ESI

The Erwin Schr¨ odinger International Institute for Mathematical Physics

Boltzmanngasse 9 A-1090 Wien, Austria

On the Ground State Energy for a Magnetic Schr¨ odinger Operator and the Effect of the De Gennes Boundary Condition

Ayman Kachmar

Vienna, Preprint ESI 1770 (2006)

Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at

January 26, 2006

ON THE GROUND STATE ENERGY FOR A MAGNETIC SCHRÖDINGER OPERATOR AND THE EFFECT OF THE DE GENNES BOUNDARY CONDITION

AYMAN KACHMAR

Abstra t. Motivated by the Ginzburg-Landau theory of super ondu tivity, we estimate in the semi- lassi al limit the ground state energy of a magneti S hrödinger operator with De Gennes boundary ondition and we study the lo alization of the ground state. We exhibit ases when the De Gennes boundary

ondition has strong ee ts on this lo alization.

1. Introdu tion Let

Ω ⊂ R2

be an open bounded domain with regular boundary. Let us onsider

a ylindri al super ondu tor sample of ross se tion erties are des ribed by the minimizers

(ψ, A)

Ω.

The super ondu ting prop-

of the Ginzburg-Landau fun tional

( f. [10, 23, 24℄) : (1.1)

Z κ2 |(∇ − iσκA)φ|2 + σ 2 κ2 |curlA − 1|2 + (|φ|2 − 1)2 dx 2 Ω Z + γ|φ(x)|2 dµ|∂Ω (x),

G(φ, A) =

∂Ω

whi h is dened for ouples

(φ, A) ∈ H 1 (Ω; C) × H 1 (Ω; R2 ).

The parameter

hara teristi of the material. A material is said to be of type I if small and it is said to be of type II when

σ

κ

κ

κ

is a

is su iently

is su iently large. The parameter

is the intensity of the applied magneti eld whi h is supposed to be onstant

and perpendi ular to

Ω.

For a minimizer (ψ, A) of the energy G , the fun tion ψ |ψ|2 measures the density of super ondu ting

is alled the order parameter and

A

Cooper ele tron paires; the ve tor eld

curlA

is alled the magneti potential and

is the indu ed magneti eld. Note that

ψ

satises the following boundary

ondition proposed by De Gennes [10℄ :

ν · (∇ − iσκA)ψ + γψ = 0,

(1.2) where

ν

is the unit outward normal of

∂Ω

and

γ ∈ R

is alled in the physi al

literature the De Gennes parameter. Note that the boundary ondition (1.2) was initially introdu ed in the theory of PDE by Robin. The physi ist De Gennes [10℄ has introdu ed the parameter

γ

in order to model

interfa es between super ondu tors and normal materials. In that ontext, γ is 1 taken to be a non-zero positive onstant and γ ( alled the extrapolation length) usually measures the penetration of the super ondu ting Cooper ele tron pairs in the normal material. Fink-Joiner [7℄ use negative values of

γ

to model the situation

when a super ondu tor is adja ent to another super ondu tor of higher transition temperature. Suppose that we have a type II super ondu tor (i.e.

κ

is large). The fun tional

G

1991 Mathemati s Subje t Classi ation. Primary 81Q10, Se ondary 35J10, 35P15, 82D55. Key words and phrases. S hrödinger operator with magneti eld, semi lassi al analysis, super ondu tivity. 1

2

AYMAN KACHMAR

has a trivial solution

(0, F)

alled a normal solution. It is then natural to study

the positivity of the hessian of

G

near a normal solution in the presen e of a strong 1 h = σκ , we have then the positivity of the quadrati form :

applied magneti eld. By dening the hange of parameter to study as

h→0

1

H (Ω) ∋ u 7→ k(h∇ − iF)uk2L2 (Ω) + hγkuk2L2 (∂Ω) − κ2 h2 kuk2L2(Ω) . Note that the semi- lassi al limit

h → 0

is now equivalent to a large eld limit

σ → +∞. For a ve tor eld and a number

A ∈ C ∞ (Ω; R2 ),

α > 0,

a regular real valued fun tion

γ ∈ C ∞(∂Ω; R)

we dene the more general quadrati form :

(1.3)

1

H (Ω) ∋ u 7→

α,γ qh,A,Ω (u)

= k(h∇

− iA)uk2L2 (Ω)

+h

1+α

Z

γ(x)|u(x)|2 dµ|∂Ω (x).

∂Ω

α,γ Observing that qh,A,Ω is semi-bounded, we onsider the self-adjoint operator assoα,γ

iated to qh,A,Ω by Friedri h's theorem. This is the magneti S hrödinger operator α,γ α,γ Ph,A,Ω with domain D(Ph,A,Ω ) dened by : (1.4)

α,γ Ph,A,Ω = −(h∇ − iA)2 , α,γ D(Ph,A,Ω ) = {u ∈ H 2 (Ω); ν · (h∇ − iA)u|∂Ω + hα γu|∂Ω = 0}.

We denote by

µ(1)(α, γ, h) the

ground state energy of

α,γ Ph,A,Ω

whi h is dened using

the min-max prin iple by : (1.5)

µ(1) (α, γ, h) :=

u∈H

α,γ qh,A,Ω (u) . 2 (Ω),u6=0 kukL2 (Ω)

inf 1

Note that due to gauge invarian e, if Ω is simply onne ted the spe trum of the α,γ Ph,A,Ω depends only on the magneti eld. For this reason we shall omit some times the referen e to the magneti potential A in our notations.

operator

In the ase when

γ≡0

(whi h orresponds to a super ondu tor surrounded by

the va uum), a lot of papers are devoted to the estimate in a semi lassi al regime α,γ of the ground state energy of Ph,A,Ω . We would like here to mention the works of Baumann-Phillips-Tang [2℄, Berno-Sternberg [3℄, del Pino-Felmer-Sternberg [18℄, Heler-Mohamed [12℄, Heler-Morame [13℄ and the re ent work of Fournais-Heler [9℄. The spe ial ase when

α=1

and

γ

is a positive onstant was onsidered by

Lu-Pan [20, 21℄. It was shown there that the ee t of the De Gennes parameter

γ

is very weak, and hen e, this ase does not orrespond to materials as those onsidered in [7℄ (where strong ee ts had been observed due to the interfa e). Note that, following the te hnique of Heler-Morame [13℄, the model operator that α,γ 1 we have to analyze is Ph,A ,Ω with h = 1, A0 = (−x2 , x1), Ω = R × R+ and γ is a 2 0

onstant. We denote by Θ(γ) the bottom of the spe trum of this model operator ( f. (2.2)). Now we state our main results.

Suppose that Ω ⊂ R2 is open, bounded, onne ted and having a smooth boundary. Suppose moreover that the magneti eld is onstant curlA = 1. α,γ Then, for α > 0 and γ ∈ C ∞ (∂Ω; R), the ground state energy of the operator Ph,A,Ω satises : Theorem 1.1.

(1.6)

µ(1)(α, γ, h) ∼ hΘ(hα−1/2 γ0 ),

where γ0 := minx∈∂Ω γ(x).

(h → 0),

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER The asymptoti s (1.6) depends strongly on

α

and

ee tively (see Propositions 2.4 and 2.5). Note that if

lim

h→0

γ0 does not always α = 12 , we get

3

appear

µ(1)(α, γ, h) = Θ(γ0 ) < 1, h

µ(1) (α,γ,h) 1 is a tually negative and tends to −∞. If 2 and γ0 < 0, h µ(1) (α,γ,h) 1 γ0 = 0 or if α > 2 , then limh→0 = Θ0 < 1. This indi ates that in the h above ases, the fun tion γ ould play a role. In the next theorem we des ribe some and when

ee t of

γ

α

0.

ξ 7→ ϕγ,ξ ∈ L2 (R+ )

Note that,

(γ, ξ + τ )ϕγ,ξ+τ (t + τ ) = H[γ, ξ] (ϕγ,ξ+τ (t + τ )) , ϕγ,ξ and then Z (1) (1) µ (γ, ξ + τ ) − µ (γ, ξ)

Taking the s alar produ t with (2.8)

∀t ∈ R+ .

integrating by parts, we get :

ϕγ,ξ+τ (t + τ )ϕγ,ξ (t) dt

R+

= ϕ′γ,ξ+τ (τ )ϕγ,ξ (0) − γϕγ,ξ+τ (τ )ϕγ,ξ (0). Re all that we have the boundary onditions :

ϕ′γ,ξ+τ (0) = γϕγ,ξ+τ (0),

ϕ′γ,ξ (0) = γϕγ,ξ (0).

Then we an rewrite (2.8) as :

Z µ(1) (γ, ξ + τ ) − µ(1)(γ, ξ) ϕγ,ξ+τ (t + τ )ϕγ,ξ (t) dt τ R+ ′ ϕγ,ξ+τ (τ ) − ϕ′γ,ξ+τ (0) ϕγ,ξ+τ (τ ) − ϕγ,ξ+τ (0) −γ · ϕγ,ξ (0). = τ τ

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER By taking the limit as

τ → 0, (1)

∂ξ µ

5

we get :

(γ, ξ) = ϕ′′γ,ξ (0) − γϕγ,ξ (0) ϕγ,ξ (0).

Finally, we make the substitutions :

ϕ′′γ,ξ (0) = ξ 2 − µ(1) (γ, ξ) ϕγ,ξ (0),

ϕ′γ,ξ (0) = γϕγ,ξ (0),

and we get the following formula,

∂ξ µ(1)(γ, ξ) = ξ 2 − µ(1) (γ, ξ) − γ 2 |ϕγ,ξ (0)|2 , F -formula.

alled usually the

An elementary analysis of the above formula permits

now to on lude the result of the Theorem ( f. [6℄). We denote by

ϕγ

the unique stri tly positive and

L2 -normalized

eigenfun tion

Θ(γ).

asso iated to the eigenvalue

We have the following relations (as in the appendix A of [13℄). Lemma 2.2.

For ea h γ ∈ R, the following relations hold : Z

(2.9)

Z

(2.10)

(t − ξ(γ))|ϕγ (t)|2 dt = 0, R+

(t − ξ(γ))2 |ϕγ (t)|2 dt =

R+

Z

(2.11)

(t − ξ(γ))3 |ϕγ (t)|2 dt =

R+

We dene now the universal onstant

M3 =

(2.12)

Z

Θ(γ) γ − |ϕγ (0)|2 , 2 4

1 (γξ(γ) + 1)2 |ϕγ (0)|2 . 6

M3 =

|ϕ0 (0)|2 . Note that (2.9) gives : 6

(t − ξ0 )3 |ϕ0 (t)|2 dt,

R+

where

ξ0 := ξ(0).

We dis uss now the regularity of the fun tions

L2 (R+ ).

γ → Θ(γ) ∈ R

and

γ 7→ ϕγ ∈

It seems for us that Kato's theory ( f. [16℄) do not apply in this ontext

at least for the reason that we do not know a priori whether the expression of the operator

H[γ] = − depends analyti ally on

γ.

d2 + (t − ξ(γ))2 dt2

Inspired by Bonnaillie [5℄, we use a modi ation of

Grushin's method [11℄ and we get the following proposition. Proposition 2.3.

The fun tions R ∋ γ 7→ Θ(γ) ∈ R and R ∋ γ 7→ ϕγ ∈ L2 (R+ )

are C . Moreover, the fun tion R ∋ γ 7→ ϕγ ∈ L∞ (R+ ) is lo ally Lips hitz. ∞

The spe i di ulty in proving the above proposition omes from the fa t that

γ . To work with χ that is equal to 1 on [0, 1] ϕ → 7 ϕ˜ = e−γtχ(t) ϕ that transforms

both the expression and the domain of the operator depend on an operator with a xed domain, we onsider a ut-o

and we apply the invertible transformation ′ the boundary ondition ϕ (0) = γϕ(0) to the usual Neumann boundary ondition ′ ϕ˜ (0) = 0 and leaves the spe trum invariant (the details are given in [15℄).

Proposition 2.4.

(2.13)

The fun tion γ 7→ Θ(γ) satises at γ = 0 : Θ′ (0) = 6M3 .

6

AYMAN KACHMAR

Proof. Let us denote the operator

H[γ, ξ(γ)] by H[γ].

We shall dene the following

trial fun tion :

u = eγt (ϕ0 + γu1 ), where

−1

u1 = (H[0] − Θ0 )

{6M3 ϕ0 + 2ϕ′0 + 2(ξ(γ) − ξ0 )(t − ξ0 )ϕ0 } .

Note that :

H[γ](u) = eγt −∂t2 + (t − ξ(γ))2 − 2γ∂t − γ 2 u.

(2.14)

Using the de omposition :

H[γ] = H[0] − 2(ξ(γ) − ξ0 )(t − ξ0 ) + (ξ(γ) − ξ0 )2 , we an rewrite (2.14) as :

(H[γ] − Θ0 − 6M3 γ) (u) = γ 2 eγt (Θ0 + 6M3 )u1 .

(2.15)

We get now by the spe tral theorem the existen e of an eigenvalue

˜ that satises Θ(γ)

the following estimate :

˜ |Θ(γ) − Θ0 − 6M3 γ| ≤ Cγ 2 ,

(2.16) for onstants

where for

H[η, ξ].

C, γ0 > 0. Using the min-max prin iple we get the following estimate : (2) µ (γ, ξ(γ)) − µ(2) (0, ξ0 ) ≤ Cγ, ∀γ ∈ [−γ0 , γ0 ],

(η, ξ) ∈ R × R, µ(2) (η, ξ)

We get nally that

denotes the se ond eigenvalue of the operator

˜ Θ(γ) = Θ(γ).

The asymptoti behavior of the eigenvalue

γ

∀γ ∈ [−γ0 , γ0 ],

Θ(γ)

with respe t to the parameter

is given in the following proposition.

Proposition 2.5.

There exist positive onstants C0 and γ0 su h that the eigenvalue

(2.17)

|Θ(γ) − 1| ≤ C0 γ exp −γ 2 ,

∀γ ∈ [γ0 , +∞[,

1 , 4γ 2

∀γ ∈] − ∞, 0[.

Θ(γ) satises :

and −γ 2 ≤ Θ(γ) ≤ −γ 2 +

(2.18)

Proof. We prove the estimate (2.17).

Theorem 2.1 we get for any

γ >0

Note that by the min-max prin iple and

:

µ(1)(0, ξ(γ)) ≤ Θ(γ) ≤ 1. The lower bound in (2.17) is proved for

µ(1)(0, ξ(γ))

by Bolley-Heler [4℄ (formula

(A.18)). The relation (2.6) gives the lower bound

Θ(γ) ≥ −γ 2 .

We get the upper bound in eγt (with γ < 0) as a trial

(2.18) from the min-max prin iple by using the fun tion fun tion for the quadrqti form dening Using Agmon's te hnique ( f. eigenfun tion

H[γ, 0].

[1℄), we get the following de ay result for the

ϕγ .

Proposition 2.6. For ea h ǫ ∈]0, 1[ there is a positive onstant Cǫ su h that, for all γ ∈ R, we have the following estimate for the eigenfun tion ϕγ :

(2.19)

2

exp − ǫ(t − ξ(γ)) ϕγ ≤ Cǫ (1 + γ− ),

1 2 H ({t∈R+ ;(t−ξ(γ))≥Cǫ })

where we use the notation γ− = max(−γ, 0).

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER

˜ R[γ]

γ ∈ R,

we re all that the regularized resolvent 2 operator dened on L (R+ ) by : For ea h

˜ R[γ]φ =

(2.20)

7

is the bounded

0 ; φ k ϕγ , −1 (H[γ] − Θ(γ)) φ ; φ⊥ϕγ ,

and extended by linearity. Again, using the Agmon's te hnique, we get that this regularized resolvent is uniformly ontinuous in suitable weighted spa es.

For ea h δ ∈]0, 1[ and η0 > 0, there exist positive onstants

Proposition 2.7.

C0 , t0 su h that,

∀u ∈ L2 (R+ ; eδ(t−ξ(γ)) dt),

∀γ ∈ [−η0 , η0],

u⊥ϕγ ,

we have,

δ(t−ξ(γ)) ˜

R[γ]u

e

(2.21)

H 1 ([t0 ,+∞[)

≤ C0 eδ(t−ξ(γ)) u

L2 (R+ )

.

3. Proof of Theorem 1.1

In this se tion we prove Theorem 1.1 by omparing with the basi model introdu ed in the pre eding se tion. We introdu e a oordinate system boundary

∂Ω

∂Ω

where

t

measures the distan e to

∂Ω

and

s

(s, t)

near the

measures the distan e in

( f. Appendix A).

(Upper bound) Under the hypothesis of Theorem 1.1, there exist positive onstants C and h0 su h that, ∀h ∈]0, h0], we have : Proposition 3.1.

µ(1) (α, γ, h) ≤ hΘ hα−1/2(γ0 + Ch1/2 ) + Ch3/2.

(3.1)

1 2 and γ0 > 0. In this ase the (1) estimate (3.1) reads simply by omparison with the rst eigenvalue λ (h) of the (1) Diri hlet realization and by using the estimate of λ (h) ( f. [12℄) and (2.17). 1 We suppose now that γ0 ≤ 0 if α < . Consider a point x0 ∈ ∂Ω su h that 2 γ(x0 ) = γ0 . We suppose that x0 = 0 in the oordinate system (s, t) near the Proof. We start with the easy ase when

boundary ( f. fun tion

uh,α

Appendix A).

α

0

and

k∈N

ϕη

in

:

tk |ϕη (t)|2 dt ≤ Ck,δ h−δk ,

∀h ∈]0, h0].

R+

After a hange of variables, thanks to (3.5) and to the de ay of

ϕη ,

formula (A.3)

gives the following upper bound :

α,˜ γ0 (uh,α ) ≤ hΘ(η) + Ch3/2, qh,A,Ω

∀h ∈]0, h0].

ϕη (Proposition 2.6), we obtain that the L2 1 as h → 0. The appli ation of the min-max (3.1).

Using formula (A.4) and the de ay of norm of

uh,α

is exponentially lose to

prin iple permits now to on lude

When α ∈] 12 , 1[ and the fun tion γ is not onstant, Proposition 3.1 gives the following upper bound for the eigenvalue µ(1)(α, γ, h), thanks to Proposition 2.4, Remark 3.2.

µ(1) (α, γ, h) ≤ hΘ0 − 6M3 γ0 hα+1/2 + O(hinf(3/2,2α)). This upper bound is a tually an asymptoti expansion as h → 0 (see Remark 5.13).

(Lower bound) Under the hypothesis of Theorem 1.1, there exist positive onstants C, C ′ and h0 su h that, ∀h ∈]0, h0], we have :

Proposition 3.3.

µ(1) (α, γ, h) ≥ hΘ hα−1/2γ0 (1 + C ′ h1/4 ) − Ch5/4 .

(3.6)

Proof. We follow the te hnique of [13℄ and we ompare with the basi model by

means of a partition of unity. We introdu e a partition of unity

(χj )

of

R2

that

satises

X

|χj |2 = 1,

j

where for

z ∈ R2

X

|∇χj |2 < +∞,

suppχj ⊂ D(zj , 1),

j

and

r > 0, we

denote by

D(z, r)

the disk of enter

z

and raduis

r.

We introdu e now the s aled partition of unity :

χhj (z) := χj (ǫ0 hρ z), where

ǫ0

and

ρ

∀z ∈ R2 ,

are two positive numbers to be hosen suitably. Note that

(χhj ) now

satises :

X

(3.7)

|χhj |2 = 1,

j

X

(3.8)

−2ρ |∇χhj |2 ≤ Cǫ−2 , 0 h

j

(3.9) where

supp

C

χhj ⊂ Qhj := D(zjh , ǫ0 hρ ),

is a positive onstant. We an also suppose that :

(3.10)

either supp

χhj ∩ ∂Ω = ∅

or

zjh ∈ ∂Ω.

Note that the alternative in (3.10) permits us to write the sum in (3.7) under the form :

X 1A tually we

=

X int

+

X

,

bnd

shall need this de ay only when α < 1/2 and γ0

< 0.

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER where the summation over int means that the support of

χhj

9

do not meet the

boundary while that over bnd means the onverse. We have now the following de omposition formula :

α,γ qh,A (u) =

(3.11)

X

α,γ qh,A (χhj u) − h2

j

X

k |∇χhj| uk2 ,

∀u ∈ H 1 (Ω),

j

usually alled in other ontexts the IMS formula.

We have now to bound from

below ea h of the terms on the right hand side of (3.11). Note that (3.8) permits to estimate the ontribution of the last term in (3.11) :

h2

(3.12)

X

2−2ρ k |∇χhj| uk2 ≤ Cǫ−2 kuk2 , 0 h

∀u ∈ H 1 (Ω).

j

If

χhj

is supported in

Ω,

then we have :

α,γ qh,A (χhj u) =

Z

R2

|(h∇ − iA)χhj u|2 dx.

Sin e the lowest eigenvalue of the S hrödinger operator with onstant magneti 2 eld in R is equal to 1, we get :

α,γ qh,A (χhj u) ≥ h

(3.13)

We have now to estimate

Z

Ω

|χhj u|2 dx,

∀u ∈ H 1 (Ω).

α,γ qh,A,Ω (χhj u) when χhj

meets the boundary. It is in this ase α,γ h that we see the ee t of the boundary ondition. Note that, by writing qh,A,Ω (χj u) in the boundary oordinates, thanks to Proposition A.1, there exist a positive

onstant

C1

independent of

h

and

(3.14) Z

Ω

j

su h that :

|(h∇ − iA)χhj u|dx ≥ (1 − C1 ǫ0 hρ )

where

A˜

Z

R×R+

is the ve tor eld asso iated to

A

˜ h u|2 dsdt, |(h∇ − iA)χ j

∀u ∈ H 1 (Ω),

by (A.2).

By a gauge transformation, we get a new magneti potential

A˜new,j

satisfying :

A˜new,j = A˜ − ∇φhj , A˜new,j (zjh ) = 0, ˜ j Anew,j (w) − A˜lin (w) ≤ C|w|2,

(3.15)

Ajlin :=

w = (s, t),

1 2 (−t, s) is the linear magneti potential and C > 0 is a onstant independent of h and j . By Cau hy-S hwarz inequality, we get for an arbitrary

where

θ

positive number

Z

and for a positive onstant

R×R+

C˜

independent of

h

and

j

˜ h u|2 dsdt |(h∇ − iA)χ j

! 2 φhj h ≥ (1 − h ) χ u dsdt (h∇ − iA˜lin ) exp −i h j R×R+

˜ −2θ |w|2 χh u 2 . −Ch j 2θ

Now we get a onstant

Z

C2 > 0

su h that after putting :

γ˜j =

γ(zjh ) − C2 hρ , 1 − C2 h2θ − C2 ǫ0 hρ

:

10

AYMAN KACHMAR

the estimate (3.14) reads nally : (3.16)

α,γ qh,A,Ω (χhj u)

ρ

≥

(1 − C2 ǫ0 h − C2 h

2θ

φhj h χ u exp −i h j

α,˜ γ )qh,A˜jj ,R×R + lin

!

−C2 ǫ20 h4ρ−2θ kχhj uk2 . Note that this permits to ompare with the model operator and to get nally the energy estimate : (3.17)

o n α,γ qh,A,Ω (χhj u) ≥ (1 − C2 ǫ0 hρ − C2 h2θ )hΘ(hα−1/2 γ˜j ) − C2 ǫ20 h4ρ−2θ kχhj uk2L2 (Ω) .

We substitute now the estimates (3.6), (3.12), and (3.13) in (3.11) and get nally : (3.18)

α,γ qh,A,Ω (u) ≥ h

XZ

Ω

int

−C h As

γ0

(3.19)

4ρ−2θ

γ,

is the minimum of

α,γ qh,A,Ω (χhj u)

XZ |χhj u|2 dx |χhj u|2 dx + hΘ hα−1/2γ˜j 2−2ρ + ǫ−2 0 h

+h

brd Ω 1+2θ

kuk2 ,

+h

∀u ∈ H 1 (Ω).

we an get instead of (3.16) the estimate :

ρ

≥

1+ρ

(1 − C2 ǫ0 h − C2 h

2θ

φhj h exp −i χ u h j

α,˜ γ0 )qh, ˜j ,R×R+ A lin

!

−C2 ǫ20 h4ρ−2θ kχhj uk2 , where

γ˜0

is dened by :

γ˜0 :=

γ0 . 1 − C2 h2θ − C2 ǫ0 hρ

We then get instead of (3.18) : (3.20)

α,γ qh,A,Ω (u) ≥ h

XZ int

−C

Ω

XZ |χhj u|2 dx + hΘ hα−1/2γ˜0 |χhj u|2 dx

2−2ρ h4ρ−2θ + ǫ−2 0 h

+ h1+ρ + h

brd Ω 1+2θ

kuk2 ,

∀u ∈ H 1 (Ω).

The advantage of (3.18) is that it gives a lower bound of the quadrati form

α,γ qh,A,Ω

in terms of a potential, see however Se tion 4. We hoose now

ǫ0 = 1, ρ = 3/8 and θ = 1/8

in (3.20) and obtain (3.6) by applying

the min-max prin iple.

Proof of Theorem 1.1. Using Propositions 2.4 and 2.5, the proof of Theorem 1.1

now follows from Propositions 3.1 and 3.3.

4. Lo alization of the ground state We work in this se tion under the hypotheses of Theorem 1.2. Due to Theorem 1.1 we have in this ase that

lim

(4.1)

h→0

µ(1)(α, γ, h) < 1. h

Then this gives, by following the same lines of the proof of Theorem 6.3 in [13℄, the following proposition. Theorem 4.1.

Under the hypotheses of Theorem 1.2, there exist positive onstants

α,γ δ, C, h0 su h that, for all h ∈]0, h0], a ground state uα,γ,h of the operator Ph,A,Ω

satises : (4.2)

exp δd(x, ∂Ω) uα,γ,h ≤ Ckuα,γ,hkL2 (Ω) ,

2

β h L (Ω)

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER

11

and

exp δd(x, ∂Ω) uα,γ,h ≤ Ch− min(1/2,β)kuα,γ,h kL2 (Ω),

1 β h H (Ω)

(4.3)

where β = 1 − α if γ0 < 0 and α < 12 , and β = 1/2 otherwise.

Φ:

2

Φ Φ α,γ (1)

qh,A,Ω exp β uα,γ,h = µ (α, γ, h) exp β uα,γ,h

2 h h L (Ω)

2

Φ 2−2β

+h

|∇Φ| exp hβ uα,γ,h 2 . L (Ω)

Proof. Integrating by parts, we get for any Lips hitz fun tion

(4.4)

Let

α,γ uα,γ,h. Using the lower bound for qh,A,Ω (u) in (3.20) together (1) bound for µ (α, γ, h) in (3.1), we get from (4.4) : XZ 1−2ρ 1 − Θ(hα−1/2 (γ0 + Ch1/2)) − Cǫ−2 − Ch4ρ−2θ−1 0 h

u = exp

the upper

Φ hβ

int

with

Ω

−Chmin(ρ,2θ) − h1−2β |∇Φ|2 × |χhj u|2 dx XZ ≤ Θ(hα−1/2 γ˜0 ) − Θ(hα−1/2 (γ0 + Ch1/2 ))+ Ω

brd

We hoose also

θ>0

ρ=β

hmin(4ρ−2θ−1,1−2ρ) + h1−2β |∇Φ|2 × |χhj u|2 dx. χhj is supported in a 4ρ − 2θ − 1 > 0 and we dene

so that ea h

su h that

disk of radius

nally the fun tion

Φ(x) = δ max dist(x, ∂Ω); ǫ0 h where

ǫ 0 hβ .

β

δ is a positive onstant to be hosen appropriately.

We hoose

Φ

by :

,

Note that

1−Θ(hα−1/2 γ˜0 )

de ays in the following way :

∃ C 0 , h0 > 0

s. t., ∀h

∈]0, h0],

1 − Θ(hα−1/2 γ˜0 ) ≥ C0 h2α−1 1 − Θ(hα−1/2 γ˜0 ) ≥ C0 Thus we an hoose

ǫ0

and

δ

if

γ0 < 0

and

α

0, ∃Cε > 0, ∀x ∈ (γ −γ0 )−1 ([ǫ, +∞[), Θ hα−1/2γ˜(x) −Θ hα−1/2γ˜0 > Cε . In the ase α−1/2

In the ase

α

0, ∃Cε > 0, ∀x ∈ (γ − γ0 )−1 ([ǫ, +∞[), Θ hα−1/2 γ˜(x) − Θ hα−1/2 γ˜0 > Cε h1−2α.

in the form :

Φh (x) = δχ(dist(x, ∂Ω))dist (x, {x ∈ ∂Ω; γ(x) = γ0 }) , δ an appropriate positive onstant and χ is the same as in (3.4), we get ǫ > 0 the following de ay near the boundary : 2 Z Φh ǫ 2 (4.6) exp hβ uα,γ,h dx ≤ Cǫ exp hβ kuα,γ,hk , ∀h ∈]0, hǫ], dist(x,∂Ω) 12 and M3 ( 12 , γ) = 41 1 − 31 (γξγ + 1)2 |ϕγ (0)|2 . (5.2)

5.2. Upper bound. We onstru t a trial fun tion by means of boundary oordinates

(s, t)

(s, t)

and we denote by

near a point

z0 ∈ ∂Ω. We suppose that z0 = 0 in the oordinate system a0 = 1 − tκ0 and η(z0 ) = hα−1/2 γ(z0 ). We then dene the

trial fun tion :

ξη(z0 ) s vh (s, t), uh = exp −i 1/2 h

(5.3) with

−1/2

vh (s, t) = h−5/16 a0

(5.4)

and where the fun tions

χ

and

f

(t)ϕη(z0 ) h−1/2 t χ(t) · f h−1/4 s ,

are as in (3.4).

We ontinue now to work in the spirit of [13℄. We work with the gauge given in Proposition A.2. An expli it al ulation, thanks to the de ay of

ϕη(z0 )

(Proposition

2.6), gives the following lemma.

Under the above notations, for ea h α ∈ [ 12 , 1] and γ ∈ C ∞ (∂Ω; R), there exist positive onstants C, h0 su h that, ∀h ∈]0, h0], we have the following estimate : Lemma 5.3.

(5.5)

Z α,γ(z0 ) h h h H U ϕη(z0 ) × U ϕη(z0 ) dt ≤ Ch13/8, qh,A,Ω (uh ) − R+

where the operators H h and U h are dened respe tively by :

2 κ0 1/2 t 1 − t H h = a−2 − h ξ(η(z )) − h2 a−1 0 0 0 ∂t (a0 ∂t ), 2 (U h g)(t) = h−1/4 g(h−1/2 t),

∀g ∈ L2 (R+ ).

We give few details of the proof. Note that in the support of

a = a0 + O(h5/8 ), 2A tually,

uh

we have

2

t A˜1 = −t 1 − κ0 + O(h9/8 ). 2

if z0 is a point of maximum of κ, the remainder is better and of order O(h3/4 ) for the rst term.

14

AYMAN KACHMAR

Then, thanks to formula (A.3), we get modulo

α,γ(z ) qh,A,Ω0 (uh )

Z

a0

R×R+

(

O(h13/8 )

:

=

|h∂t vh |

2

+ a−2 0

+h1+α γ(z0 )|vh (0)|2 .

Integrating with respe t to

s,

2 ) t 1/2 t 1 − κ0 − h ξ(η(z0 )) vh dsdt 2

we get after an integration by parts the estimate in

(5.5). Similar omputations give also the following lemma. Lemma 5.4.

Under the hypotheses of Lemma 5.3, there exist positive onstants

C, h0 su h that, ∀h ∈]0, h0], we have :

h

(H − H h − H h )U h ϕη(z ) 2 (5.6) 0 1 0 L (R

+)

≤ Ch2 ,

where the operators H0h and H1h are dened respe tively by : 2 H0h = −h2 ∂t2 + t − h1/2 ξ(η(z0 )) ,

2 H1h = 2tκ0 t − h1/2 ξ(η(z0 )) − κ0 t2 t − h1/2 ξ(η(z0 )) + h2 κ0 ∂t .

(1) The next lemma permits to on lude an upper bound for the eigenvalue µ (α, γ, h).

Under the above notations, there exist positive onstants C, h0 su h that, when h ∈]0, h0], we have the following estimate :

Lemma 5.5.

α,γ qh,A,Ω (uh ) − {Θ(η(z0 )) − 2M3 (α, γ)κ0 h3/2 }kuh k2L2 (Ω) ≤ Chǫα ,

where ǫα = inf(13/8, 2α + 12 ) for α >

Proof. Noti e that in the support of

1 2

and ǫ1/2 = 13/8.

uh

we have

γ(z) = γ(z0 ) + O(h1/8 ).

Then

this gives :

α,γ(z )

α,γ qh,A,Ω (uh ) − qh,A,Ω0 (uh ) = O(h9/8+α ). In view of Lemmas 5.3 and 5.4, we get the following estimate :

(5.7)

Z h α,γ h h h H1 + H0 U ϕη(z0 ) × U ϕη(z0 ) dt ≤ Ch13/8. qh,A,Ω (uh ) − R+

We note also that we have the following relations :

o n 2 (U h )⋆ H0h U h = h −∂t2 + (t − ξ(η(z0 ))) ,

(5.8)

(U h )⋆ H1h U h = κ0 h3/2 H1 ,

(5.9) where the operator

H1

is dened by :

3

By dening (5.10)

H1 = (t − ξ(η(z0 ))) − ξ(η(z0 ))2 (t − ξ(η(z0 ))) + ∂t . R K3 (α, h) := R+ H1 ϕη(z0 ) · ϕη(z0 ) dt, the estimate (5.7) reads o n α,γ qh,A,Ω (uh ) − hΘ(η(z0 )) + K3 (α, h)κ0 h3/2 ≤ Ch13/8.

1 1 , we get by using (2.2) that K3 ( , h) 2 2 thanks to Propositions 2.3 and 2.4, we get that

Now, for

α=

= −2M3 ( 12 , γ(z0 )).

K3 (α, h) = −2M3 + O(h2α−1 ).

as :

For

α>

1 , 2

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER Finally, the de ay of

lose to

1.

ϕη(z0 )

in Proposition 2.6 gives that

kuh kL2(Ω)

15

is exponentially

This a hieves the proof of the lemma.

The min-max prin iple gives now, thanks to Lemma 5.5, an upper bound for

µ(1) (α, γ, h).

Under the hypothesis of Theorem 5.1 we take

z0

su h that

(κ − 3γ)(z0 ) = (κ − 3γ)max and we use the estimate in (2.16). Under the hypothesis of Theorem 5.2 we hoose

z0

su h that

κ(z0 ) = κmax .

5.3. Lower bound. As in the proof of Proposition 3.3, we onsider a standard 3 χ 2 s aled partition of unity j,h1/6 j∈Z2 of R that satises :

X

(5.11)

|χj,h1/6 (z)|2 = 1,

j∈J

X

|∇χj,h1/6 (z)|2 ≤ Ch−1/3,

j∈J

(5.12)

supp

χj,h1/6 ⊂ jh1/6 + [−h1/6 , h1/6]2 .

We dene the following set of indi es :

1 Jτ(h) := {j ∈ Z2 ; suppχj,h1/6 ∩ Ω 6= φ, dist(supp χj,h1/6 , ∂Ω) ≤ τ (h)}, where the number

τ (h)

is dened by :

τ (h) = hδ ,

(5.13) and the number

δ

with

1 1 ≤δ≤ , 6 2

will be hosen in a suitable manner.

We onsider also another s aled partition of unity in

2 2 ψ0,τ(h) (t) + ψ1,τ(h) (t) = 1,

(5.14)

supp ψ0,τ(h) ⊂ [

(5.15)

Note that, for ea h

preted, by means of boundary

ψ1,τ(h) (t)χj,h1/6 (s, t)

′ |ψj,τ(h) (t)| ≤

:

C , τ (h)

j = 0, 1,

τ (h) ]. 10 the fun tion ψ1,τ(h) (t)χj,h1/6 (s, t) ould be inter oordinates, as a fun tion in Ω. Moreover, ea h

τ (h) , +∞[, 20

1 j ∈ Jτ(h) ,

R

supp ψ1,τ(h) ⊂] − ∞,

is supported in a re tangle

K(j, h) =] − h1/6 + sj , sj + h1/6 [×[0, hδ [ ∂Ω. The role of δ is then to ontrol the size K(j, h). Due to the exponential de ay of a ground near

of the width of ea h re tangle state away from the boundary

(Theorem 4.1), we get the following lemma.

Suppose that α > 12 . Under the above notations, a L2 -normalized α,γ satises : ground state uα,γ,h of the operator Ph,A,Ω

Lemma 5.6.

(5.16)

X α,γ (1) qh,A,Ω χj,h1/6 ψ1,τ(h) uγ,h − µ (α, γ, h) ≤ Ch5/3 . 1 j∈Jτ(h)

The proof of (5.16) follows the same lines of that in [13℄ (formulas (10.4), (10.5) and (10.6)).

1 j ∈ Jτ(h) , we dene a unique point zj ∈ ∂Ω by the relation s(zj ) = sj . We t j denote then by κj = κ(zj ), aj (t) = 1 − κj t, A (t) = −t 1 − κj , and γj = γ(zj ). 2 We onsider now the k -family of one dimensional dierential operators : For ea h

(5.17)

3The

j 2 Hh,j,k = −h2 a−1 j ∂t (aj ∂t ) + (1 + 2κj t)(hk − A ) ,

only dieren e is that we hoose the partitions support in squares rather than dis s.

16

AYMAN KACHMAR α,γ,D k is a real parameter. We denote by Hh,j,k the δ ]0, h [; aj (t)dt of Hh,j,k whose domain is given by :

where

2

L

self-adjoint realization on

γ,D D(Hh,j,k ) = {v ∈ H 2 (]0, hδ [); v′ (0) = hα γ˜j v(0), v(hδ ) = 0}.

(5.18)

The parameter

γ˜j

is dened by :

γ˜j = γj + ε(h), with

ε(h) = 0

if the fun tion

γ

is onstant and

ε(h) = O(h1/6 )

otherwise. We then

introdu e :

γ,D µj1 (α, γ, h) := inf inf Sp(Hh,j,k ).

(5.19)

k∈R

We have now the following Lemma whi h is proved in [15℄. Lemma 5.7.

For ea h α ∈ [ 12 , +∞[, we have under the above notations : (1)

µ

(5.20)

(α, γ, h) ≥

inf

1 j∈Jτ(h)

!

µj1 (α, γ, h)

+ O(h5/3 ).

Again the proof follows the same lines of [13℄ (Se tion 11), but let us explain α,γ qh,A,Ω (ψ1,τ(h) χj,h1/6 uα,γ,h) in boundary oordinates. We work with the lo al hoi e of gauge given in Proposition A.2. briey the main steps. We express ea h term

We expand now all terms by Taylor's formula near

(sj , 0).

After ontrolling the

remainder terms, thanks to the exponential de ay of the ground states away from the boundary, we apply a partial Fourier transformation in the tangential variable

s

and we get nally the result of the lemma.

We have now to nd, uniformly over k ∈ R, a lower bound for the rst eigenvalue α,γ,D 1/2 of the operator Hh,j,k . Putting β = κj , ξ = −h k and η = γ˜j , we get by a s aling argument :

µj1 (k; α, γ, h)

α,η,D µj1 (k; α, γ, h) = hµ1 (Hh,β,ξ ), where

α,η,D µ1 (Hh,β,ξ )

is the rst eigenvalue of the one dimensional operator :

α,η,D Hh,β,ξ = −∂t2 + (t − ξ)2 + βh1/2 (1 − βh1/2 t)−1 ∂t 2 t2 t4 +2βh1/2 t t − ξ − βh1/2 − βh1/2 t2 (t − ξ) + β 2 h , 2 4

(5.21)

whose domain is dened by :

α,η,D D(Hh,β,ξ ) = {u ∈ H 2 (]0, hδ−1/2[); u′ (0) = hα−1/2 η u(0), u(hδ−1/2 ) = 0}.

η, β ∈] − M, M [ and M a given positive onstant), α,η,D ξ ∈ R, a lower bound for the eigenvalue µ1 (Hh,β,ξ ). The

We have then to nd (when uniformly with respe t to

min-max prin iple gives the following preliminary lo alization of the spe trum of α,η,D Hh,β,ξ :

the operator

For ea h M > 0 and α ∈ [ 21 , +∞[, there exist positive onstants C, h0 su h that, ∀η, β ∈] − M, M [, ∀ξ ∈ R, we have when h ∈]0, h0], Lemma 5.8.

(5.22)

α,η,D α,η,D α,η,D µj (Hh,β,ξ ) − µj (H0,ξ ) ≤ Ch2δ−1/2 1 + µj (H0,ξ ) ,

where, for an operator T having a ompa t resolvent, µj (T ) denotes the in reasing sequen e of eigenvalues of T .

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER Remark 5.9.

17

Note that the min-max prin iple gives now that α,η,D µj (H0,ξ ) ≥ µ(j)(hα−1/2 η, ξ),

where, for η˜ ∈ R, µ(j) (˜ η , ξ) is the in reasing sequen e of eigenvalues of the operator H[η, ξ] introdu ed in Se tion 2.3. The following lemma deals with the ase when

ξ

is not lo alized very lose to

ξ(hα−1/2 η).

Suppose that δ ∈]1/4, 1/2[. For ea h α ≥ 12 , there exists ρ ∈]0, δ− 14 ], and for ea h M > 0, there exist positive onstants ζ, h0 > 0 su h that,

Lemma 5.10.

∀ξ su h that |ξ − ξ(hα−1/2 η)| ≥ ζhρ ,

∀η, β ∈] − M, M [,

we have, α,η,D µ1 (Hh,β,ξ ) ≥ Θ(hα−1/2 η) + h2ρ ,

(5.23)

∀h ∈]0, h0].

µ(1) (hα−1/2η, ξ), thanks to Lemma 5.8 1 and Remark 5.9. We start with the ase when α = 2 and η ∈] − M, M [. Writing (1) Taylors formula up to the se ond order for the fun tion ξ 7→ µ (η, ξ), we get positive onstants θ, C1 su h that when |ξ − ξ(η)| ≤ θ, we have : Proof. It is su ient to obtain (5.23) for

µ(1) (η, ξ) ≥ Θ(η) + C1 |ξ − ξ(η)|2 . Then by taking

ζ

C1 ζ > ζ0 , where ζ0 > 1 ζhρ ≤ |ξ − ξ(η)| ≤ θ,

su h that

appropriately, we get when

is a onstant to be hosen

µ(1) (η, ξ) ≥ Θ(η) + ζ0 h2ρ, where

ρ

is also a positive onstant to be hosen later. When

a positive onstant

ǫθ

|ξ − ξ(η)| > θ,

we get

su h that :

µ(1) (η, ξ) ≥ Θ(η) + ǫθ . Then by hoosing

h0

su h that

ζ0 hρ0 < ǫθ , we get for |ξ −ξ(η)| ≥ ζhρ

and

h ∈]0, h0] :

µ(1) (η, ξ) ≥ Θ(η) + ζ0 h2ρ.

(5.24)

α > 1/2. η ∈] − M, M [,

We treat now the ase when uniformly for all

ξ∈R

and

Note that the min-max prin iple gives

µ(1)(hα−1/2 η, ξ) ≥ (1 − Cη−hα−1/2 )µ(1)(0, ξ). η = 0 and ρ = inf(δ − 41 , α − 12 ), we an hoose ζ0 |ξ − ξ0 | ≥ ζhρ :

Then using (5.24) for so that we have for

µ(1) (hα−1/2 η, ξ) ≥ Θ(hα−1/2 η) + To nish the proof, we repla e

ξ0

large enough

ζ0 ρ h . 2

ξ(hα−1/2 η) getting an error of order O(hα−1/2 ).

by

α−1/2 η)| < ζhρ . |ξ − ξ(h

Now we deal with the ase when look for a formal solution (5.25) in the form :

α,η µ, fh,β,ξ

α,η α,η α,η Hh,β,ξ fh,β,ξ = µfh,β,ξ ,

Let

η˜ = hα−1/2 η .

of the spe tral problem

α,η fh,β,ξ

′

α,η (0) = hα−1/2fh,β,ξ (0),

2

(5.26)

µ = d0 + d1 (ξ − ξ(˜ η )) + d2 (ξ − ξ(˜ η )) + d3 h1/2 ,

(5.27)

α,η fh,β,ξ = u0 + (ξ − ξ(˜ η )) u1 + (ξ − ξ(˜ η )) u2 + h1/2 u3 ,

2

We

18

AYMAN KACHMAR

d0 , d1, d2 , d3 and the fun tions u0 , u1, u2 , u3 are to be deterα,η,D Hh,β,ξ in powers of (ξ −ξ(˜ η )) and then we identify the terms of orders (ξ − ξ(˜ η ))j (j = 0, 1, 2) and h1/2 . We then

where the oe ients

mined. We expand the operator the oe ients of

obtain for the oe ients :

(5.28)

d0 = Θ(˜ η ), u0 = ϕη˜ ˜ η ] {(t − ξ(˜ d1 = 0, u1 = 2R[˜ η ))ϕη˜ } R d2 =: d2 (α, η) = 1 − 2 R+ (t − ξ(˜ η ))ϕη˜ u1 dt, o n ˜ η ] 4 (t − ξ(˜ ˜ η ] [(t − ξ(˜ u2 = R[˜ η )) R[˜ η )) ϕη˜] − d2 R d3 =: d3 (α, η) = β R+ ϕη˜ ∂t + (t − ξ(˜ η ))3 ϕη˜dt, i h 3 u = −R[˜ ˜ η] β ∂t + (t − ξ(˜ η )) − ξ(˜ η )2 (t − ξ(˜ η )) − d3 u0 . 3

Using the fun tion

α,η t χ( hδ−1/2 )fh,β,ξ

(where

χ is the same as in (3.4)) as a quasi-mode,

we get by the spe tral theorem, thanks to the de ay results in Propositions 2.6 and 2.7 and to the lo alization of the spe trum in Lemma 5.8, the following lemma.

Suppose that δ ∈] 41 , 12 [. For ea h M > 0 and α ∈ [ 21 , 1], there exist positive onstants C > 0, h0 su h that, Lemma 5.11.

∀η, β ∈] − M, M [,

∀ξ su h that |ξ − ξ(˜ η )| ≤ ζhρ ,

we have when h ∈]0, h0], (5.29)

o n α,η,D η ) + d2 (α, η)(ξ − ξ(˜ η ))2 + d3 (α, η)h1/2 µ1 (Hh,β,ξ ) − Θ(˜ i h ≤ C h1/2 |ξ − ξ(˜ η )| + hδ+1/2 .

where d2 (α, η) and d3 (α, η) are dened by (5.28) respe tively.

Hen e we have obtained by this analysis a lower bound for the rst eigenvalue µ(1) (α, γ, h). We omplete the pi ture by showing that the term d2 (α, η) is positive. Lemma 5.12.

h0 su h that :

For ea h α ∈ [ 12 , +∞[ and M > 0, there exists a positive onstant d2 (α, η) > 0,

∀h ∈]0, h0],

∀η ∈] − M, M [.

α = 12 . 1 If α > , we repla e d2 (α, η) by its approximation up to the rst order, thanks to 2 Proposition 2.3, and we obtain that : Preuve. It is a tually su ient to prove the on lusion of the lemma when

1 d2 (α, η) = d2 ( , 0) + O(hα−1/2 ), 2 whi h gives the on lusion of the lemma. 1 For the parti ular ase α = 2 , we show that :

whi h is stri tly positive.

1 d2 ( , η) = ∂ξ2 µ(1) (η, ·) (ξ(η)) 2

We are now able to on lude the asymptoti s given in Theorems 5.1 and 5.2. 5 1 δ = 12 . When α > 2 we repla e Θ(˜ η ) and d3 by their approximations up to the se ond and rst orders respe tively, thanks to Propositions 2.4 and First we hoose

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER 2.3. For

1 we get by (2.11) that 2

α=

d3

is indeed equal to

19

−2M3 ( 12 , γ).

When α ∈] 21 , 1[ and the fun tion γ is not onstant, we get from the above analysis that the upper bound in Remark 3.2 is a tually an asymptoti expansion. That is, as h → 0, we have, Remark 5.13.

µ(1) (α, γ, h) = hΘ0 + 6M3 γ0 hα+1/2 + O(hinf(3/2,2α)). α,γ We get also that the quadrati form qh,A,Ω

an be bounded from below by means of a potential W : α,γ qh,A,Ω (u) ≥

Z

W (x)|u(x)|2dx,

∀u ∈ H 1 (Ω),

Ω

where W is dened for some positive onstant C0 by : W (x) =

h hΘ0 + 6M3 γ(x)h

α+1/2

− C0 h

inf(3/2,2α)

; if ; if

dist(x, ∂Ω) > h1/6 dist(x, ∂Ω) < h1/6 .

Then, as in Se tion 4, we get by Agmon's te hnique that a ground state de ays exponentially away from the boundary points where γ is minimum and hen e we have ompleted the proof of Theorem 1.2. Note also that the above analysis permits, under the hypotheses α,γ of Theorems 5.1 and 5.2, to bound the quadrati form qh,A,Ω from below using a potential W dened either by means of the fun tion κ − 3γ (when α = 1) or by the s alar urvature κ (when γ is onstant). Then, by using Agmon's te hnique, we nish the proofs of Theorems 1.3 and 1.4. Remark 5.14.

6. Con lusion We have extended in Theorems 5.1 and 5.2 the expansion announ ed by Pan [17℄ in the parti ular ase when is a spe i di ulty when

γ

α=1

and

γ

is a positive onstant. However, there

is negative and the systemati analysis in the spirit

of [13℄ had allowed us to understand the role of the boundary ondition imposed by De Gennes. We have not been able to obtain the lo alization of the groundstate when

α < 1/2

and

γ0 < 0.

This is strongly related to the question of the lo al-

ization of the ground state of the Diri hlet realization of the S hrödinger operator with onstant magneti eld whi h is open. Finally, in the spirit of [8, 14, 20℄, we hope to apply this analysis to the onset of super ondu tivity and to omplete the analysis of [19℄ ( f. [15℄). A knowledgements. I am deeply grateful to Professor B. Heler for the on-

stant attention to this work, his help, advi es and omments. I would like also to thank S. Fournais for his attentive reading. I a knowledge the ESI at Vienna where I found good onditions to prepare a part of this work and the ESF whi h had supported the visit under the SPECT program. This work had been done by the nan ial support of the

Agen e universitaire de la fran ophonie

(AUF).

Appendix A. Coordinates near the boundary

We re all in this appendix well-known oordinates that straightens a portion of |∂Ω| |∂Ω| ∂Ω. Let s ∈] − 2 , 2 ] 7→ M (s) ∈ ∂Ω be a regular parametrization of ∂Ω. For ea h x ∈ Ω and ǫ > 0 we denote by :

the boundary

t(x) = dist(x, ∂Ω)

and

Ωǫ = {x ∈ Ω; dist(x, ∂Ω) < ǫ}.

20

AYMAN KACHMAR

Then there exist a positive onstant

x ∈ Ωt 0 ,

t0 > 0 depending (s(x), t(x)) by :

on

Ω

su h that, for ea h

we an dene the oordinates

t(x) = |x − M (s(x))|, and su h that the transformation :

ψ : Ωt0 ∋ x 7→ (s(x), t(x)) ∈ S1|∂Ω|/2π × [0, t0[ is a dieomorphisim. The Ja obian of this oordinate transformation is given by :

a(s, t) = det(Dψ) = 1 − tκ(s).

(A.1)

∞ 2 To a ve tor eld A = (A1 , A2 ) ∈ C (Ω; R ), we asso iate the ve tor eld ˜2 ) ∈ C ∞ (S2 (A˜1 , A |∂Ω|/2π × [0, t0[) by the following relation :

A˜ =

A˜1 ds + A˜2 dt = A1 dx1 + A2 dx2 .

(A.2)

We get then the following hange of variable formulas.

Let u ∈ H 1 (Ω) be supported in Ωt0 . Then we have :

Proposition A.1.

(A.3)

Z

2

|(h∇ − iA)u| dx =

S1|∂Ω|/2π ×[0,t0 [

Ωt0

and (A.4)

Z

Z

|u(x)|2 dx =

h

i |(h∂t − iA˜2 )v|2 + a−2 |(h∂s − iA˜1 )v|2 a dsdt.

Z

|v(s, t)|2 a dsdt,

S1|∂Ω|/2π ×[0,ǫ0 [

Ωt0

where v(s, t) = u(ψ−1 (s, t)). We have also the relation :

whi h gives,

(∂x1 A2 − ∂x2 A1 ) dx1 ∧ dx2 = ∂s A˜2 − ∂t A˜2 a−1 ds ∧ dt, curl A˜ = (1 − tκ(s)) curl A.

We give in the next proposition a standard hoi e of gauge.

Consider a ve tor eld A = (A1 , A2 ) ∈ C ∞(Ω; R2 ) su h that curlA = 1. For ea h point x0 ∈ ∂Ω, there exist a neighborhood Vx0 ⊂ Ωt0 of x0 and a smooth real-valued fun tion φx0 su h the ve tor eld Anew := A − ∇φx0 satises : Proposition A.2.

(A.5)

t A˜1new = −t 1 − κ(s) and A˜2new = 0 in Vx0 . 2 Referen es

[1℄ S. Agmon : Le tures on exponential de ay of solutions of se ond order ellipti equations, Math. Notes, T. 29, Prin eton University Press (1982). [2℄ P. Baumann, D. Phillip, and Q. Tang : Stable nu leation for the Ginzburg-Landau model system with an applied magneti eld. Ar h. Rational Me h. Anal. 142, p. 1-43 (1998). [3℄ A. Bernoff and P. Sternberg : Onset of super ondu tivity in de reasing elds for general domains. J. Math. Phys. 39, p. 1272-1284 (1998). [4℄ C. Bolley and B. Helffer : An appli ation of semi- lassi al analysis to the asymptoti study of the super ooling eld of a super ondu ting material. Ann. Inst. Henri Poin aré, se tion Physique théorique 58 (2), p. 189-233 (1993). [5℄ V. Bonnaillie : Analyse mathématique de la supra ondu tivité dans un domaine à oins : Méthodes semi- lassique et numériques. Thèse à l'université Paris-Sud (2003). [6℄ M. Dauge and B. Helffer : Eigenvalues variation I, Neumann problem for SturmLiouville operators, J. of Dierential Equations, 104 (2), p. 243-262 (1993). [7℄ H.J. Fink and W.C.H. Joiner : Surfa e nu leation and boundary ondition in super ondu tors. Physi al review letters 23 (3) (1969). [8℄ S. Fournais and B. Helffer :On the third riti al eld in the Ginzburg-Landau theory. Preprint (2005).

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER

21

[9℄ S. Fournais and B. Helffer : A

urate eigenvalue asymptoti s for the magneti S hrödinger Lapla ian. To appear in Ann. Inst. Fourier (2004). [10℄ P.G. De Gennes : Super ondu tivity of metals and alloys. Benjamin (1966). [11℄ V. V. Grushin : On a lass of hypoellipti operators. Mat. Sb. (N.S.) 12, p. 458-475 (1970). [12℄ B. Helffer and A. Mohamed : Semi lassi al analysis of a S hrödinger operator with magneti wells. J. Fun t. Anal. 138 (1), p. 40-81 (1996). [13℄ B. Helffer and A. Morame: Magneti bottles in onne tion with super ondu tivity. J. Fun t. Anal. 185 (2), p. 604-680 (2001). [14℄ B. Helffer and X.-B. Pan : Upper riti al eld and lo ation of surfa e nu leation of super ondu tivity. Ann. Inst. H. Poin aré (Se tion analyse non-linéaire) 20 (1), p. 145-181 (2003). [15℄ A. Ka hmar : Thèse à l'université Paris-Sud. In preparation. [16℄ T. Kato : Perturbation theory for linear operators. New York, Springer-Verlag (1996). [17℄ X.-B. Pan : Super ondu ting lms in perpendi ular elds and the ee t of the De Gennes parameter, Siam J. Math. Anal. 34 (4), p. 957-991 (2003). [18℄ M. Del Pino, P.L. Felmer and P. Sternberg : Boundary on entration for eigenvalue problems related to the onset of super ondu tivity. Comm. Math. Phys. 210, p. 413-446 (2000). [19℄ K. Lu and X.-B. Pan : Ginzburg-Landau equation with De Gennes boundary ondition. J. Dierential Equations 129 (1), p. 136-165 (1996). [20℄ K. Lu and X.-B. Pan : Estimates of the upper riti al eld of the Ginzburg-Landau equations of super ondu tivity. Physi a D, p. 73-104 (1999). [21℄ K. Lu and X.-B. Pan : Eigenvalue problems of Ginzburg-Landau operator in bounded domains. J. Math. Phys. 40 (6), p. 2647-2670 (1999). [22℄ M. Reed and B. Simon : Methods of modern mathemati al physi s. A ademi press (1979). [23℄ D. Saint-James, G. Sarma, E.J. Thomas : Type II super ondu tivity. Pergamon, Oxford 1969. [24℄ M. Tinkham : Introdu tion to Super ondu tivity. M Graw-Hill In ., New York (1975). A. Ka hmar Université Paris-Sud, Bât. 425, F-91405 Orsay Université Libanaise, Hadeth, Beyrouth, Lebanon

E-mail address : ayman.ka hmarmath.u-psud.fr

The Erwin Schr¨ odinger International Institute for Mathematical Physics

Boltzmanngasse 9 A-1090 Wien, Austria

On the Ground State Energy for a Magnetic Schr¨ odinger Operator and the Effect of the De Gennes Boundary Condition

Ayman Kachmar

Vienna, Preprint ESI 1770 (2006)

Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at

January 26, 2006

ON THE GROUND STATE ENERGY FOR A MAGNETIC SCHRÖDINGER OPERATOR AND THE EFFECT OF THE DE GENNES BOUNDARY CONDITION

AYMAN KACHMAR

Abstra t. Motivated by the Ginzburg-Landau theory of super ondu tivity, we estimate in the semi- lassi al limit the ground state energy of a magneti S hrödinger operator with De Gennes boundary ondition and we study the lo alization of the ground state. We exhibit ases when the De Gennes boundary

ondition has strong ee ts on this lo alization.

1. Introdu tion Let

Ω ⊂ R2

be an open bounded domain with regular boundary. Let us onsider

a ylindri al super ondu tor sample of ross se tion erties are des ribed by the minimizers

(ψ, A)

Ω.

The super ondu ting prop-

of the Ginzburg-Landau fun tional

( f. [10, 23, 24℄) : (1.1)

Z κ2 |(∇ − iσκA)φ|2 + σ 2 κ2 |curlA − 1|2 + (|φ|2 − 1)2 dx 2 Ω Z + γ|φ(x)|2 dµ|∂Ω (x),

G(φ, A) =

∂Ω

whi h is dened for ouples

(φ, A) ∈ H 1 (Ω; C) × H 1 (Ω; R2 ).

The parameter

hara teristi of the material. A material is said to be of type I if small and it is said to be of type II when

σ

κ

κ

κ

is a

is su iently

is su iently large. The parameter

is the intensity of the applied magneti eld whi h is supposed to be onstant

and perpendi ular to

Ω.

For a minimizer (ψ, A) of the energy G , the fun tion ψ |ψ|2 measures the density of super ondu ting

is alled the order parameter and

A

Cooper ele tron paires; the ve tor eld

curlA

is alled the magneti potential and

is the indu ed magneti eld. Note that

ψ

satises the following boundary

ondition proposed by De Gennes [10℄ :

ν · (∇ − iσκA)ψ + γψ = 0,

(1.2) where

ν

is the unit outward normal of

∂Ω

and

γ ∈ R

is alled in the physi al

literature the De Gennes parameter. Note that the boundary ondition (1.2) was initially introdu ed in the theory of PDE by Robin. The physi ist De Gennes [10℄ has introdu ed the parameter

γ

in order to model

interfa es between super ondu tors and normal materials. In that ontext, γ is 1 taken to be a non-zero positive onstant and γ ( alled the extrapolation length) usually measures the penetration of the super ondu ting Cooper ele tron pairs in the normal material. Fink-Joiner [7℄ use negative values of

γ

to model the situation

when a super ondu tor is adja ent to another super ondu tor of higher transition temperature. Suppose that we have a type II super ondu tor (i.e.

κ

is large). The fun tional

G

1991 Mathemati s Subje t Classi ation. Primary 81Q10, Se ondary 35J10, 35P15, 82D55. Key words and phrases. S hrödinger operator with magneti eld, semi lassi al analysis, super ondu tivity. 1

2

AYMAN KACHMAR

has a trivial solution

(0, F)

alled a normal solution. It is then natural to study

the positivity of the hessian of

G

near a normal solution in the presen e of a strong 1 h = σκ , we have then the positivity of the quadrati form :

applied magneti eld. By dening the hange of parameter to study as

h→0

1

H (Ω) ∋ u 7→ k(h∇ − iF)uk2L2 (Ω) + hγkuk2L2 (∂Ω) − κ2 h2 kuk2L2(Ω) . Note that the semi- lassi al limit

h → 0

is now equivalent to a large eld limit

σ → +∞. For a ve tor eld and a number

A ∈ C ∞ (Ω; R2 ),

α > 0,

a regular real valued fun tion

γ ∈ C ∞(∂Ω; R)

we dene the more general quadrati form :

(1.3)

1

H (Ω) ∋ u 7→

α,γ qh,A,Ω (u)

= k(h∇

− iA)uk2L2 (Ω)

+h

1+α

Z

γ(x)|u(x)|2 dµ|∂Ω (x).

∂Ω

α,γ Observing that qh,A,Ω is semi-bounded, we onsider the self-adjoint operator assoα,γ

iated to qh,A,Ω by Friedri h's theorem. This is the magneti S hrödinger operator α,γ α,γ Ph,A,Ω with domain D(Ph,A,Ω ) dened by : (1.4)

α,γ Ph,A,Ω = −(h∇ − iA)2 , α,γ D(Ph,A,Ω ) = {u ∈ H 2 (Ω); ν · (h∇ − iA)u|∂Ω + hα γu|∂Ω = 0}.

We denote by

µ(1)(α, γ, h) the

ground state energy of

α,γ Ph,A,Ω

whi h is dened using

the min-max prin iple by : (1.5)

µ(1) (α, γ, h) :=

u∈H

α,γ qh,A,Ω (u) . 2 (Ω),u6=0 kukL2 (Ω)

inf 1

Note that due to gauge invarian e, if Ω is simply onne ted the spe trum of the α,γ Ph,A,Ω depends only on the magneti eld. For this reason we shall omit some times the referen e to the magneti potential A in our notations.

operator

In the ase when

γ≡0

(whi h orresponds to a super ondu tor surrounded by

the va uum), a lot of papers are devoted to the estimate in a semi lassi al regime α,γ of the ground state energy of Ph,A,Ω . We would like here to mention the works of Baumann-Phillips-Tang [2℄, Berno-Sternberg [3℄, del Pino-Felmer-Sternberg [18℄, Heler-Mohamed [12℄, Heler-Morame [13℄ and the re ent work of Fournais-Heler [9℄. The spe ial ase when

α=1

and

γ

is a positive onstant was onsidered by

Lu-Pan [20, 21℄. It was shown there that the ee t of the De Gennes parameter

γ

is very weak, and hen e, this ase does not orrespond to materials as those onsidered in [7℄ (where strong ee ts had been observed due to the interfa e). Note that, following the te hnique of Heler-Morame [13℄, the model operator that α,γ 1 we have to analyze is Ph,A ,Ω with h = 1, A0 = (−x2 , x1), Ω = R × R+ and γ is a 2 0

onstant. We denote by Θ(γ) the bottom of the spe trum of this model operator ( f. (2.2)). Now we state our main results.

Suppose that Ω ⊂ R2 is open, bounded, onne ted and having a smooth boundary. Suppose moreover that the magneti eld is onstant curlA = 1. α,γ Then, for α > 0 and γ ∈ C ∞ (∂Ω; R), the ground state energy of the operator Ph,A,Ω satises : Theorem 1.1.

(1.6)

µ(1)(α, γ, h) ∼ hΘ(hα−1/2 γ0 ),

where γ0 := minx∈∂Ω γ(x).

(h → 0),

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER The asymptoti s (1.6) depends strongly on

α

and

ee tively (see Propositions 2.4 and 2.5). Note that if

lim

h→0

γ0 does not always α = 12 , we get

3

appear

µ(1)(α, γ, h) = Θ(γ0 ) < 1, h

µ(1) (α,γ,h) 1 is a tually negative and tends to −∞. If 2 and γ0 < 0, h µ(1) (α,γ,h) 1 γ0 = 0 or if α > 2 , then limh→0 = Θ0 < 1. This indi ates that in the h above ases, the fun tion γ ould play a role. In the next theorem we des ribe some and when

ee t of

γ

α

0.

ξ 7→ ϕγ,ξ ∈ L2 (R+ )

Note that,

(γ, ξ + τ )ϕγ,ξ+τ (t + τ ) = H[γ, ξ] (ϕγ,ξ+τ (t + τ )) , ϕγ,ξ and then Z (1) (1) µ (γ, ξ + τ ) − µ (γ, ξ)

Taking the s alar produ t with (2.8)

∀t ∈ R+ .

integrating by parts, we get :

ϕγ,ξ+τ (t + τ )ϕγ,ξ (t) dt

R+

= ϕ′γ,ξ+τ (τ )ϕγ,ξ (0) − γϕγ,ξ+τ (τ )ϕγ,ξ (0). Re all that we have the boundary onditions :

ϕ′γ,ξ+τ (0) = γϕγ,ξ+τ (0),

ϕ′γ,ξ (0) = γϕγ,ξ (0).

Then we an rewrite (2.8) as :

Z µ(1) (γ, ξ + τ ) − µ(1)(γ, ξ) ϕγ,ξ+τ (t + τ )ϕγ,ξ (t) dt τ R+ ′ ϕγ,ξ+τ (τ ) − ϕ′γ,ξ+τ (0) ϕγ,ξ+τ (τ ) − ϕγ,ξ+τ (0) −γ · ϕγ,ξ (0). = τ τ

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER By taking the limit as

τ → 0, (1)

∂ξ µ

5

we get :

(γ, ξ) = ϕ′′γ,ξ (0) − γϕγ,ξ (0) ϕγ,ξ (0).

Finally, we make the substitutions :

ϕ′′γ,ξ (0) = ξ 2 − µ(1) (γ, ξ) ϕγ,ξ (0),

ϕ′γ,ξ (0) = γϕγ,ξ (0),

and we get the following formula,

∂ξ µ(1)(γ, ξ) = ξ 2 − µ(1) (γ, ξ) − γ 2 |ϕγ,ξ (0)|2 , F -formula.

alled usually the

An elementary analysis of the above formula permits

now to on lude the result of the Theorem ( f. [6℄). We denote by

ϕγ

the unique stri tly positive and

L2 -normalized

eigenfun tion

Θ(γ).

asso iated to the eigenvalue

We have the following relations (as in the appendix A of [13℄). Lemma 2.2.

For ea h γ ∈ R, the following relations hold : Z

(2.9)

Z

(2.10)

(t − ξ(γ))|ϕγ (t)|2 dt = 0, R+

(t − ξ(γ))2 |ϕγ (t)|2 dt =

R+

Z

(2.11)

(t − ξ(γ))3 |ϕγ (t)|2 dt =

R+

We dene now the universal onstant

M3 =

(2.12)

Z

Θ(γ) γ − |ϕγ (0)|2 , 2 4

1 (γξ(γ) + 1)2 |ϕγ (0)|2 . 6

M3 =

|ϕ0 (0)|2 . Note that (2.9) gives : 6

(t − ξ0 )3 |ϕ0 (t)|2 dt,

R+

where

ξ0 := ξ(0).

We dis uss now the regularity of the fun tions

L2 (R+ ).

γ → Θ(γ) ∈ R

and

γ 7→ ϕγ ∈

It seems for us that Kato's theory ( f. [16℄) do not apply in this ontext

at least for the reason that we do not know a priori whether the expression of the operator

H[γ] = − depends analyti ally on

γ.

d2 + (t − ξ(γ))2 dt2

Inspired by Bonnaillie [5℄, we use a modi ation of

Grushin's method [11℄ and we get the following proposition. Proposition 2.3.

The fun tions R ∋ γ 7→ Θ(γ) ∈ R and R ∋ γ 7→ ϕγ ∈ L2 (R+ )

are C . Moreover, the fun tion R ∋ γ 7→ ϕγ ∈ L∞ (R+ ) is lo ally Lips hitz. ∞

The spe i di ulty in proving the above proposition omes from the fa t that

γ . To work with χ that is equal to 1 on [0, 1] ϕ → 7 ϕ˜ = e−γtχ(t) ϕ that transforms

both the expression and the domain of the operator depend on an operator with a xed domain, we onsider a ut-o

and we apply the invertible transformation ′ the boundary ondition ϕ (0) = γϕ(0) to the usual Neumann boundary ondition ′ ϕ˜ (0) = 0 and leaves the spe trum invariant (the details are given in [15℄).

Proposition 2.4.

(2.13)

The fun tion γ 7→ Θ(γ) satises at γ = 0 : Θ′ (0) = 6M3 .

6

AYMAN KACHMAR

Proof. Let us denote the operator

H[γ, ξ(γ)] by H[γ].

We shall dene the following

trial fun tion :

u = eγt (ϕ0 + γu1 ), where

−1

u1 = (H[0] − Θ0 )

{6M3 ϕ0 + 2ϕ′0 + 2(ξ(γ) − ξ0 )(t − ξ0 )ϕ0 } .

Note that :

H[γ](u) = eγt −∂t2 + (t − ξ(γ))2 − 2γ∂t − γ 2 u.

(2.14)

Using the de omposition :

H[γ] = H[0] − 2(ξ(γ) − ξ0 )(t − ξ0 ) + (ξ(γ) − ξ0 )2 , we an rewrite (2.14) as :

(H[γ] − Θ0 − 6M3 γ) (u) = γ 2 eγt (Θ0 + 6M3 )u1 .

(2.15)

We get now by the spe tral theorem the existen e of an eigenvalue

˜ that satises Θ(γ)

the following estimate :

˜ |Θ(γ) − Θ0 − 6M3 γ| ≤ Cγ 2 ,

(2.16) for onstants

where for

H[η, ξ].

C, γ0 > 0. Using the min-max prin iple we get the following estimate : (2) µ (γ, ξ(γ)) − µ(2) (0, ξ0 ) ≤ Cγ, ∀γ ∈ [−γ0 , γ0 ],

(η, ξ) ∈ R × R, µ(2) (η, ξ)

We get nally that

denotes the se ond eigenvalue of the operator

˜ Θ(γ) = Θ(γ).

The asymptoti behavior of the eigenvalue

γ

∀γ ∈ [−γ0 , γ0 ],

Θ(γ)

with respe t to the parameter

is given in the following proposition.

Proposition 2.5.

There exist positive onstants C0 and γ0 su h that the eigenvalue

(2.17)

|Θ(γ) − 1| ≤ C0 γ exp −γ 2 ,

∀γ ∈ [γ0 , +∞[,

1 , 4γ 2

∀γ ∈] − ∞, 0[.

Θ(γ) satises :

and −γ 2 ≤ Θ(γ) ≤ −γ 2 +

(2.18)

Proof. We prove the estimate (2.17).

Theorem 2.1 we get for any

γ >0

Note that by the min-max prin iple and

:

µ(1)(0, ξ(γ)) ≤ Θ(γ) ≤ 1. The lower bound in (2.17) is proved for

µ(1)(0, ξ(γ))

by Bolley-Heler [4℄ (formula

(A.18)). The relation (2.6) gives the lower bound

Θ(γ) ≥ −γ 2 .

We get the upper bound in eγt (with γ < 0) as a trial

(2.18) from the min-max prin iple by using the fun tion fun tion for the quadrqti form dening Using Agmon's te hnique ( f. eigenfun tion

H[γ, 0].

[1℄), we get the following de ay result for the

ϕγ .

Proposition 2.6. For ea h ǫ ∈]0, 1[ there is a positive onstant Cǫ su h that, for all γ ∈ R, we have the following estimate for the eigenfun tion ϕγ :

(2.19)

2

exp − ǫ(t − ξ(γ)) ϕγ ≤ Cǫ (1 + γ− ),

1 2 H ({t∈R+ ;(t−ξ(γ))≥Cǫ })

where we use the notation γ− = max(−γ, 0).

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER

˜ R[γ]

γ ∈ R,

we re all that the regularized resolvent 2 operator dened on L (R+ ) by : For ea h

˜ R[γ]φ =

(2.20)

7

is the bounded

0 ; φ k ϕγ , −1 (H[γ] − Θ(γ)) φ ; φ⊥ϕγ ,

and extended by linearity. Again, using the Agmon's te hnique, we get that this regularized resolvent is uniformly ontinuous in suitable weighted spa es.

For ea h δ ∈]0, 1[ and η0 > 0, there exist positive onstants

Proposition 2.7.

C0 , t0 su h that,

∀u ∈ L2 (R+ ; eδ(t−ξ(γ)) dt),

∀γ ∈ [−η0 , η0],

u⊥ϕγ ,

we have,

δ(t−ξ(γ)) ˜

R[γ]u

e

(2.21)

H 1 ([t0 ,+∞[)

≤ C0 eδ(t−ξ(γ)) u

L2 (R+ )

.

3. Proof of Theorem 1.1

In this se tion we prove Theorem 1.1 by omparing with the basi model introdu ed in the pre eding se tion. We introdu e a oordinate system boundary

∂Ω

∂Ω

where

t

measures the distan e to

∂Ω

and

s

(s, t)

near the

measures the distan e in

( f. Appendix A).

(Upper bound) Under the hypothesis of Theorem 1.1, there exist positive onstants C and h0 su h that, ∀h ∈]0, h0], we have : Proposition 3.1.

µ(1) (α, γ, h) ≤ hΘ hα−1/2(γ0 + Ch1/2 ) + Ch3/2.

(3.1)

1 2 and γ0 > 0. In this ase the (1) estimate (3.1) reads simply by omparison with the rst eigenvalue λ (h) of the (1) Diri hlet realization and by using the estimate of λ (h) ( f. [12℄) and (2.17). 1 We suppose now that γ0 ≤ 0 if α < . Consider a point x0 ∈ ∂Ω su h that 2 γ(x0 ) = γ0 . We suppose that x0 = 0 in the oordinate system (s, t) near the Proof. We start with the easy ase when

boundary ( f. fun tion

uh,α

Appendix A).

α

0

and

k∈N

ϕη

in

:

tk |ϕη (t)|2 dt ≤ Ck,δ h−δk ,

∀h ∈]0, h0].

R+

After a hange of variables, thanks to (3.5) and to the de ay of

ϕη ,

formula (A.3)

gives the following upper bound :

α,˜ γ0 (uh,α ) ≤ hΘ(η) + Ch3/2, qh,A,Ω

∀h ∈]0, h0].

ϕη (Proposition 2.6), we obtain that the L2 1 as h → 0. The appli ation of the min-max (3.1).

Using formula (A.4) and the de ay of norm of

uh,α

is exponentially lose to

prin iple permits now to on lude

When α ∈] 12 , 1[ and the fun tion γ is not onstant, Proposition 3.1 gives the following upper bound for the eigenvalue µ(1)(α, γ, h), thanks to Proposition 2.4, Remark 3.2.

µ(1) (α, γ, h) ≤ hΘ0 − 6M3 γ0 hα+1/2 + O(hinf(3/2,2α)). This upper bound is a tually an asymptoti expansion as h → 0 (see Remark 5.13).

(Lower bound) Under the hypothesis of Theorem 1.1, there exist positive onstants C, C ′ and h0 su h that, ∀h ∈]0, h0], we have :

Proposition 3.3.

µ(1) (α, γ, h) ≥ hΘ hα−1/2γ0 (1 + C ′ h1/4 ) − Ch5/4 .

(3.6)

Proof. We follow the te hnique of [13℄ and we ompare with the basi model by

means of a partition of unity. We introdu e a partition of unity

(χj )

of

R2

that

satises

X

|χj |2 = 1,

j

where for

z ∈ R2

X

|∇χj |2 < +∞,

suppχj ⊂ D(zj , 1),

j

and

r > 0, we

denote by

D(z, r)

the disk of enter

z

and raduis

r.

We introdu e now the s aled partition of unity :

χhj (z) := χj (ǫ0 hρ z), where

ǫ0

and

ρ

∀z ∈ R2 ,

are two positive numbers to be hosen suitably. Note that

(χhj ) now

satises :

X

(3.7)

|χhj |2 = 1,

j

X

(3.8)

−2ρ |∇χhj |2 ≤ Cǫ−2 , 0 h

j

(3.9) where

supp

C

χhj ⊂ Qhj := D(zjh , ǫ0 hρ ),

is a positive onstant. We an also suppose that :

(3.10)

either supp

χhj ∩ ∂Ω = ∅

or

zjh ∈ ∂Ω.

Note that the alternative in (3.10) permits us to write the sum in (3.7) under the form :

X 1A tually we

=

X int

+

X

,

bnd

shall need this de ay only when α < 1/2 and γ0

< 0.

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER where the summation over int means that the support of

χhj

9

do not meet the

boundary while that over bnd means the onverse. We have now the following de omposition formula :

α,γ qh,A (u) =

(3.11)

X

α,γ qh,A (χhj u) − h2

j

X

k |∇χhj| uk2 ,

∀u ∈ H 1 (Ω),

j

usually alled in other ontexts the IMS formula.

We have now to bound from

below ea h of the terms on the right hand side of (3.11). Note that (3.8) permits to estimate the ontribution of the last term in (3.11) :

h2

(3.12)

X

2−2ρ k |∇χhj| uk2 ≤ Cǫ−2 kuk2 , 0 h

∀u ∈ H 1 (Ω).

j

If

χhj

is supported in

Ω,

then we have :

α,γ qh,A (χhj u) =

Z

R2

|(h∇ − iA)χhj u|2 dx.

Sin e the lowest eigenvalue of the S hrödinger operator with onstant magneti 2 eld in R is equal to 1, we get :

α,γ qh,A (χhj u) ≥ h

(3.13)

We have now to estimate

Z

Ω

|χhj u|2 dx,

∀u ∈ H 1 (Ω).

α,γ qh,A,Ω (χhj u) when χhj

meets the boundary. It is in this ase α,γ h that we see the ee t of the boundary ondition. Note that, by writing qh,A,Ω (χj u) in the boundary oordinates, thanks to Proposition A.1, there exist a positive

onstant

C1

independent of

h

and

(3.14) Z

Ω

j

su h that :

|(h∇ − iA)χhj u|dx ≥ (1 − C1 ǫ0 hρ )

where

A˜

Z

R×R+

is the ve tor eld asso iated to

A

˜ h u|2 dsdt, |(h∇ − iA)χ j

∀u ∈ H 1 (Ω),

by (A.2).

By a gauge transformation, we get a new magneti potential

A˜new,j

satisfying :

A˜new,j = A˜ − ∇φhj , A˜new,j (zjh ) = 0, ˜ j Anew,j (w) − A˜lin (w) ≤ C|w|2,

(3.15)

Ajlin :=

w = (s, t),

1 2 (−t, s) is the linear magneti potential and C > 0 is a onstant independent of h and j . By Cau hy-S hwarz inequality, we get for an arbitrary

where

θ

positive number

Z

and for a positive onstant

R×R+

C˜

independent of

h

and

j

˜ h u|2 dsdt |(h∇ − iA)χ j

! 2 φhj h ≥ (1 − h ) χ u dsdt (h∇ − iA˜lin ) exp −i h j R×R+

˜ −2θ |w|2 χh u 2 . −Ch j 2θ

Now we get a onstant

Z

C2 > 0

su h that after putting :

γ˜j =

γ(zjh ) − C2 hρ , 1 − C2 h2θ − C2 ǫ0 hρ

:

10

AYMAN KACHMAR

the estimate (3.14) reads nally : (3.16)

α,γ qh,A,Ω (χhj u)

ρ

≥

(1 − C2 ǫ0 h − C2 h

2θ

φhj h χ u exp −i h j

α,˜ γ )qh,A˜jj ,R×R + lin

!

−C2 ǫ20 h4ρ−2θ kχhj uk2 . Note that this permits to ompare with the model operator and to get nally the energy estimate : (3.17)

o n α,γ qh,A,Ω (χhj u) ≥ (1 − C2 ǫ0 hρ − C2 h2θ )hΘ(hα−1/2 γ˜j ) − C2 ǫ20 h4ρ−2θ kχhj uk2L2 (Ω) .

We substitute now the estimates (3.6), (3.12), and (3.13) in (3.11) and get nally : (3.18)

α,γ qh,A,Ω (u) ≥ h

XZ

Ω

int

−C h As

γ0

(3.19)

4ρ−2θ

γ,

is the minimum of

α,γ qh,A,Ω (χhj u)

XZ |χhj u|2 dx |χhj u|2 dx + hΘ hα−1/2γ˜j 2−2ρ + ǫ−2 0 h

+h

brd Ω 1+2θ

kuk2 ,

+h

∀u ∈ H 1 (Ω).

we an get instead of (3.16) the estimate :

ρ

≥

1+ρ

(1 − C2 ǫ0 h − C2 h

2θ

φhj h exp −i χ u h j

α,˜ γ0 )qh, ˜j ,R×R+ A lin

!

−C2 ǫ20 h4ρ−2θ kχhj uk2 , where

γ˜0

is dened by :

γ˜0 :=

γ0 . 1 − C2 h2θ − C2 ǫ0 hρ

We then get instead of (3.18) : (3.20)

α,γ qh,A,Ω (u) ≥ h

XZ int

−C

Ω

XZ |χhj u|2 dx + hΘ hα−1/2γ˜0 |χhj u|2 dx

2−2ρ h4ρ−2θ + ǫ−2 0 h

+ h1+ρ + h

brd Ω 1+2θ

kuk2 ,

∀u ∈ H 1 (Ω).

The advantage of (3.18) is that it gives a lower bound of the quadrati form

α,γ qh,A,Ω

in terms of a potential, see however Se tion 4. We hoose now

ǫ0 = 1, ρ = 3/8 and θ = 1/8

in (3.20) and obtain (3.6) by applying

the min-max prin iple.

Proof of Theorem 1.1. Using Propositions 2.4 and 2.5, the proof of Theorem 1.1

now follows from Propositions 3.1 and 3.3.

4. Lo alization of the ground state We work in this se tion under the hypotheses of Theorem 1.2. Due to Theorem 1.1 we have in this ase that

lim

(4.1)

h→0

µ(1)(α, γ, h) < 1. h

Then this gives, by following the same lines of the proof of Theorem 6.3 in [13℄, the following proposition. Theorem 4.1.

Under the hypotheses of Theorem 1.2, there exist positive onstants

α,γ δ, C, h0 su h that, for all h ∈]0, h0], a ground state uα,γ,h of the operator Ph,A,Ω

satises : (4.2)

exp δd(x, ∂Ω) uα,γ,h ≤ Ckuα,γ,hkL2 (Ω) ,

2

β h L (Ω)

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER

11

and

exp δd(x, ∂Ω) uα,γ,h ≤ Ch− min(1/2,β)kuα,γ,h kL2 (Ω),

1 β h H (Ω)

(4.3)

where β = 1 − α if γ0 < 0 and α < 12 , and β = 1/2 otherwise.

Φ:

2

Φ Φ α,γ (1)

qh,A,Ω exp β uα,γ,h = µ (α, γ, h) exp β uα,γ,h

2 h h L (Ω)

2

Φ 2−2β

+h

|∇Φ| exp hβ uα,γ,h 2 . L (Ω)

Proof. Integrating by parts, we get for any Lips hitz fun tion

(4.4)

Let

α,γ uα,γ,h. Using the lower bound for qh,A,Ω (u) in (3.20) together (1) bound for µ (α, γ, h) in (3.1), we get from (4.4) : XZ 1−2ρ 1 − Θ(hα−1/2 (γ0 + Ch1/2)) − Cǫ−2 − Ch4ρ−2θ−1 0 h

u = exp

the upper

Φ hβ

int

with

Ω

−Chmin(ρ,2θ) − h1−2β |∇Φ|2 × |χhj u|2 dx XZ ≤ Θ(hα−1/2 γ˜0 ) − Θ(hα−1/2 (γ0 + Ch1/2 ))+ Ω

brd

We hoose also

θ>0

ρ=β

hmin(4ρ−2θ−1,1−2ρ) + h1−2β |∇Φ|2 × |χhj u|2 dx. χhj is supported in a 4ρ − 2θ − 1 > 0 and we dene

so that ea h

su h that

disk of radius

nally the fun tion

Φ(x) = δ max dist(x, ∂Ω); ǫ0 h where

ǫ 0 hβ .

β

δ is a positive onstant to be hosen appropriately.

We hoose

Φ

by :

,

Note that

1−Θ(hα−1/2 γ˜0 )

de ays in the following way :

∃ C 0 , h0 > 0

s. t., ∀h

∈]0, h0],

1 − Θ(hα−1/2 γ˜0 ) ≥ C0 h2α−1 1 − Θ(hα−1/2 γ˜0 ) ≥ C0 Thus we an hoose

ǫ0

and

δ

if

γ0 < 0

and

α

0, ∃Cε > 0, ∀x ∈ (γ −γ0 )−1 ([ǫ, +∞[), Θ hα−1/2γ˜(x) −Θ hα−1/2γ˜0 > Cε . In the ase α−1/2

In the ase

α

0, ∃Cε > 0, ∀x ∈ (γ − γ0 )−1 ([ǫ, +∞[), Θ hα−1/2 γ˜(x) − Θ hα−1/2 γ˜0 > Cε h1−2α.

in the form :

Φh (x) = δχ(dist(x, ∂Ω))dist (x, {x ∈ ∂Ω; γ(x) = γ0 }) , δ an appropriate positive onstant and χ is the same as in (3.4), we get ǫ > 0 the following de ay near the boundary : 2 Z Φh ǫ 2 (4.6) exp hβ uα,γ,h dx ≤ Cǫ exp hβ kuα,γ,hk , ∀h ∈]0, hǫ], dist(x,∂Ω) 12 and M3 ( 12 , γ) = 41 1 − 31 (γξγ + 1)2 |ϕγ (0)|2 . (5.2)

5.2. Upper bound. We onstru t a trial fun tion by means of boundary oordinates

(s, t)

(s, t)

and we denote by

near a point

z0 ∈ ∂Ω. We suppose that z0 = 0 in the oordinate system a0 = 1 − tκ0 and η(z0 ) = hα−1/2 γ(z0 ). We then dene the

trial fun tion :

ξη(z0 ) s vh (s, t), uh = exp −i 1/2 h

(5.3) with

−1/2

vh (s, t) = h−5/16 a0

(5.4)

and where the fun tions

χ

and

f

(t)ϕη(z0 ) h−1/2 t χ(t) · f h−1/4 s ,

are as in (3.4).

We ontinue now to work in the spirit of [13℄. We work with the gauge given in Proposition A.2. An expli it al ulation, thanks to the de ay of

ϕη(z0 )

(Proposition

2.6), gives the following lemma.

Under the above notations, for ea h α ∈ [ 12 , 1] and γ ∈ C ∞ (∂Ω; R), there exist positive onstants C, h0 su h that, ∀h ∈]0, h0], we have the following estimate : Lemma 5.3.

(5.5)

Z α,γ(z0 ) h h h H U ϕη(z0 ) × U ϕη(z0 ) dt ≤ Ch13/8, qh,A,Ω (uh ) − R+

where the operators H h and U h are dened respe tively by :

2 κ0 1/2 t 1 − t H h = a−2 − h ξ(η(z )) − h2 a−1 0 0 0 ∂t (a0 ∂t ), 2 (U h g)(t) = h−1/4 g(h−1/2 t),

∀g ∈ L2 (R+ ).

We give few details of the proof. Note that in the support of

a = a0 + O(h5/8 ), 2A tually,

uh

we have

2

t A˜1 = −t 1 − κ0 + O(h9/8 ). 2

if z0 is a point of maximum of κ, the remainder is better and of order O(h3/4 ) for the rst term.

14

AYMAN KACHMAR

Then, thanks to formula (A.3), we get modulo

α,γ(z ) qh,A,Ω0 (uh )

Z

a0

R×R+

(

O(h13/8 )

:

=

|h∂t vh |

2

+ a−2 0

+h1+α γ(z0 )|vh (0)|2 .

Integrating with respe t to

s,

2 ) t 1/2 t 1 − κ0 − h ξ(η(z0 )) vh dsdt 2

we get after an integration by parts the estimate in

(5.5). Similar omputations give also the following lemma. Lemma 5.4.

Under the hypotheses of Lemma 5.3, there exist positive onstants

C, h0 su h that, ∀h ∈]0, h0], we have :

h

(H − H h − H h )U h ϕη(z ) 2 (5.6) 0 1 0 L (R

+)

≤ Ch2 ,

where the operators H0h and H1h are dened respe tively by : 2 H0h = −h2 ∂t2 + t − h1/2 ξ(η(z0 )) ,

2 H1h = 2tκ0 t − h1/2 ξ(η(z0 )) − κ0 t2 t − h1/2 ξ(η(z0 )) + h2 κ0 ∂t .

(1) The next lemma permits to on lude an upper bound for the eigenvalue µ (α, γ, h).

Under the above notations, there exist positive onstants C, h0 su h that, when h ∈]0, h0], we have the following estimate :

Lemma 5.5.

α,γ qh,A,Ω (uh ) − {Θ(η(z0 )) − 2M3 (α, γ)κ0 h3/2 }kuh k2L2 (Ω) ≤ Chǫα ,

where ǫα = inf(13/8, 2α + 12 ) for α >

Proof. Noti e that in the support of

1 2

and ǫ1/2 = 13/8.

uh

we have

γ(z) = γ(z0 ) + O(h1/8 ).

Then

this gives :

α,γ(z )

α,γ qh,A,Ω (uh ) − qh,A,Ω0 (uh ) = O(h9/8+α ). In view of Lemmas 5.3 and 5.4, we get the following estimate :

(5.7)

Z h α,γ h h h H1 + H0 U ϕη(z0 ) × U ϕη(z0 ) dt ≤ Ch13/8. qh,A,Ω (uh ) − R+

We note also that we have the following relations :

o n 2 (U h )⋆ H0h U h = h −∂t2 + (t − ξ(η(z0 ))) ,

(5.8)

(U h )⋆ H1h U h = κ0 h3/2 H1 ,

(5.9) where the operator

H1

is dened by :

3

By dening (5.10)

H1 = (t − ξ(η(z0 ))) − ξ(η(z0 ))2 (t − ξ(η(z0 ))) + ∂t . R K3 (α, h) := R+ H1 ϕη(z0 ) · ϕη(z0 ) dt, the estimate (5.7) reads o n α,γ qh,A,Ω (uh ) − hΘ(η(z0 )) + K3 (α, h)κ0 h3/2 ≤ Ch13/8.

1 1 , we get by using (2.2) that K3 ( , h) 2 2 thanks to Propositions 2.3 and 2.4, we get that

Now, for

α=

= −2M3 ( 12 , γ(z0 )).

K3 (α, h) = −2M3 + O(h2α−1 ).

as :

For

α>

1 , 2

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER Finally, the de ay of

lose to

1.

ϕη(z0 )

in Proposition 2.6 gives that

kuh kL2(Ω)

15

is exponentially

This a hieves the proof of the lemma.

The min-max prin iple gives now, thanks to Lemma 5.5, an upper bound for

µ(1) (α, γ, h).

Under the hypothesis of Theorem 5.1 we take

z0

su h that

(κ − 3γ)(z0 ) = (κ − 3γ)max and we use the estimate in (2.16). Under the hypothesis of Theorem 5.2 we hoose

z0

su h that

κ(z0 ) = κmax .

5.3. Lower bound. As in the proof of Proposition 3.3, we onsider a standard 3 χ 2 s aled partition of unity j,h1/6 j∈Z2 of R that satises :

X

(5.11)

|χj,h1/6 (z)|2 = 1,

j∈J

X

|∇χj,h1/6 (z)|2 ≤ Ch−1/3,

j∈J

(5.12)

supp

χj,h1/6 ⊂ jh1/6 + [−h1/6 , h1/6]2 .

We dene the following set of indi es :

1 Jτ(h) := {j ∈ Z2 ; suppχj,h1/6 ∩ Ω 6= φ, dist(supp χj,h1/6 , ∂Ω) ≤ τ (h)}, where the number

τ (h)

is dened by :

τ (h) = hδ ,

(5.13) and the number

δ

with

1 1 ≤δ≤ , 6 2

will be hosen in a suitable manner.

We onsider also another s aled partition of unity in

2 2 ψ0,τ(h) (t) + ψ1,τ(h) (t) = 1,

(5.14)

supp ψ0,τ(h) ⊂ [

(5.15)

Note that, for ea h

preted, by means of boundary

ψ1,τ(h) (t)χj,h1/6 (s, t)

′ |ψj,τ(h) (t)| ≤

:

C , τ (h)

j = 0, 1,

τ (h) ]. 10 the fun tion ψ1,τ(h) (t)χj,h1/6 (s, t) ould be inter oordinates, as a fun tion in Ω. Moreover, ea h

τ (h) , +∞[, 20

1 j ∈ Jτ(h) ,

R

supp ψ1,τ(h) ⊂] − ∞,

is supported in a re tangle

K(j, h) =] − h1/6 + sj , sj + h1/6 [×[0, hδ [ ∂Ω. The role of δ is then to ontrol the size K(j, h). Due to the exponential de ay of a ground near

of the width of ea h re tangle state away from the boundary

(Theorem 4.1), we get the following lemma.

Suppose that α > 12 . Under the above notations, a L2 -normalized α,γ satises : ground state uα,γ,h of the operator Ph,A,Ω

Lemma 5.6.

(5.16)

X α,γ (1) qh,A,Ω χj,h1/6 ψ1,τ(h) uγ,h − µ (α, γ, h) ≤ Ch5/3 . 1 j∈Jτ(h)

The proof of (5.16) follows the same lines of that in [13℄ (formulas (10.4), (10.5) and (10.6)).

1 j ∈ Jτ(h) , we dene a unique point zj ∈ ∂Ω by the relation s(zj ) = sj . We t j denote then by κj = κ(zj ), aj (t) = 1 − κj t, A (t) = −t 1 − κj , and γj = γ(zj ). 2 We onsider now the k -family of one dimensional dierential operators : For ea h

(5.17)

3The

j 2 Hh,j,k = −h2 a−1 j ∂t (aj ∂t ) + (1 + 2κj t)(hk − A ) ,

only dieren e is that we hoose the partitions support in squares rather than dis s.

16

AYMAN KACHMAR α,γ,D k is a real parameter. We denote by Hh,j,k the δ ]0, h [; aj (t)dt of Hh,j,k whose domain is given by :

where

2

L

self-adjoint realization on

γ,D D(Hh,j,k ) = {v ∈ H 2 (]0, hδ [); v′ (0) = hα γ˜j v(0), v(hδ ) = 0}.

(5.18)

The parameter

γ˜j

is dened by :

γ˜j = γj + ε(h), with

ε(h) = 0

if the fun tion

γ

is onstant and

ε(h) = O(h1/6 )

otherwise. We then

introdu e :

γ,D µj1 (α, γ, h) := inf inf Sp(Hh,j,k ).

(5.19)

k∈R

We have now the following Lemma whi h is proved in [15℄. Lemma 5.7.

For ea h α ∈ [ 12 , +∞[, we have under the above notations : (1)

µ

(5.20)

(α, γ, h) ≥

inf

1 j∈Jτ(h)

!

µj1 (α, γ, h)

+ O(h5/3 ).

Again the proof follows the same lines of [13℄ (Se tion 11), but let us explain α,γ qh,A,Ω (ψ1,τ(h) χj,h1/6 uα,γ,h) in boundary oordinates. We work with the lo al hoi e of gauge given in Proposition A.2. briey the main steps. We express ea h term

We expand now all terms by Taylor's formula near

(sj , 0).

After ontrolling the

remainder terms, thanks to the exponential de ay of the ground states away from the boundary, we apply a partial Fourier transformation in the tangential variable

s

and we get nally the result of the lemma.

We have now to nd, uniformly over k ∈ R, a lower bound for the rst eigenvalue α,γ,D 1/2 of the operator Hh,j,k . Putting β = κj , ξ = −h k and η = γ˜j , we get by a s aling argument :

µj1 (k; α, γ, h)

α,η,D µj1 (k; α, γ, h) = hµ1 (Hh,β,ξ ), where

α,η,D µ1 (Hh,β,ξ )

is the rst eigenvalue of the one dimensional operator :

α,η,D Hh,β,ξ = −∂t2 + (t − ξ)2 + βh1/2 (1 − βh1/2 t)−1 ∂t 2 t2 t4 +2βh1/2 t t − ξ − βh1/2 − βh1/2 t2 (t − ξ) + β 2 h , 2 4

(5.21)

whose domain is dened by :

α,η,D D(Hh,β,ξ ) = {u ∈ H 2 (]0, hδ−1/2[); u′ (0) = hα−1/2 η u(0), u(hδ−1/2 ) = 0}.

η, β ∈] − M, M [ and M a given positive onstant), α,η,D ξ ∈ R, a lower bound for the eigenvalue µ1 (Hh,β,ξ ). The

We have then to nd (when uniformly with respe t to

min-max prin iple gives the following preliminary lo alization of the spe trum of α,η,D Hh,β,ξ :

the operator

For ea h M > 0 and α ∈ [ 21 , +∞[, there exist positive onstants C, h0 su h that, ∀η, β ∈] − M, M [, ∀ξ ∈ R, we have when h ∈]0, h0], Lemma 5.8.

(5.22)

α,η,D α,η,D α,η,D µj (Hh,β,ξ ) − µj (H0,ξ ) ≤ Ch2δ−1/2 1 + µj (H0,ξ ) ,

where, for an operator T having a ompa t resolvent, µj (T ) denotes the in reasing sequen e of eigenvalues of T .

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER Remark 5.9.

17

Note that the min-max prin iple gives now that α,η,D µj (H0,ξ ) ≥ µ(j)(hα−1/2 η, ξ),

where, for η˜ ∈ R, µ(j) (˜ η , ξ) is the in reasing sequen e of eigenvalues of the operator H[η, ξ] introdu ed in Se tion 2.3. The following lemma deals with the ase when

ξ

is not lo alized very lose to

ξ(hα−1/2 η).

Suppose that δ ∈]1/4, 1/2[. For ea h α ≥ 12 , there exists ρ ∈]0, δ− 14 ], and for ea h M > 0, there exist positive onstants ζ, h0 > 0 su h that,

Lemma 5.10.

∀ξ su h that |ξ − ξ(hα−1/2 η)| ≥ ζhρ ,

∀η, β ∈] − M, M [,

we have, α,η,D µ1 (Hh,β,ξ ) ≥ Θ(hα−1/2 η) + h2ρ ,

(5.23)

∀h ∈]0, h0].

µ(1) (hα−1/2η, ξ), thanks to Lemma 5.8 1 and Remark 5.9. We start with the ase when α = 2 and η ∈] − M, M [. Writing (1) Taylors formula up to the se ond order for the fun tion ξ 7→ µ (η, ξ), we get positive onstants θ, C1 su h that when |ξ − ξ(η)| ≤ θ, we have : Proof. It is su ient to obtain (5.23) for

µ(1) (η, ξ) ≥ Θ(η) + C1 |ξ − ξ(η)|2 . Then by taking

ζ

C1 ζ > ζ0 , where ζ0 > 1 ζhρ ≤ |ξ − ξ(η)| ≤ θ,

su h that

appropriately, we get when

is a onstant to be hosen

µ(1) (η, ξ) ≥ Θ(η) + ζ0 h2ρ, where

ρ

is also a positive onstant to be hosen later. When

a positive onstant

ǫθ

|ξ − ξ(η)| > θ,

we get

su h that :

µ(1) (η, ξ) ≥ Θ(η) + ǫθ . Then by hoosing

h0

su h that

ζ0 hρ0 < ǫθ , we get for |ξ −ξ(η)| ≥ ζhρ

and

h ∈]0, h0] :

µ(1) (η, ξ) ≥ Θ(η) + ζ0 h2ρ.

(5.24)

α > 1/2. η ∈] − M, M [,

We treat now the ase when uniformly for all

ξ∈R

and

Note that the min-max prin iple gives

µ(1)(hα−1/2 η, ξ) ≥ (1 − Cη−hα−1/2 )µ(1)(0, ξ). η = 0 and ρ = inf(δ − 41 , α − 12 ), we an hoose ζ0 |ξ − ξ0 | ≥ ζhρ :

Then using (5.24) for so that we have for

µ(1) (hα−1/2 η, ξ) ≥ Θ(hα−1/2 η) + To nish the proof, we repla e

ξ0

large enough

ζ0 ρ h . 2

ξ(hα−1/2 η) getting an error of order O(hα−1/2 ).

by

α−1/2 η)| < ζhρ . |ξ − ξ(h

Now we deal with the ase when look for a formal solution (5.25) in the form :

α,η µ, fh,β,ξ

α,η α,η α,η Hh,β,ξ fh,β,ξ = µfh,β,ξ ,

Let

η˜ = hα−1/2 η .

of the spe tral problem

α,η fh,β,ξ

′

α,η (0) = hα−1/2fh,β,ξ (0),

2

(5.26)

µ = d0 + d1 (ξ − ξ(˜ η )) + d2 (ξ − ξ(˜ η )) + d3 h1/2 ,

(5.27)

α,η fh,β,ξ = u0 + (ξ − ξ(˜ η )) u1 + (ξ − ξ(˜ η )) u2 + h1/2 u3 ,

2

We

18

AYMAN KACHMAR

d0 , d1, d2 , d3 and the fun tions u0 , u1, u2 , u3 are to be deterα,η,D Hh,β,ξ in powers of (ξ −ξ(˜ η )) and then we identify the terms of orders (ξ − ξ(˜ η ))j (j = 0, 1, 2) and h1/2 . We then

where the oe ients

mined. We expand the operator the oe ients of

obtain for the oe ients :

(5.28)

d0 = Θ(˜ η ), u0 = ϕη˜ ˜ η ] {(t − ξ(˜ d1 = 0, u1 = 2R[˜ η ))ϕη˜ } R d2 =: d2 (α, η) = 1 − 2 R+ (t − ξ(˜ η ))ϕη˜ u1 dt, o n ˜ η ] 4 (t − ξ(˜ ˜ η ] [(t − ξ(˜ u2 = R[˜ η )) R[˜ η )) ϕη˜] − d2 R d3 =: d3 (α, η) = β R+ ϕη˜ ∂t + (t − ξ(˜ η ))3 ϕη˜dt, i h 3 u = −R[˜ ˜ η] β ∂t + (t − ξ(˜ η )) − ξ(˜ η )2 (t − ξ(˜ η )) − d3 u0 . 3

Using the fun tion

α,η t χ( hδ−1/2 )fh,β,ξ

(where

χ is the same as in (3.4)) as a quasi-mode,

we get by the spe tral theorem, thanks to the de ay results in Propositions 2.6 and 2.7 and to the lo alization of the spe trum in Lemma 5.8, the following lemma.

Suppose that δ ∈] 41 , 12 [. For ea h M > 0 and α ∈ [ 21 , 1], there exist positive onstants C > 0, h0 su h that, Lemma 5.11.

∀η, β ∈] − M, M [,

∀ξ su h that |ξ − ξ(˜ η )| ≤ ζhρ ,

we have when h ∈]0, h0], (5.29)

o n α,η,D η ) + d2 (α, η)(ξ − ξ(˜ η ))2 + d3 (α, η)h1/2 µ1 (Hh,β,ξ ) − Θ(˜ i h ≤ C h1/2 |ξ − ξ(˜ η )| + hδ+1/2 .

where d2 (α, η) and d3 (α, η) are dened by (5.28) respe tively.

Hen e we have obtained by this analysis a lower bound for the rst eigenvalue µ(1) (α, γ, h). We omplete the pi ture by showing that the term d2 (α, η) is positive. Lemma 5.12.

h0 su h that :

For ea h α ∈ [ 12 , +∞[ and M > 0, there exists a positive onstant d2 (α, η) > 0,

∀h ∈]0, h0],

∀η ∈] − M, M [.

α = 12 . 1 If α > , we repla e d2 (α, η) by its approximation up to the rst order, thanks to 2 Proposition 2.3, and we obtain that : Preuve. It is a tually su ient to prove the on lusion of the lemma when

1 d2 (α, η) = d2 ( , 0) + O(hα−1/2 ), 2 whi h gives the on lusion of the lemma. 1 For the parti ular ase α = 2 , we show that :

whi h is stri tly positive.

1 d2 ( , η) = ∂ξ2 µ(1) (η, ·) (ξ(η)) 2

We are now able to on lude the asymptoti s given in Theorems 5.1 and 5.2. 5 1 δ = 12 . When α > 2 we repla e Θ(˜ η ) and d3 by their approximations up to the se ond and rst orders respe tively, thanks to Propositions 2.4 and First we hoose

MAGNETIC SCHRÖDINGER OPERATOR AND THE DE GENNES PARAMETER 2.3. For

1 we get by (2.11) that 2

α=

d3

is indeed equal to

19

−2M3 ( 12 , γ).

When α ∈] 21 , 1[ and the fun tion γ is not onstant, we get from the above analysis that the upper bound in Remark 3.2 is a tually an asymptoti expansion. That is, as h → 0, we have, Remark 5.13.

µ(1) (α, γ, h) = hΘ0 + 6M3 γ0 hα+1/2 + O(hinf(3/2,2α)). α,γ We get also that the quadrati form qh,A,Ω

an be bounded from below by means of a potential W : α,γ qh,A,Ω (u) ≥

Z

W (x)|u(x)|2dx,

∀u ∈ H 1 (Ω),

Ω

where W is dened for some positive onstant C0 by : W (x) =

h hΘ0 + 6M3 γ(x)h

α+1/2

− C0 h

inf(3/2,2α)

; if ; if

dist(x, ∂Ω) > h1/6 dist(x, ∂Ω) < h1/6 .

Then, as in Se tion 4, we get by Agmon's te hnique that a ground state de ays exponentially away from the boundary points where γ is minimum and hen e we have ompleted the proof of Theorem 1.2. Note also that the above analysis permits, under the hypotheses α,γ of Theorems 5.1 and 5.2, to bound the quadrati form qh,A,Ω from below using a potential W dened either by means of the fun tion κ − 3γ (when α = 1) or by the s alar urvature κ (when γ is onstant). Then, by using Agmon's te hnique, we nish the proofs of Theorems 1.3 and 1.4. Remark 5.14.

6. Con lusion We have extended in Theorems 5.1 and 5.2 the expansion announ ed by Pan [17℄ in the parti ular ase when is a spe i di ulty when

γ

α=1

and

γ

is a positive onstant. However, there

is negative and the systemati analysis in the spirit

of [13℄ had allowed us to understand the role of the boundary ondition imposed by De Gennes. We have not been able to obtain the lo alization of the groundstate when

α < 1/2

and

γ0 < 0.

This is strongly related to the question of the lo al-

ization of the ground state of the Diri hlet realization of the S hrödinger operator with onstant magneti eld whi h is open. Finally, in the spirit of [8, 14, 20℄, we hope to apply this analysis to the onset of super ondu tivity and to omplete the analysis of [19℄ ( f. [15℄). A knowledgements. I am deeply grateful to Professor B. Heler for the on-

stant attention to this work, his help, advi es and omments. I would like also to thank S. Fournais for his attentive reading. I a knowledge the ESI at Vienna where I found good onditions to prepare a part of this work and the ESF whi h had supported the visit under the SPECT program. This work had been done by the nan ial support of the

Agen e universitaire de la fran ophonie

(AUF).

Appendix A. Coordinates near the boundary

We re all in this appendix well-known oordinates that straightens a portion of |∂Ω| |∂Ω| ∂Ω. Let s ∈] − 2 , 2 ] 7→ M (s) ∈ ∂Ω be a regular parametrization of ∂Ω. For ea h x ∈ Ω and ǫ > 0 we denote by :

the boundary

t(x) = dist(x, ∂Ω)

and

Ωǫ = {x ∈ Ω; dist(x, ∂Ω) < ǫ}.

20

AYMAN KACHMAR

Then there exist a positive onstant

x ∈ Ωt 0 ,

t0 > 0 depending (s(x), t(x)) by :

on

Ω

su h that, for ea h

we an dene the oordinates

t(x) = |x − M (s(x))|, and su h that the transformation :

ψ : Ωt0 ∋ x 7→ (s(x), t(x)) ∈ S1|∂Ω|/2π × [0, t0[ is a dieomorphisim. The Ja obian of this oordinate transformation is given by :

a(s, t) = det(Dψ) = 1 − tκ(s).

(A.1)

∞ 2 To a ve tor eld A = (A1 , A2 ) ∈ C (Ω; R ), we asso iate the ve tor eld ˜2 ) ∈ C ∞ (S2 (A˜1 , A |∂Ω|/2π × [0, t0[) by the following relation :

A˜ =

A˜1 ds + A˜2 dt = A1 dx1 + A2 dx2 .

(A.2)

We get then the following hange of variable formulas.

Let u ∈ H 1 (Ω) be supported in Ωt0 . Then we have :

Proposition A.1.

(A.3)

Z

2

|(h∇ − iA)u| dx =

S1|∂Ω|/2π ×[0,t0 [

Ωt0

and (A.4)

Z

Z

|u(x)|2 dx =

h

i |(h∂t − iA˜2 )v|2 + a−2 |(h∂s − iA˜1 )v|2 a dsdt.

Z

|v(s, t)|2 a dsdt,

S1|∂Ω|/2π ×[0,ǫ0 [

Ωt0

where v(s, t) = u(ψ−1 (s, t)). We have also the relation :

whi h gives,

(∂x1 A2 − ∂x2 A1 ) dx1 ∧ dx2 = ∂s A˜2 − ∂t A˜2 a−1 ds ∧ dt, curl A˜ = (1 − tκ(s)) curl A.

We give in the next proposition a standard hoi e of gauge.

Consider a ve tor eld A = (A1 , A2 ) ∈ C ∞(Ω; R2 ) su h that curlA = 1. For ea h point x0 ∈ ∂Ω, there exist a neighborhood Vx0 ⊂ Ωt0 of x0 and a smooth real-valued fun tion φx0 su h the ve tor eld Anew := A − ∇φx0 satises : Proposition A.2.

(A.5)

t A˜1new = −t 1 − κ(s) and A˜2new = 0 in Vx0 . 2 Referen es

[1℄ S. Agmon : Le tures on exponential de ay of solutions of se ond order ellipti equations, Math. Notes, T. 29, Prin eton University Press (1982). [2℄ P. Baumann, D. Phillip, and Q. Tang : Stable nu leation for the Ginzburg-Landau model system with an applied magneti eld. Ar h. Rational Me h. Anal. 142, p. 1-43 (1998). [3℄ A. Bernoff and P. Sternberg : Onset of super ondu tivity in de reasing elds for general domains. J. Math. Phys. 39, p. 1272-1284 (1998). [4℄ C. Bolley and B. Helffer : An appli ation of semi- lassi al analysis to the asymptoti study of the super ooling eld of a super ondu ting material. Ann. Inst. Henri Poin aré, se tion Physique théorique 58 (2), p. 189-233 (1993). [5℄ V. Bonnaillie : Analyse mathématique de la supra ondu tivité dans un domaine à oins : Méthodes semi- lassique et numériques. Thèse à l'université Paris-Sud (2003). [6℄ M. Dauge and B. Helffer : Eigenvalues variation I, Neumann problem for SturmLiouville operators, J. of Dierential Equations, 104 (2), p. 243-262 (1993). [7℄ H.J. Fink and W.C.H. Joiner : Surfa e nu leation and boundary ondition in super ondu tors. Physi al review letters 23 (3) (1969). [8℄ S. Fournais and B. Helffer :On the third riti al eld in the Ginzburg-Landau theory. Preprint (2005).

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[9℄ S. Fournais and B. Helffer : A

urate eigenvalue asymptoti s for the magneti S hrödinger Lapla ian. To appear in Ann. Inst. Fourier (2004). [10℄ P.G. De Gennes : Super ondu tivity of metals and alloys. Benjamin (1966). [11℄ V. V. Grushin : On a lass of hypoellipti operators. Mat. Sb. (N.S.) 12, p. 458-475 (1970). [12℄ B. Helffer and A. Mohamed : Semi lassi al analysis of a S hrödinger operator with magneti wells. J. Fun t. Anal. 138 (1), p. 40-81 (1996). [13℄ B. Helffer and A. Morame: Magneti bottles in onne tion with super ondu tivity. J. Fun t. Anal. 185 (2), p. 604-680 (2001). [14℄ B. Helffer and X.-B. Pan : Upper riti al eld and lo ation of surfa e nu leation of super ondu tivity. Ann. Inst. H. Poin aré (Se tion analyse non-linéaire) 20 (1), p. 145-181 (2003). [15℄ A. Ka hmar : Thèse à l'université Paris-Sud. In preparation. [16℄ T. Kato : Perturbation theory for linear operators. New York, Springer-Verlag (1996). [17℄ X.-B. Pan : Super ondu ting lms in perpendi ular elds and the ee t of the De Gennes parameter, Siam J. Math. Anal. 34 (4), p. 957-991 (2003). [18℄ M. Del Pino, P.L. Felmer and P. Sternberg : Boundary on entration for eigenvalue problems related to the onset of super ondu tivity. Comm. Math. Phys. 210, p. 413-446 (2000). [19℄ K. Lu and X.-B. Pan : Ginzburg-Landau equation with De Gennes boundary ondition. J. Dierential Equations 129 (1), p. 136-165 (1996). [20℄ K. Lu and X.-B. Pan : Estimates of the upper riti al eld of the Ginzburg-Landau equations of super ondu tivity. Physi a D, p. 73-104 (1999). [21℄ K. Lu and X.-B. Pan : Eigenvalue problems of Ginzburg-Landau operator in bounded domains. J. Math. Phys. 40 (6), p. 2647-2670 (1999). [22℄ M. Reed and B. Simon : Methods of modern mathemati al physi s. A ademi press (1979). [23℄ D. Saint-James, G. Sarma, E.J. Thomas : Type II super ondu tivity. Pergamon, Oxford 1969. [24℄ M. Tinkham : Introdu tion to Super ondu tivity. M Graw-Hill In ., New York (1975). A. Ka hmar Université Paris-Sud, Bât. 425, F-91405 Orsay Université Libanaise, Hadeth, Beyrouth, Lebanon

E-mail address : ayman.ka hmarmath.u-psud.fr