ|ϕh,θ > ≤ Ce S0/h. (4.15)
(cid:12)kϕh,θk2 L2(Wn;C2)(cid:12) −
(cid:12) L2(Wn;C2) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The proof comes easily from (4.12) and (4.13) as in [Ou] or in [He].
(cid:12) (cid:12)
Using the same argument of [He-Sj-1] as in the proof of (4.10), we get that
|Eh,θ − Eh −τh,θ| ≤ Ce (S0+η0)/h), (4.16)
Ω −
1
with τh,θ = < rh,θ|ϕh,θ > .
kϕh,θk2 L2(Wn;C2)
L2(Wn;C2)
(cid:3)
Theorem 4.2 follows from (4.15) and (4.16)
10