On a stationary Schrödinger equation with periodic magnetic potential

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On a stationary Schrödinger equation with periodic magnetic potential

Pascal Bégout, Ian Schindler

To cite this version:

Pascal Bégout, Ian Schindler. On a stationary Schrödinger equation with periodic magnetic potential.

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales - Serie A. Matemáticas, Springer,

2021, 115 (2), pp.72. �10.1007/s13398-021-01011-9�. �hal-03170320v3�

On a Stationary Schr¨ odinger Equation with Periodic Magnetic Potential

Pascal B´ egout

^∗

and Ian Schindler

^∗,†

Abstract

We prove existence results for a stationary Schr¨odinger equation with periodic magnetic po- tential satisfying a local integrability condition on the whole space using a critical value function.

Contents

1 Introduction and main result 2

2 The space H

_A,V¹

(R

^N

) and an equivalent definition of H

¹

(R

^N

) 7

3 The set of dislocations 15

4 Cocompactness 20

5 An associated critical value function and proof of the main result 21

6 Applications 27

A Some proofs 28

B Topological vector spaces over the field of complex numbers restricted to the field of real numbers 35

References 37

∗Institut de Math´ematiques de Toulouse & TSE, Universit´e Toulouse I Capitole, 1 Esplanade de l’Universit´e, 31080 Toulouse, Cedex 06, France

∗E-mail:[email protected]

†E-mail:[email protected]

Pascal B´egout and Ian Schindler acknowledge funding from ANR under grant ANR-17-EUR-0010 (Investissements d’Avenir program)

2020 Mathematics Subject Classification: 35Q55 (35A01, 35D30)

Key Words: stationary Schr¨odinger equation, periodic magnetic potential, weak solution, cocompactness

1 Introduction and main result

We wish to investigate for which λ > 0 there is a weak solution to the stationary Schr¨odinger equation with magnetic potential:

 



( − i ∇ + A)

²

u + V (x)u = λf (x, u(x)), in R

^N

, u ∈ H

_A,V¹

(R

^N

),

(1.1)

where N > 2, A : R

^N

−→ R

^N

is the magnetic potential, B = curl A is the magnetic field, V : R

^N

−→

R, f satisfy some suitable assumptions, λ > 0 and H

_A,V¹

(R

^N

)

^def

= n

u ∈ L

²

(R

^N

); V u

²

∈ L

¹

(R

^N

) and ( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

) o

. (1.2)

Here, i

²

= − 1 and in what follows, unless specified, all functions are complex-valued (H

¹

(R

^N

) = H

¹

(R

^N

; C), L

^p

(R

^N

) = L

^p

(R

^N

; C), D (R

^N

) = D (R

^N

; C), etc). If N = 1 then (1.1) is equivalent to the case A = 0. Indeed, Assume that A, V ∈ L

¹_loc

(R; R) and f ( . , e

^iθ

z) = e

^iθ

f ( . , z), for any (θ, z) ∈ R × C (which is the case in this paper). Set for any x ∈ R, ϕ(x) = R

x

0

A(s)ds. If u ∈ H

_A,V¹

(R) is a solution to (1.1) then by the gauge transformation u 7−→ v = e

^iϕ

u, a straighforward calculation gives that v ∈ H

¹

(R) is a solution to − ∆v + V (x)v = λf (x, v(x)), which is (1.1) with A = 0. We thus restrict our study to the case N > 2.

We make assumptions that insure the functional associated with (1.1) is invariant with respect to the transformations τ

_y

: u 7−→ e

^iϕy

u( . + y), where ϕ

_y

is defined in (3.4) and y ∈ Z

^N

. In [11], the authors stated that this set of transformations was a group of dislocations as defined in [14] which is false. In Section 3 we prove (directly) that the set D of such transformations is a set of dislocations permitting us to use the profile decomposition theorem [14, Theorem 3.1, p.62-63]. In Devillanova and Tintarev [4, Appendix] this was proved by embedding the set of dislocations into a group via multiplication of τ

_y

τ

_z

by e

^iα

, α ∈ R in such a way that the composition agrees with τ

y+z

.

Arioli and Szulkin [1] treated a similar problem with more general conditions on V (the spectrum of the operator ( − i ∇ + A)

²

+ V (x) can be negative), but they assume the Rabinowitz condition on the right hand side. We make less restrictive assumptions on the right hand side and introduce a parameter λ and an unbounded interval I

_γ

⊂ (0, ∞ ) such that for almost every λ ∈ I

_γ

there is a solution to (1.1). In [4] a magnetic Schr¨odinger equation with bounded non-periodic magnetic field is studied.

In Section 2 we show that if the magnetic potential A ∈ L

^N_loc

(R

^N

; R

^N

) and V ∈ L

_loc^N2

(R

^N

; R) then

H

_A,V¹

(R

^N

) = H

¹

(R

^N

). In Section 3, we introduce the set of invariant dislocations acting on (1.1) and

prove necessary results to the dislocation theorem in [14]. In Section 4 we prove a cocompactness result. In Section 5 we introduce a related critical value function the study of which allows us to obtain our main result. In Section 6 we give some examples of nonlinearities to which our result applies.

Throughout this paper, we use the following notation. We denote by z the conjugate of the complex number z and by Re(z) its real part. For 1 6 p 6 ∞ , p

^′

denotes the conjugate of p defined by

1

p

+

_p^1′

= 1. By { Q

_j

}

^j>1

we will denote a countable covering of R

^N

\ Z

^N

by open unit cubes, thus R

^N

= S

j>1

Q

j

, and Q = (0, 1)

^N

. All vectors spaces considered will be over the field R. For a Banach space X (over R), we denote by X

^⋆def

= L (X ; R) its topological dual and by h . , . i

^X⋆,X

∈ R the X

^⋆

− X duality product and for a Hilbert space H, its (real) scalar product will be denoted by h . , . i

H

. In particular, for any T ∈ L

^p′

(Ω) and ϕ ∈ L

^p

(Ω) with 1 6 p < ∞ , h T, ϕ i

L^p′(Ω),L^p(Ω)

= Re R

Ω

T (x)ϕ(x)dx.

See Appendix B for more details. If u ∈ L

^p

(R

^N

), with 1 6 p 6 ∞ , and if Ω is an open subset of R

^N

, with some abuse of notation, expression k u k

L^p(Ω)

will stand for k u

_|Ω

k

L^p(Ω)

. This convention also holds for the others functional spaces. The subscript “c” on a functional space indicates that the functions have compact support. For instance, if Ω ⊆ R

^N

is an open subset then u ∈ L

^p_c

(Ω) means that u ∈ L

^p

(Ω), supp u ⊂ Ω and supp u is a compact subset of R

^N

. For a Banach space E, the notation E

w

designates the space E endowed with the weak topology σ(E, E

^⋆

) and E

w^⋆⋆

the space E

^⋆

endowed with the weak⋆ topology σ(E

^⋆

, E). We denote by C auxiliary positive constants which may change from a line to another one, and sometimes, for positive parameters a

1

, . . . , a

_n

, write C(a

1

, . . . , a

_n

) to indicate that the constant C continuously depends only on a

1

, . . . , a

n

(this convention also holds for constants which are not denoted by “C”). Finally, we denote by 2

^⋆

=

_N−2^2N

the critical exponent of the embedding H

¹

(R

^N

) ֒ → L

^2⋆

(R

^N

), with the convention that 2

^⋆

= ∞ , if N 6 2.

We shall make the following assumptions on A : R

^N

−→ R

^N

. Assumption 1.1. Let (e

1

, . . . , e

_n

) be the canonical basis of R

^N

.

1. The magnetic potential A : R

^N

−→ R

^N

satisfies,

 



A ∈ L

^N_loc

(R

^N

; R

^N

), if N > 3, A ∈ L

^2+ε_loc

(R

²

; R

²

), for some ε > 0, if N = 2.

(1.3) and

α

A def

=

 

 

 

 sup

j∈N

k A k

L^N(Qj)

< ∞ , if N > 3, sup

j∈N

k A k

L^2+ε(Qj)

< ∞ , if N = 2.

(1.4)

If N > 3 then there exists ε > 0 such that,

A ∈ L

^N_loc^+ε

(R

^N

; R

^N

). (1.5)

2. A is a Z

^N

− periodic magnetic potential:

∀ j ∈ J1, N K, curl A(x + e

_j

)

^D

′(R^N)

= curl A(x), (1.6)

where curl A ∈ M

_N

D

^′

(R

^N

; R)

is the skew-symmetric, matrix-valued distribution with A

ij

=

∂

_i

A

_j

− ∂

_j

A

_i

.

Remark 1.2. It is easy to see that in Assumption 1.1, (1.6) is equivalent to the condition: for any y ∈ Z

^N

, curl A(x + y)

^D

′(R^N)

= curl A(x). By Lemma 1.1 in Leinfelder [7], (1.6) is also equivalent to:

for any y ∈ Z

^N

, there exists ϕ

_y

∈ W

_loc^1,N+ε

(R

^N

; R) such that for almost every x ∈ R

^N

, A(x + y) = A(x) + ∇ ϕ

y

(x).

Assumption 1.3. We will use the following assumptions on V and f. Let f : R

^N

× [0, ∞ ) −→ R be such that f (x, t) is measurable in x and continuous in t and let F (x, t)

^def

=

Z

t 0

f (x, s)ds, for almost every x ∈ R

^N

and any t > 0. We extend f to the complex plane by setting for almost every x ∈ R

^N

and any z ∈ C \ { 0 } , f (x, z) = f (x, | z | )

_|z|^z

, and f (x, 0) = f (x, 0). Finally, we set for any measurable function u : R

^N

−→ C and almost every x ∈ R

^N

, g(u)(x) = f (x, u(x)) and,

∀ u ∈ H

¹

(R

^N

), ψ(u) = Z

R^N

F (x, | u(x) | )dx. (1.7)

1. For every ε > 0, there exist p

_ε

∈ (2, 2

^⋆

) and C

_ε

> 0 such that for almost every x ∈ R

^N

and any t > 0,

| f (x, t) | 6 ε(t + t

^2⋆−1

) + C

_ε

t

^pε−1

, (1.8) if N > 3 and

| f (x, t) | 6 εt + C

ε

t

^pε−1

, (1.9) if N 6 2.

∃ u ∈ H

¹

(R

^N

) such that ψ(u) > 0. (1.10) 2. The function f and the electric potential V : R

^N

−→ R are Z

^N

-periodic, that is for almost

every (x, y) ∈ R

^N

× Z

^N

and any t > 0, f (x + y, t) = f (x, t) and V (x + y) = V (x).

3. We have,

V ∈ L

¹_loc

(R

^N

; R) and ν

^def

= ess inf

x∈R^N

V (x) > 0. (1.11) Remark 1.4. If N 6 2 then Assumption (1.9) is equivalent to the following:

t→0

lim f (x, t)

t = 0, uniformly in x ∈ R

^N

, (1.12)

and there exist p ∈ (2, 2

^⋆

) and C > 0 such that,

for a.e. x ∈ R

^N

, ∀ t > 0, | f (x, t) | 6 C + Ct

^p−1

. (1.13) If N > 3 then Assumption (1.8) is equivalent to (1.12), (1.13) with p = 2

^⋆

and,

t→∞

lim f (x, t)

t

^2⋆−1

= 0, uniformly in x ∈ R

^N

.

Assumption 1.5. The electric potential V : R

^N

−→ R satisfies,

 

 

 



V ∈ L

_loc^N2

(R

^N

; R) and α

_V ^def

= sup

j∈N

k V k

_L^N_2(Q

j)

< ∞ , if N > 3, V ∈ L

^1+ε_loc

(R

²

; R) and α

_V ^def

= sup

j∈N

k V k

L^1+ε(Qj)

< ∞ , for some ε > 0, if N = 2.

(1.14)

Remark 1.6. Note that if V has the local integrability (1.14) and if furthermore V is Z

^N

-periodic then we necessarily have α

V

< ∞ since α

V

= k V k

L¹(Q)

.

Notation 1.7. Let A and V satisfying (1.3)–(1.4) and (1.11), respectively, and let H

_A,V¹

(R

^N

) be defined by (1.2). We shall denote by H

_A,V⁻¹

(R

^N

) the topological dual of the space H

_A,V¹

(R

^N

). This dual space is identified with a real vector subspace of the space of distributions D

^′

(R

^N

) (see Theorem 2.3 below).

Definition 1.8. Let A and V satisfying (1.3)–(1.4) and (1.11), respectively, and let H

_A,V¹

(R

^N

) be defined by (1.2). We shall write that u is a weak solution of (1.1) if u ∈ H

_A,V¹

(R

^N

) and if u satisfies (1.1) in H

_A,V⁻¹

(R

^N

).

Remark 1.9. With respect to Definition 1.8 we note:

1. If u ∈ H

¹

(R

^N

) then ( − i ∇ + A)

²

u ∈ H

⁻¹

(R

^N

) and,

( − i ∇ + A)

²

u = − ∆u − i ∇ .(Au) − iA. ∇ u + | A |

²

u, in H

⁻¹

(R

^N

), (1.15)

h iA. ∇ u, v i

H⁻¹(R^N),H¹(R^N)

= h i ∇ u, Av i

L²(R^N),L²(R^N)

, (1.16)

h| A |

²

u, v i

H⁻¹(R^N),H¹(R^N)

= h Au, Av i

L²(R^N),L²(R^N)

, (1.17)

for any v ∈ H

¹

(R

^N

). Indeed, if u ∈ H

¹

(R

^N

) then by Lemma 2.13 below, Au ∈ L

²

(R

^N

; C

^N

) so that − ∆u ∈ H

⁻¹

(R

^N

) and ∇ .(Au) ∈ H

⁻¹

(R

^N

). In addition, by H¨ older’s inequality, A. ∇ u, | A |

²

u ∈ L

¹_loc

(R

^N

) ֒ → D

^′

(R

^N

) and for any ϕ ∈ D (R

^N

),

h iA. ∇ u, ϕ i

^D′,D

= h i ∇ u, Aϕ i

L²,L²

and h| A |

²

u, ϕ i

^D′,D

= h Au, Aϕ i

L²L²

.

By density and estimates in Property 1 of Lemma 2.14 below, it follows that A. ∇ u ∈ H

⁻¹

(R

^N

),

| A |

²

u ∈ H

⁻¹

(R

^N

) and (1.15)–(1.17) follow.

2. Let u ∈ H

_A,V¹

(R

^N

). Let R > 0. We have, Z

B(0,R)

| V u | dx 6 Z

B(0,R)∩{|u|61}

| V || u | dx + Z

{|u|>1}

| V || u |

²

dx < ∞ ,

since V ∈ L

¹_loc

(R

^N

; R) and V u

²

∈ L

¹

(R

^N

). It follows that V u ∈ L

¹_loc

(R

^N

) ֒ → D

^′

(R

^N

) and for any ϕ ∈ D (R

^N

),

h V u, ϕ i

^D′,D

= Re Z

R^N

V uϕdx,

|h V u, ϕ i

^D′,D

| 6 k √

V u k

L²

k √

V ϕ k

L²

6 k u k

H_A,V¹

k ϕ k

H¹_A,V

,

by the Cauchy-Schwarz inequality (see Definition 2.1 below for the definition of k . k

H_A,V¹

). By the density of D (R

^N

) in H

_A,V¹

(R

^N

) (Theorem 2.3 below), it follows that V u ∈ H

_A,V⁻¹

(R

^N

) and for any v ∈ H

_A,V¹

(R

^N

),

h V u, v i

H_A,V⁻¹ (R^N),H_A,V¹ (R^N)

= Re Z

R^N

V u v dx. (1.18)

Finally, by Proposition 5.1 below, g(u) ∈ H

⁻¹

(R

^N

). In conclusion, since H

⁻¹

(R

^N

) ֒ → H

_A,V⁻¹

(R

^N

) (Theorem 2.3 below), it follows from (1.15) and (1.18) that

( − i ∇ + A)

²

u ∈ H

_A,V⁻¹

(R

^N

), V u ∈ H

_A,V⁻¹

(R

^N

) and g(u) ∈ H

_A,V⁻¹

(R

^N

).

Thus Definition 1.8 makes sense.

Our main result follows.

Theorem 1.10. Let N > 2 and let Assumptions 1.1 and 1.3 be satisfied. Let H

_A,V¹

(R

^N

) be defined by (1.2). Then for almost every λ > 0 sufficiently large, there exists, at least one non zero weak solution to,

 



− ∆

A

u + V (x)u = λg(u) in R

^N

, u ∈ H

_A,V¹

(R

^N

),

(1.19)

where − ∆

A

u = ( − i ∇ + A)

²

u.

2 The space H _A,V ¹ (R ^N ) and an equivalent definition of H ¹ (R ^N )

In this section, we study the H

_A,V¹

(R

^N

), including the one-dimensional case N = 1 because we believe that it is of interest for itself. For N = 1, the corresponding assumptions to (1.4) and (1.14) are

A ∈ L

²_loc

(R; R) and α

_A^def

= sup

j∈N

k A k

L²(Qj)

< ∞ , (2.1) V ∈ L

¹_loc

(R; R) and α

V

def

= sup

j∈N

k V k

L¹(Qj)

< ∞ , (2.2) respectively.

Definition 2.1. Let N > 1 and let A ∈ L

²_loc

(R

^N

; R

^N

) and V ∈ L

¹_loc

(R

^N

; R) satisfy (1.3) and (1.11), respectively. We recall that H

_A,V¹

(R

^N

) is defined by,

H

_A,V¹

(R

^N

) = n

u ∈ L

²

(R

^N

); V u

²

∈ L

¹

(R

^N

) and ∇

^A

u ∈ L

²

(R

^N

; C

^N

) o

. (2.3)

where ∇

^A

u = ( ∇ +iA)u. We endow H

_A,V¹

(R

^N

) with the following scalar product and its corresponding norm,

∀ u, v ∈ H

_A,V¹

(R

^N

), h u, v i

H_A,V¹ (R^N)

= Re Z

R^N

V u v dx + Re Z

R^N

∇

^A

u. ∇

^A

vdx,

∀ u ∈ H

_A,V¹

(R

^N

), k u k

²H_A,V¹ (R^N)

= h u, u i

H_A,V¹ (R^N)

= Z

R^N

V | u |

²

dx + k∇

^A

u k

²L²(R^N)

,

making this space a real pre-Hilbert space. Indeed, it follows from (1.11) that h . , . i

H_A,V¹ (R^N)

is a bilinear symmetric positive definite form on H

_A,V¹

(R

^N

) × H

_A,V¹

(R

^N

).

Remark 2.2. Below are some comments about the definition of H

_A,V¹

(R

^N

).

1. If u ∈ H

_A,V¹

(R

^N

) then ∇

^A

u

^def

= ( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

) but, a priori, we do not assume that

∇ u or Au belong separately in L

²

(R

^N

).

2. Frequently, in the literature (see for instance Sections 7.19–7.22, p.191–195, of Lieb and Loss [8]),

it is assumed that A ∈ L

²_loc

(R

^N

; R

^N

) rather than A ∈ L

^N_loc

(R

^N

; R

^N

) and V ≡ 1. With these

assumptions it can be shown that H

_A,1¹

(R

^N

) is a Hilbert space having D (R

^N

) as a dense

subset. Moreover, if u ∈ H

_A,1¹

(R

^N

) then | u | ∈ H

¹

(R

^N

) and the so-called diamagnetic in-

equality (2.11) below holds. However if A 6∈ L

^N_loc

(R

^N

; R

^N

) then H

¹

(R

^N

) 6⊂ H

_A,1¹

(R

^N

) and

H

_A,1¹

(R

^N

) 6⊂ H

¹

(R

^N

). We show that if A ∈ L

^N_loc

(R

^N

; R

^N

) then H

_A,1¹

(R

^N

) = H

¹

(R

^N

) (see

Theorem 2.5 below).

3. Arioli and Szulkin showed (Lemma 2.3 in [1]) that if N > 2 and A ∈ L

^N_loc

(R

^N

; R

^N

) (A ∈ L

^2+ε_loc

(R

^N

; R

^N

), if N = 2) then H

_A,1¹

(Ω) = H

¹

(Ω) with equivalent norms for open bounded subsets Ω of R

^N

with smooth boundaries. We extend their result to the case Ω = R

^N

for any N > 1, under assumptions (1.3)–(1.4) (Theorem 2.5 below).

Theorem 2.3. Let A ∈ L

²_loc

(R

^N

; R

^N

) and V ∈ L

¹_loc

(R

^N

; R) satisfy (1.3)–(1.4) ((2.1), if N = 1) and (1.11), respectively, and let H

_A,V¹

(R

^N

) be defined by (2.3). Then,

H

_A,V¹

(R

^N

) = n

u ∈ H

¹

(R

^N

); V u

²

∈ L

¹

(R

^N

) and Au ∈ L

²

(R

^N

; C

^N

) o

, (2.4)

H

_A,V¹

(R

^N

) is a separable Hilbert space, (2.5) D (R

^N

) ֒ → H

_A,V¹

(R

^N

) ֒ → H

¹

(R

^N

), (2.6) with both dense embeddings. In particular, each term in the integrals of h . , . i

H¹_A,V(R^N)

belongs to L

¹

(R

^N

). In addition,

H

⁻¹

(R

^N

) ֒ → H

_A,V⁻¹

(R

^N

) ֒ → D

^′

(R

^N

), (2.7) where H

_A,V⁻¹

(R

^N

)

^def

= H

_A,V¹

(R

^N

)

⋆

and both dense embeddings.

Remark 2.4. By (2.6), for any u ∈ H

_A,V¹

(R

^N

), there exists (ϕ

n

)

n∈N

⊂ D (R

^N

) such that ϕ

_n ^H

1

−−−−→

_n→∞A,V

u.

As with the classical proofs of density we have for any n ∈ N, k ϕ

n

k

L^p(R^N)

6 k u k

L^p(R^N)

, and p ∈ [1, ∞ ], if u ∈ L

^p

(R

^N

). See Lemma 2.16.

Theorem 2.5. Let A ∈ L

²_loc

(R

^N

; R

^N

) satisfy (1.3)–(1.4) ((2.1), if N = 1) and let V ∈ L

¹_loc

(R

^N

; R) satisfy (1.11) and (1.14) ((1.11) and (2.2), if N = 1). Then,

H

_A,V¹

(R

^N

) = H

¹

(R

^N

), with equivalent norms.

Remark 2.6. To find examples such that H

_A,V¹

(R

^N

) ( H

¹

(R

^N

), note that by (2.15) below, assuming (1.3)–(1.4), for any u ∈ H

¹

(R

^N

), u ∈ H

_A,V¹

(R

^N

) if, and only if, V u

²

∈ L

¹

(R

^N

). So we look for a V that does not satisfy (1.14). If N = 1 we must have α

V

= ∞ . In other words, V cannot be Z

^N

- periodic. Below, for each N, we give an example of a V and a u ∈ H

¹

(R

^N

) such that u 6∈ H

_A,V¹

(R

^N

).

For N > 2, u is a positive continuous function over R

^N

\ { 0 } such that for | x | > 10, u(x) = | x |

^−N

. We give its definition for x near 0 below.

1. For N = 1. Define for any x ∈ R, V (x) = x

⁴

+ 1 and u(x) = (x

²

+ 1)

⁻¹

. Then V satisfies (1.11),

u ∈ H

¹

(R

^N

; R) but V u

²

6∈ L

¹

(R

^N

; R).

2. For N = 2. For | x | < e

^−e

, let u(x) = ln | ln | x || and let V be Z

^N

-periodic such that for any x ∈ Q,

V (x) = 1

| x |

²

| ln | x || (ln | ln | x || )

²

1

{0<|x|<e^−e}

(x) + 1

{|x|>e^−e}∩Q

(x).

Then V satisfies (1.11) but for any p ∈ (1, ∞ ], V 6∈ L

^p_loc

(R

²

; R), so that V does not verify (1.14).

In addition, u ∈ H

¹

(R

^N

; R) but V u

²

6∈ L

¹

(R

^N

; R).

3. For N > 3. Let for | x | < 1, u(x) = | x |

^−N−24

and let V be Z

^N

-periodic such that for any x ∈ Q, V (x) = | x |

^−N+22

. Then V satisfies (1.11) but V 6∈ L

N 2

loc

(R

^N

; R), so that V does not verify (1.14).

In addition, u ∈ H

¹

(R

^N

; R) but V u

²

6∈ L

¹

(R

^N

; R).

We recall that − ∆

A

= ( − i ∇ + A)

²

and ∇

^A

= ∇ + iA.

Theorem 2.7. Let A ∈ L

²_loc

(R

^N

; R

^N

) satisfy (1.3)–(1.4) ((2.1), if N = 1). If u ∈ H

¹

(R

^N

) then

− ∆

A

u ∈ H

⁻¹

(R

^N

) and for any v ∈ H

¹

(R

^N

),

h− ∆

A

u, v i

H⁻¹(R^N),H¹(R^N)

= h∇

^A

u, ∇

^A

v i

L²(R^N),L²(R^N)

. (2.8) If in addition V satisfies (1.11) and if H

_A,V¹

(R

^N

) is defined by (2.3) then for any T ∈ H

⁻¹

(R

^N

), T ∈ H

_A,V⁻¹

(R

^N

) and

h T, u i

H_A,V⁻¹ (R^N),H_A,V¹ (R^N)

= h T, u i

H⁻¹(R^N),H¹(R^N)

. (2.9) for any u ∈ H

_A,V¹

(R

^N

),

Remark 2.8. Let λ > 0 and let u be a solution to (1.19). By Definition 1.8, we may take the H

_A,V⁻¹

− H

_A,V¹

duality product of (1.19) with u. We have by (2.9), (2.8) and (1.18),

k u k

²H_A,V¹ (R^N)

= λ h g(u), u i

H_A,V⁻¹ (R^N),H_A,V¹ (R^N)

> 0. (2.10) It follows that if g ≡ 0 then necessarily u ≡ 0. Note that if ψ, defined by (1.7), satisfies (1.10) then g 6≡ 0.

Remark 2.9. Let A ∈ L

²_loc

(R

^N

; R

^N

) satisfy (1.3)–(1.4) ((2.1), if N = 1). It follows from Theorem 2.7 and (2.16) below that − ∆

A

∈ L H

¹

(R

^N

); H

⁻¹

(R

^N

)

. If, in addition, V ∈ L

¹_loc

(R

^N

; R) satisfies (1.11) then by Theorems 2.3 and 2.7, − ∆

A|H_A,V¹ (R^N)

∈ L H

_A,V¹

(R

^N

); H

_A,V⁻¹

(R

^N

)

.

We split the proof of Theorem 2.3 in several lemmas. We begin by recalling the diamagnetic inequality for functions belonging in H

_A,V¹

(R

^N

). Its proof is well-known (Lieb and Loss [8], Theorem 7.21, p.193).

For the sake of completeness, we sketch the proof.

Theorem 2.10 (Diamagnetic inequality, [8]). Let A ∈ L

²_loc

(R

^N

; R

^N

). Let u ∈ L

²

(R

^N

) be such that ( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

). Then ∇ u, Au ∈ L

¹_loc

(R

^N

; C

^N

), | u | ∈ H

¹

(R

^N

), ( ∇ + iA)u

^a.e.

= ∇ u + iAu and

|∇| u | |

^a.e.

6 |∇ u + iAu | . (2.11)

Lemma 2.11 ([8]). If u ∈ H

¹

(R

^N

) then | u | ∈ H

¹

(R

^N

; R) and |∇| u | |

^a.e.

6 |∇ u | .

Proof of Theorem 2.10 and Lemma 2.11. We recall that if u ∈ W

_loc^1,p

(R

^N

), for some 1 6 p 6 ∞ , then | u | ∈ W

_loc^1,p

(R

^N

; R) and ∇| u |

^a.e.

= Re

u

|u|

∇ u

1

(Theorem 6.17, p.152, in Lieb and Loss [8]). This proves Lemma 2.11. Now, let u ∈ L

²

(R

^N

) be such that ( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

).

Then ∇ u ∈ H

⁻¹

(R

^N

; C

^N

) and, by the Cauchy-Schwarz inequality, Au ∈ L

¹_loc

(R

^N

; C

^N

). This implies that ∇ u ∈ L

¹_loc

(R

^N

; C

^N

). We then infer, u ∈ W

_loc^1,1

(R

^N

) and ( ∇ + iA)u

^a.e.

= ∇ u + iAu. And since Re

u

|u|

( ∇ u + iAu)

_a.e.

= Re

u

|u|

∇ u

_a.e.

= ∇| u | , one obtains that | u | ∈ H

¹

(R

^N

) and (2.11).

Lemma 2.12. Let u ∈ L

²

(R

^N

) be such that | u | ∈ H

¹

(R

^N

; R).

1. If A ∈ L

²_loc

(R

^N

; R

^N

) satisfies (1.3)–(1.4) ((2.1), if N = 1) then Au ∈ L

²

(R

^N

; C

^N

) and,

k Au k

L²(R^N)

6 Cα

A

k | u | k

H¹(R^N)

, (2.12) where C = C(N) (C = C(ε), if N = 2).

2. If V satisfies (1.14) ((2.2), if N = 1) then V u

²

∈ L

¹

(R

^N

) and,

k V u

²

k

L¹(R^N)

6 Cα

V

k | u | k

²H¹(R^N)

, (2.13) where C = C(N) (C = C(ε), if N = 2).

Proof. Let u ∈ L

²

(R

^N

) be such that | u | ∈ H

¹

(R

^N

; R). We start by proving Property 1 with N > 3.

By the Sobolev embedding H

¹

(Q

j

) ֒ → L

^2⋆

(Q

j

), there exists C = C(N, | Q

j

| ) such that for any j ∈ N, k u k

L^2⋆(Qj)

6 C k | u | k

H¹(Qj)

. Actually, C only depends on N since for any j ∈ N, | Q

j

| = 1. It follows from H¨ older’s inequality that,

Z

R^N

| Au |

²

dx = X

j∈N

Z

Qj

| Au |

²

dx

6 X

j∈N

k A k

²L^N(Qj)

k u k

²L^2⋆(Qj)

6 C

²

α

²_A

X

j∈N

k | u | k

²H¹(Qj)

= C

²

α

²_A

k | u | k

²H¹(R^N)

.

1∇|u|= 0,almost everywhere whereu= 0.

If N = 2 then the second line is replaced with P

j∈N

k A k

²L^2+ε(Qj)

k u k

²

L

2(2+ε)

ε (Qj)

and we use the embed- ding H

¹

(Q

j

) ֒ → L

^2(2+ε)ε

(Q

j

), while if N = 1 then the second line is replaced with P

j∈N

k A k

²L²(Qj)

k u k

²L^∞(Qj)

and we use the embedding H

¹

(Q

j

) ֒ → L

^∞

(Q

j

). Hence 1. Property 2 follows in the same way: replace A with p

| V | in the above estimates.

Lemma 2.13. Let A ∈ L

²_loc

(R

^N

; R

^N

) satisfy (1.3)–(1.4) ((2.1), if N = 1). Then,

( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

) ⇐⇒ u ∈ H

¹

(R

^N

), (2.14)

∇ u ∈ L

²

(R

^N

; C

^N

) = ⇒ Au ∈ L

²

(R

^N

; C

^N

). (2.15) Finally, if u ∈ H

¹

(R

^N

) then ( ∇ + iA)u = ∇ u + iAu, in L

²

(R

^N

; C

^N

) and

k∇ u + iAu k

L²(R^N)

6 C k u k

H¹(R^N)

, (2.16) where C = C(α

A

, N) (C = C(α

A

, N, ε), if N = 2).

Proof. Let A satisfy (1.3)–(1.4) ((2.1), if N = 1). Let u ∈ L

²

(R

^N

).

• If ∇ u ∈ L

²

(R

^N

; C

^N

) then by Lemmas 2.11 and 2.12, Au ∈ L

²

(R

^N

; C

^N

). Hence (2.15) and ⇐ = in (2.14).

• If ( ∇ + iA)u ∈ L

²

(R

^N

; C

^N

) then by Theorem 2.10 and Lemma 2.12, Au ∈ L

²

(R

^N

; C

^N

) and ( ∇ + iA)u

^a.e.

= ∇ u + iAu. Hence, ∇ u ∈ L

²

(R

^N

; C

^N

) and = ⇒ in (2.14) is proved.

• By (2.12) and Lemma 2.11, we have

k∇ u + iAu k

L²(R^N)

6 k∇ u k

L²(R^N)

+ k Au k

L²(R^N)

6 (Cα

A

+ 1) k u k

H¹(R^N)

. Hence the result.

Lemma 2.14. Let u, v ∈ H

¹

(R

^N

).

1. Let A ∈ L

²_loc

(R

^N

; R

^N

) satisfy (1.3)–(1.4) ((2.1), if N = 1). Then (Au). ∇ v ∈ L

¹

(R

^N

), | A |

²

uv ∈ L

¹

(R

^N

) and we have,

Z

R^N

| Au | |∇ v | dx 6 Cα

A

k u k

H¹(R^N)

k v k

H¹(R^N)

, Z

R^N

| A |

²

| uv | dx 6 C

²

α

²_A

k u k

H¹(R^N)

k v k

H¹(R^N)

.

where the constant C is given by (2.12).

2. Let V satisfy (1.14) ((2.2), if N = 1). Then V uv ∈ L

¹

(R

^N

) and we have, Z

R^N

| V || uv | dx 6 Cα

V

k u k

H¹(R^N)

k v k

H¹(R^N)

, where the constant C is given by (2.13).

Proof. The results come from Lemma 2.11, Lemma 2.12, and the Cauchy-Schwarz inequality.

From now and until the end of this section, we shall suppose that the assumptions of Theorem 2.3 are fulfilled.

Lemma 2.15. Let us define, E = n

u ∈ H

¹

(R

^N

); V u

²

∈ L

¹

(R

^N

) and Au ∈ L

²

(R

^N

; C

^N

) o .

Then, H

_A,V¹

(R

^N

) = E and D (R

^N

) ֒ → H

_A,V¹

(R

^N

)

_dense

֒ → H

¹

(R

^N

). In particular, each term in the integrals of h . , . i

H¹_A,V(R^N)

belongs to L

¹

(R

^N

).

Proof. It is clear that E ⊂ H

_A,V¹

(R

^N

). By Lemma 2.13, H

_A,V¹

(R

^N

) ⊂ E ⊂ H

¹

(R

^N

). It follows that H

_A,V¹

(R

^N

) = E, which gives the last part of the lemma, with help of Lemma 2.14. Let u ∈ H

_A,V¹

(R

^N

).

We have by (1.11), (2.12) and (2.11), k u k

²L²(R^N)

6 1

ν Z

R^N

V | u |

²

dx 6 1

ν k u k

²H_A,V¹ (R^N)

, and

k∇ u k

L²(R^N)

6 k∇ u + iAu k

L²(R^N)

+ k Au k

L²(R^N)

6 k∇ u + iAu k

L²(R^N)

+ Cα

_A

k | u | k

H¹(R^N)

6 k∇ u + iAu k

L²(R^N)

+ Cα

A

k u k

L²(R^N)

+ k∇ u + iAu k

L²(R^N)

6 C k u k

H_A,V¹ (R^N)

.

Hence, H

_A,V¹

(R

^N

) ֒ → H

¹

(R

^N

). Let ϕ ∈ D (R

^N

). Let R > 0 be such that supp ϕ ⊂ B(0, R). By H¨ older’s inequality, V ϕ

²

∈ L

¹

(R

^N

) and Aϕ ∈ L

²

(R

^N

; C

^N

). It follows that ϕ ∈ H

_A,V¹

(R

^N

). Again by H¨ older’s inequality and (2.16), we have

k ϕ k

²H_A,V¹ (R^N)

6 k V k

^L1(B(0,R))

k ϕ k

²L^∞(R^N)

+ C k ϕ k

²H¹(R^N)

,

where C does not depend on ϕ. Hence, D (R

^N

) ֒ → H

_A,V¹

(R

^N

). Finally, since D (R

^N

) ⊂ H

_A,V¹

(R

^N

) and D (R

^N

) ֒ →

dense

H

¹

(R

^N

), we conclude that H

_A,V¹

(R

^N

) ֒ →

dense

H

¹

(R

^N

).

Lemma 2.16. It holds that D (R

^N

) ⊂ H

_A,V¹

(R

^N

) and for any u ∈ H

_A,V¹

(R

^N

), there exists (ϕ

n

)

n∈N

⊂ D (R

^N

) such that ϕ

_n ^H

1

−−−−→

_n→∞A,V

u. In addition, we have for any n ∈ N, k ϕ

_n

k

L^p(R^N)

6 k u k

L^p(R^N)

, for any p ∈ [1, ∞ ], as soon as u ∈ L

^p

(R

^N

).

Proof. We adapt the proof of Theorem 7.22, p.194, in Lieb and Loss [8] to handle the presence of the potential V in the integral R

V | u |

²

dx. By Lemma 2.16, we already know that D (R

^N

) ⊂ H

_A,V¹

(R

^N

).

Let ξ ∈ C

^∞

(R; R) be such that 0 6 ξ 6 1, ξ(t) = 1, if | t | 6 1 and ξ(t) = 0, if | t | > 2. Let n ∈ N. Set for any x ∈ R

^N

, ξ

n

(x) = ξ

|x|

n

. We denote by (ρ

n

)

n∈N

⊂ D (R

^N

) any standard sequence of mollifiers.

Let u ∈ H

_A,V¹

(R

^N

). Let ε > 0. Let p ∈ [1, ∞ ] be such that u ∈ L

^p

(R

^N

). We proceed in three steps.

Step 1: There exists v ∈ H

_A,V¹

(R

^N

) ∩ L

^∞_c

(R

^N

) such that k u − v k

H¹_A,V(R^N)

<

^ε₂

and | v |

^a.e.

6 | u | . Let n ∈ N. Let for x ∈ R

^N

, u

n

(x) = ξ

n

(x)ξ

_|u(x)|

n

u(x). Then, supp u

n

⊂ B(0, 2n), k u

n

k

L^∞(R^N)

6 2n, u

n

a.e. inR^N

−−−−−−→

_n→∞

u, √ V u

n

a.e. inR^N

−−−−−−→

_n→∞

√

V u, | u

n

| 6 | u | ∈ L

²

(R

^N

) and √

V | u

n

| 6 √

V | u | ∈ L

²

. It follows that (u

n

)

n∈N

⊂ L

^∞_c

(R

ⁿ

) and by the dominated convergence Theorem,

n→∞

lim Z

R^N

V | u − u

_n

|

²

dx = 0 and u

_n ^L

2(R^N)

−−−−−→

_n→∞

u. (2.17)

In addition,

∇ u

_n

= 1 n ξ

^′

| . | n

ξ

| u | n

u x

| x | + 1 n ξ

_n

ξ

^′

| u | n

u ∇| u | + ξ

_n

ξ | u |

n

∇ u −−−−−−→

^{a.e. in}_n→∞^RN

∇ u.

But, ξ

^′

_|u|

n

= 0, if | u | > 2n so that,

|∇ u

n

| 6 k ξ

^′

k

L^∞(R)

| u | + (2 k ξ

^′

k

L^∞(R)

+ 1) |∇ u | ∈ L

²

(R

^N

),

by Lemmas 2.15 and 2.11. By the dominated convergence Theorem and (2.17), we then infer that u

_n ^H

1(R^N)

−−−−−→

_n→∞

u. It follows from (2.16)–(2.17) that (u

n

)

n∈N

⊂ H

_A,V¹

(R

^N

) and u

_n ^H

1 A,V(R^N)

−−−−−−−→

_n→∞

u. Pick any n

0

∈ N large enough to have k u − u

n0

k

H¹_A,V(RN)

<

^ε₂

. Hence the result with v = u

n0

.

Step 2: There exists ϕ ∈ D (R

^N

) such that k v − ϕ k

H_A,V¹ (R^N)

<

₂^ε

and k ϕ k

L^p(R^N)

6 k v k

L^p(R^N)

. Let n ∈ N. Let R > 1 be such that B(0, R) ⊃ supp v. Set, v

_n

= ρ

_n

⋆ v. Since v ∈ H

_c¹

(R

^N

) it is well-known that v

n

∈ D (R

^N

), k v

n

k

L^p(R^N)

6 k v k

L^p(R^N)

(by Young’s inequality),

supp v

n

⊂ supp ρ

n

+ supp v ⊂ B(0, 2R), (2.18) v

n

H¹(R^N)

−−−−−→

_n→∞

v, (2.19)

(see for instance Brezis [3]: Proposition 4.18, p.106; Proposition 4.20, p.107; Theorem 4.22, p.109;

Lemma 9.1, p.266). Then, v

n

∈ H

_A,V¹

(R

^N

) and by (2.16),

∇ v

n

+ iAv

n L²(R^N)

−−−−−→

_n→∞

∇ v + iAv. (2.20)

By (2.19), we may extract a subsequence, that we still denote by (v

n

)

n∈N

, such that v

_n

−−−−−−→

^{a.e. in}_n→∞^RN

v.

As a consequence, √ V v

n

a.e. inR^N

−−−−−−→

_n→∞

√

V v. Applying Young’s inequality and (2.18), we see that

√ V | v

n

| 6 √

V k v k

L^∞(R^N)

1

B(0,2R)

∈ L

²

(R

^N

).

It follows from the dominated convergence Theorem that lim

n→∞

Z

R^N

V | v − v

_n

|

²

dx = 0, which gives with (2.20),

n→∞

lim k v − v

_n

k

H_A,V¹ (R^N)

= 0.

We then choose ϕ = v

n1

, where n

1

∈ N is sufficiently large to have k v − v

n1

k

H_A,V¹ (R^N

<

^ε₂

. Hence Step 2.

Step 3: Conclusion.

The result follows from Steps 1 and 2.

Lemma 2.17. The space H

_A,V¹

(R

^N

) is complete.

Proof. Let (u

n

)

n∈N

⊂ H

_A,V¹

(R

^N

) be a Cauchy sequence. Since H

_A,V¹

(R

^N

) ֒ → H

¹

(R

^N

) which is complete (Lemma 2.15), there exists u ∈ H

¹

(R

^N

) such that u

n

H¹(R^N)

−−−−−→

_n→∞

u. By Lemma 2.13, Au ∈ L

²

(R

^N

; C

^N

) and ∇ u

n

+ iAu

n

L²(R^N)

−−−−−→

_n→∞

∇ u+ iAu. To conclude, it remains to show that √

V u ∈ L

²

(R

^N

) and √

V u

_n ^L

2(R^N)

−−−−−→

_n→∞

√

V u. The sequence ( √

V u

_n

)

n∈N

being Cauchy in L

²

(R

^N

), it is bounded and there exists v ∈ L

²

(R

^N

) such that √

V u

_n ^L

2(R^N)

−−−−−→

_n→∞

v. There exists a subsequence (u

nk

)

k∈N

⊂ (u

n

)

n∈N

such that u

nk

a.e. inR^N

−−−−−−→

_k→∞

u. It follows that √ V u

nk

a.e. inR^N

−−−−−−→

_k→∞

√

V u and by Fatou’s Lemma, √

V u ∈ L

²

(R

^N

).

Let ϕ ∈ D (R

^N

). By H¨ older’s inequality, √

V ϕ ∈ L

²

(R

^N

). We have for any n ∈ N, h √

V u

n

, ϕ i

L²(R^N),L²(R^N)

= h u

n

, √

V ϕ i

L²(R^N),L²(R^N)

.

By the above convergences, we can pass to the limit and we get for any ϕ ∈ D (R

^N

), h v, ϕ i

L²(R^N),L²(R^N)

= h u, √

V ϕ i

L²(R^N),L²(R^N)

= h √

V u, ϕ i

L²(R^N),L²(R^N)

. It follows that, v = √

V u in D

^′

(R

^N

) and so in L

²

(R

^N

). The lemma is proved.

Proof of Theorem 2.5. By Lemma 2.15, H

_A,V¹

(R

^N

) ֒ → H

¹

(R

^N

). It remains to show that, H

¹

(R

^N

) ֒ → H

_A,V¹

(R

^N

). Let u ∈ H

¹

(R

^N

). Then, | u | ∈ H

¹

(R

^N

) and k | u | k

H¹(R^N)

6 k u k

H¹(R^N)

(Lemma 2.11). By Lemma 2.12, Au ∈ L

²

(R

^N

; C

^N

) and V u

²

∈ L

¹

(R

^N

). As a consequence, u ∈ H

_A,V¹

(R

^N

) and by (2.13) and (2.16),

k u k

H_A,V¹ (R^N)

6 C k u k

H¹(R^N)

,

where C does not depend on u.

Lemma 2.18. The space H

_A,V¹

(R

^N

) is separable.

Proof. By Theorem 2.5, H

_A,1¹

(R

^N

) = H

¹

(R

^N

) with equivalent norms. Let H = H

¹

(R

^N

), with k . k

H

= k . k

H¹_A,1(R^N)

. Since (H

¹

(R

^N

), k . k

H¹(R^N)

) is separable, so is (H, k . k

H

). Let us define the linear operator T by,

T : H

_A,V¹

(R

^N

) −→ L

²

(R

^N

) × H u 7−→ ( √

V u, u),

with k (u, v) k

²_L²(R^N)×H

= k u k

²_L²(R^N)

+ k v k

²H

, for any (u, v) ∈ L

²

(R

^N

) × H. Clearly, L

²

(R

^N

) × H is separable. Thus, T H

_A,V¹

(R

^N

)

is also separable (Brezis [3]: Proposition 3.25, p.73). But for any u ∈ H

_A,V¹

(R

^N

),

k T (u) k

L²(R^N)×H

> k u k

H_A,V¹ (R^N)

, so that H

_A,V¹

(R

^N

) is separable.

Proof of Theorem 2.3. By Lemmas 2.15–2.18, it remains to show the continuous embeddings and the densities in (2.7). This comes from the fact that the embeddings in (2.6) are dense and from the reflexivity of the spaces D (R

^N

) and H

_A,V¹

(R

^N

).

Proof of Theorem 2.7. Estimate (2.8) comes from (1.15)–(1.17) and a straightforward calculation, while (2.9) is a consequence of the embeddings H

_A,V¹

(R

^N

) ֒ → H

¹

(R

^N

) and H

⁻¹

(R

^N

) ֒ → H

_A,V⁻¹

(R

^N

), due to (2.6) and (2.7).

3 The set of dislocations

Lemma 3.1. Let A satisfy (1.3), (1.5) and (1.6). Then for any y ∈ Z

^N

, there exists a unique continuous function ψ

y

∈ W

_loc^1,N+ε

(R

^N

; R) (ψ

y

∈ H

_loc¹

(R; R), if N = 1) such that

ψ

_y

(0) = 0, (3.1)

∀ x ∈ R

^N

, ψ

y

(x − y) + ψ

−y

(x) = ψ

y

( − y) = ψ

−y

(y), (3.2)

A(x + y) = A(x) + ∇ ψ

y

(x), (3.3)

for almost every x ∈ R

^N

. In particular, ψ

0

= 0 over R

^N

.

Proof. Let y ∈ Z

^N

. Uniqueness for ψ

_y

comes from (3.1) and (3.3), once continuity is proved.

By Remark 1.2 and the Sobolev embedding, there exists ψ f

_y

∈ W

_loc^1,N+ε

(R

^N

; R) satisfying (3.3) and continuous over R

^N

. Setting ψ

y

= ψ f

y

− ψ f

y

(0), we see that ψ

y

verifies (3.1) and (3.3). Notice that the function x 7−→ 0 satisfies (3.3) for y = 0, so that ψ

0

= 0, by uniqueness. It remains to establish (3.2). Applying (3.3) with y at the point x − y and a second time with − y, we obtain for almost every x ∈ R

^N

,

A(x − y) = A(x) − ∇ ψ

_y

(x − y) = A(x) + ∇ ψ

−y

(x).

It follows that there exists c ∈ R such that,

∀ x ∈ R

^N

, ψ

_y

(x − y) + ψ

−y

(x) = c.

Substituting first x = 0, then x = y and using (3.1) we obtain (3.2).

Lemma 3.2. Let A satisfy (1.3), (1.5) and (1.6). Let ψ

_y

y∈ZN

be given by Lemma 3.1. For any y ∈ Z

^N

, let ϕ

y

∈ W

_loc^1,N+ε

(R

^N

; R) be defined by,

ϕ

_y^def

= ψ

_y

− 1

2 ψ

_y

( − y), (3.4)

Then ϕ

y

∈ C(R

^N

; R) and verifies,

∀ x ∈ R

^N

, ϕ

_y

(x − y) + ϕ

−y

(x) = 0, (3.5)

A(x + y) = A(x) + ∇ ϕ

y

(x), (3.6)

for almost every x ∈ R

^N

. Finally, ϕ

0

= 0 over R

^N

.

Proof. By Lemma 3.1 and (3.4), we only have to check (3.5). The result then comes from (3.4) and (3.2).

Assume that A and V satisfy Assumptions 1.1 and 1.3, respectively. For any y ∈ Z

^N

, we define τ

_y

∈ L H

_A,V¹

(R

^N

)

as follows.

τ

y

: H

_A,V¹

(R

^N

) −→ H

_A,V¹

(R

^N

) u 7−→ e

^iϕy

u( · + y),

where ϕ

_y

is given by (3.4). Indeed, it is clear that τ

_y

: H

_A,V¹

(R

^N

) −→ L

²

(R

^N

) is linear and, Z

R^N

V | τ

y

u |

²

dx = Z

R^N

V | u |

²

dx, (3.7)

for any u ∈ H

_A,V¹

(R

^N

). In addition, by (3.6), we have for any y ∈ Z

^N

, u ∈ H

_A,V¹

(R

^N

) and almost every x ∈ R

^N

,

∇ (τ

y

u)(x) + iA(x)(τ

y

u)(x)

= ∇ u(x + y) + iA(x)u(x + y) + i ∇ ϕ

y

(x)u(x + y) e

^iϕy(x)

= ∇ u(x + y) + iA(x + y)u(x + y) e

^iϕy(x)

.

We deduce that τ

y

: H

_A,V¹

(R

^N

) −→ H

_A,V¹

(R

^N

) is well-defined, linear and k∇ (τ

y

u) + iA(τ

y

u) k

L²(R^N)

= k∇ u + iAu k

L²(R^N)

.

The above estimates and (3.7) permit us to see that for any y ∈ Z

^N

, τ

y

∈ L H

_A,V¹

(R

^N

) with k τ

_y

k

^L(H_A,V¹ (R^N))

= 1. Let

D

^def

=

τ

y

; y ∈ Z

^N

. (3.8)

Proposition 3.3. Let A and V satisfy Assumptions 1.1 and 1.3, respectively, and let D be defined by (3.8). Then D is a set of unitary operators on H

_A,V¹

(R

^N

). In addition,

τ

0

= Id, (3.9)

τ

_y⁻¹

= τ

_y^⋆

= τ

−y

, (3.10)

h τ

y

u, τ

y

v i

H_A,V¹ (RN)

= h u, v i

H¹_A,V(RN)

, (3.11) for any y ∈ Z

^N

and u, v ∈ H

_A,V¹

(R

^N

).

Proof of Lemma 3.3. Recall that D is set of bounded linear operators on H

_A,V¹

(R

^N

). By Lemma 3.2, ϕ

0

= 0 so that τ

0

= Id . Let y ∈ Z

^N

and let u ∈ H

_A,V¹

(R

^N

). For almost every x ∈ R

^N

, one has,

τ

y

τ

−y

u

(x) = e

^iϕy(x)

τ

−y

u

(x + y) = e

^iϕy(x)

e

^iϕ−y(x+y)

u(x) = u(x), where we have used (3.5) in the last equality. Still with (3.5), we show that τ

−y

τ

_y

u

= u. It follows that τ

y

is invertible and τ

_y⁻¹

= τ

−y

. Now, let v ∈ H

_A,V¹

(R

^N

). By a straightforward calculation and with help of (3.5) again and (3.6), we obtain

h u, τ

_y^⋆

v i

H_A,V¹ (R^N)

def

= h τ

y

u, v i

H_A,V¹ (R^N)

= h u, τ

_y⁻¹

v i

H_A,V¹ (R^N)

, so that, τ

_y^⋆

= τ

_y⁻¹

which concludes the proof.

Let us recall the following definition (see Definition 3.1, p.60, in Tintarev and Fieseler [14] and

Proposition 3.3).

Definition 3.4. Let A and V satisfy Assumptions 1.1 and 1.3, respectively, and let D be defined by (3.8). Let (u

n

)

n∈N

⊂ H

_A,V¹

(R

^N

) and u ∈ H

_A,V¹

(R

^N

). We shall say that the sequence (u

n

)

n∈N

converges to u D-weakly, which we will denote as, u

_n

−−

^D

⇀

n→∞

u, if

n→∞

lim h u

n

− u, τ

yn

v i

H¹_A,V(R^N)

= 0, for any sequence (τ

yn

)

n∈N

⊂ D and v ∈ H

_A,V¹

(R

^N

).

Notation 3.5. Let A and V satisfy Assumptions 1.1 and 1.3, respectively, and let D be defined by (3.8). Let (τ

yn

)

n∈N

⊂ D. We shall write,

τ

yn

−− ⇀

n→∞

0, to mean that for any u ∈ H

_A,V¹

(R

^N

), τ

yn

u −− ⇀

n→∞

0 in H

_A,V¹

-weakly, or equivalently,

n→∞

lim h τ

yn

u

,

v i

H_A,V¹ (R^N)

= 0, for any u, v ∈ H

_A,V¹

(R

^N

).

Remark 3.6. If u

_n

−−

^D

⇀

n→∞

0 then u

_n

−− ⇀

n→∞

0 in H

_A,V¹

-weakly. In particular, u

_n ^H

1

−− ⇀

w

n→∞

0 and for any sequence (τ

yn

)

n∈N

⊂ D, τ

yn

u

n

H¹w

−− ⇀

n→∞

0. Indeed, this follows from (3.9), (3.10) and (2.6).

Lemma 3.7. Let A and V satisfy Assumptions 1.1 and 1.3, respectively. Let (y

k

)

k

, (z

k

)

k

⊂ Z

^N

. Then,

τ

yk

τ

zk

−− ⇀

k→∞

0 ⇐⇒ | y

k

+ z

k

| −−−−→

_k→∞

∞ . (3.12) Moreover, if τ

_yk

τ

_zk

✚ −− ✚ ⇀ ✚

k→∞

0 then τ

_yk

τ

_zk

k

admits a constant subsequence.

Proof. Let (y

k

)

k

, (z

k

)

k

⊂ Z

^N

. Step 1: If lim inf

k→∞

| y

k

+ z

k

| < ∞ then (y

k

+ z

k

)

k

admits a constant subsequence.

Indeed, if lim inf

k→∞

| y

k

+ z

k

| < ∞ then (y

k

+ z

k

)

k

admits a bounded subsequence, from which we extract a convergent subsequence y

kℓ

+ z

kℓ

ℓ

. Since y

kℓ

+ z

kℓ

ℓ

converges in Z

^N

, Step 1 follows.

Step 2: Proof of = ⇒ .

We show the contraposition. Assume that lim inf

k→∞

| y

k

+z

k

| < ∞ . By Step 1, there exists y

kℓ

+ z

kℓ

ℓ

⊂