2D random magnetic Laplacian with white noise magnetic field

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2D random magnetic Laplacian with white noise magnetic field

Léo Morin, Antoine Mouzard

To cite this version:

Léo Morin, Antoine Mouzard. 2D random magnetic Laplacian with white noise mag- netic field. Stochastic Processes and their Applications, Elsevier, 2022, 142, pp.160-184.

�10.1016/j.spa.2021.10.004�. �hal-03107403v2�

2D random magnetic Laplacian with white noise magnetic field

Léo MORIN and Antoine MOUZARD

Abstract

We define the random magnetic Laplacian with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L

²

. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.

MSC 2020–35J10; 60H25; 47A52

Keywords –Magnetic Laplacian; White noise; Paracontrolled calculus; Spectral theory.

Contents

1 Definition of the operator 5

1.1 Construction of the domain . . . . 5

1.2 First properties of H . . . . 7

1.3 Form comparison between H and ∆ . . . 10

2 Self-adjointness and spectrum 11 2.1 The operator is symmetric . . . 11

2.2 The operator is self-adjoint . . . 12

2.3 H is the resolvent-limit of H

ε

. . . 13

2.4 Comparison between the spectrum of H and ∆ . . . 14 3 Stochastic renormalisation of the singular potential 15

A Paracontrolled calculus 17

Introduction

The magnetic Laplacian associated to a magnetic potential A in two dimensions H = (i∂

1

+ A

1

)

²

+ (i∂

2

+ A

2

)

²

is of interest in the description of a number of physical models. For example, it describes the behavior of a particule in a magnetic field B related to A via

B = ∇ × A = ∂

2

A

1

− ∂

1

A

2

.

While the case of constant magnetic field has been largely studied, the analysis of the magnetic Laplacian with nonconstant magnetic field gives rise to a number of interesting questions. Apart from the analogy with the electric Laplacian −∆ + V , the spectral study of the magnetic Laplacian is also motivated by supraconductivity, where H plays a specific role in the third critical field of Ginzburg-Landau theory, see [7]. This work is dedicated to the study of the magnetic Laplacian with random singular magnetic field given by the space white noise B = ξ on the two-dimensional torus T

²

. It can be constructed as a distribution with independant random Fourier coefficients with centered normal law of unit variance. In two dimensions, the space white noise belongs almost surely to the Sobolev spaces H

^−1−κ

or the Besov-Hölder spaces C

^−1−κ

for any κ > 0 . Since the associated potential A satisfies the equation

ξ = ∂

₂

A

₁

− ∂

₁

A

₂

,

each component A

1

, A

2

is expected to belong to C

^−κ

for any κ > 0 . Thus the potential is not a measurable function and the associated magnetic Laplacian falls out of the range of the classical theory: It is a singular random operator. The study of PDEs or operators where rough stochastic terms give rise to ill-defined operations such as product of distributions has intensified since the 2010 ’s with the parallel introduction of regularity structures in [10] by Hairer and the paracontrolled calculus in [8] by Gubinelli, Imkeller and Perkowski. A famous example of such singular operator is the Anderson Hamiltonian

−∆ + ξ

which naturally appears in the parabolic Anderson model describing random walks in random environment, see [13] for a survey on the topic. The random operator introduced and studied in this work is the magnetic analog of the Anderson Hamiltonian with a first order perturbation of the Laplacian.

The class of singular stochastic PDEs consists of equations with rough stochastic terms which give rise to ill-defined terms, classicaly products of distributions. Examples are given by the KPZ equation

(∂

t

− ∂

_x²

)u = ζ + (∂

x

u)

²

with ζ a spacetime white noise in dimension 1 + 1 or the Φ

⁴_d

equation (∂

t

− ∆)φ = −φ

³

+ ξ

with ξ a space white noise in dimension d ≥ 2 . These equations do not make sense a priori since solutions are expected to be too irregular for the terms (∂

_x

u)

²

or φ

³

to make sense. The recently developped approach consists in the construction of a random subspace of classical function spaces through a stochastic renormalisation procedure from the noise in which one can make sense almost surely of the equation. The different approaches of regularity structures and paracontrolled calculus differ by the tools used to study the singular products. In the first one, distributions are locally described by generalised Taylor expansions while the second one relies on tools from harmonic analysis with a description of distributions by frequency “schemes”. In both approaches, solutions are described by a richer stochastic object than the noise called the enhanced noise, the renormalisation procedure corresponds to its construction. This idea takes root in the theory of Lyons’ rough paths and Gubinelli’s controlled paths developped as a pathwise approach to stochastic integration. This has been used to study the Anderson Hamiltonian

−∆ + ξ

which is singular in dimension d ≥ 2 , the first time by Allez and Chouk using paracontrolled calculus on the two-dimensional torus, see [1]. See also [9, 16, 18] for more general settings including dimension d ∈ {2, 3} , different boundary conditions or a more general geometrical framework.

Following the approach for singular SPDEs, the idea is to construct a random subspace D

_Ξ

of L

²

as domain for the operator given an enhancement Ξ of the noise ξ such that (H, D

_Ξ

) is a well-defined unbounded operator in L

²

. The singularity is dealed with through a renormalisation procedure in the construction of the enhanced noise Ξ . In this work, we construct such a domain for the random magnetic Laplacian with white noise magnetic field.

Since the Anderson Hamiltonian appears, at least formally, as the continuum limit of discrete Anderson Hamiltonians with independant and identically distributed random potentials with finite variance, the random magnetic Laplacian naturally describes the formal continuous limit of Hamil- tonians with magnetic random fields. It is also of interest to the study of different PDEs as for the deterministic magnetic Laplacian. For example, it can be used to solve dispersive PDEs like the Schrödinger equation

∂

_t

u = (i∂

₁

+ A

₁

)

²

u + (i∂

₂

+ A

₂

)

²

u − |u|

²

u

or the wave equation. In particular, see [19] where Strichartz inequalities are obtained for the Anderson Hamiltonian as well as the random magnetic Laplacian constructed in the present work.

Main results

The 2D magnetic Laplacian with potential A : T

²

→ R

²

is formally given by H = −∆ + 2iA · ∇ + A · A + i div A

with associated magnetic field

B = ∇ × A.

From the physical point of view, one considers a given magnetic field B : T

²

→ R and chooses a potential A satisfying this equation. Different choices of A yield different operators H

A

, this is the choice of gauge. However, the physical quantities are expected to be independent of this choice and there is a partial invariance by gauge change. Indeed, if A : T

²

→ R

²

is a given potential associated to B , then the new potential A + ∇f is still a potential associated to B for functions f : T

²

→ R.

The two operators are related formally by the relation H

_A+∇f

= e

^if

H

_A

e

^−if

hence the spectral properties of H

_A

and H

_A+∇f

are the same. While on R

²

, every pair (A, A

⁰

) of smooth potential that satifies

∇ × A = ∇ × A

⁰

are related by

A = A

⁰

+ ∇f,

this is no longer the case on T

²

due to topological holes. One can show that the choice of a potential A can influence the spectrum of H

A

, see for example the Aharonov-Bohm effect in [12]. In this work, we consider the magnetic field B = ξ the space white noise and we choose the Lorentz gauge defined as follows. Take

A = ∇

^⊥

ϕ where ϕ := ∆

⁻¹

ξ.

Then A satifies indeed

ξ = ∇ × A

with the addition of div (A) = 0 . Our method could also deal with a gauge A + ∇f with A the Lorentz gauge and f a smooth deterministic function, see Section 3 for a discussion about this. It would be interesting to study the influence of a more general gauge but the implict definition

B = ∇ × A

makes more difficult the study of singularities. This leaves us with H = −∆ + 2iA · ∇ + A · A

where A ∈ C

^α−1

almost surely for any α < 1 . The term A · A is singular and one has to give a meaning to it using probabilistic arguments. This is done in Section 3 and yields the associated enhanced potential

A = (A, A

²

) ∈ C

^α−1

( T

²

, R

²

) × C

^2α−2

( T

²

, R ).

Remark that since A is a distribution of negative Hölder regularity, the singular product A · A is expected to worsen the regularity. Given such an enhanced potential A , we construct in Section 1 a dense subspace D

_A

⊂ L

²

such that

u ∈ D

A

⊂ L

²

= ⇒ Hu ∈ L

²

.

In Section 2, we show that (H, D

A

) is almost surely a self-adjoint operator with pure point spectrum.

We also prove that it is the resolvent-limit of

H

ε

= −∆ + 2iA

ε

· ∇ + A

²_ε

for any regularisation A

ε

= (A

ε

, A

²_ε

) ∈ C

^∞

( T

²

, R

²

) × C

^∞

( T

²

, R ) such that

ε→0

lim kA − A

ε

k

_C^α−2

+ kA

²

− A

²_ε

k

_C^2α−2

= 0.

Finally, we construct in Section 3 the enhanced potential A associated to the random magnetic field B = ξ . In particular, it is described by the limit in probability of

A

ε

, A

²_ε

− c

ε

as ε goes to 0 where A

ε

is a regularisation of A and

c

ε

= E

A

ε

(0) · A

ε

(0) .

In particular, the almost sure singularity of the product A · A implies the need to substract a

diverging constant c

_ε

as ε goes to 0 and a singular random operator has to be interpreted as the

description of the limiting behavior of a diverging system. One is interested in the fluctuations of

this system in this diverging frame, as the central limit Theorem for a simple random walk. For the case of the Anderson Hamiltonian, see the work [17] of Martin and Perkowski for a nice example.

Our construction is done in the same spirit as in [18] for the Anderson Hamiltonian with the additional difficulty of the first order perturbation here. In particular, this work illustrates that the heat paracontrolled calculus deals naturally with such terms and its flexibility to solve singular stochastic PDEs. In particular, this shows that the method could be used to deal with a general class of operators of the form

−∆ + a

1

· ∇ + a

2

with rough stochastic fields a

1

, a

2

and therefore associated time-dependant PDEs. A very important example of such operators in probability corresponds to the case a

2

= 0 where one recovers the infinitesimal generator of an SDE with drift a

1

. For example, it makes possible the study of processes as the Brox diffusion in a more general framework, see for example [15] by Kremp and Perkowski. This is current investigation and will be done in a subsequent paper. Finally since we only define and prove self-adjointness of the operator, no major differences appear in the study of the Anderson Hamiltonian or the random magnetic Laplacian with white noise magnetic field. As one does for smooth potentials, it is natural to pursue the study of both operators to understand their differences. For example, an interesting first question it to investigate the asymptotics of the eigenvalues as the volume of the torus goes to infinity.

In both the Anderson and the magnetic case, it would be interesting to study the influence of a scaling λ > 0 in the noise. For instance, the behavior of

−∆ + λξ

as λ to 0 should not be a problem however the limit λ to ∞ is much more delicate and of great interest. It is equivalent to the large-volume limit obtained by taking a torus of large diameter √

λ . It was studied for example in [5] by Chouk and Van Zuijlen using the large deviation principle, see also [14] for motivations from Probability. This is also equivalent to the semiclassical limit h goes to 0 in

−h∆ + ξ.

For regular potentials V , the semiclassical limit is deeply related to the classical dynamics described by the Hamiltonian

(x, v) ∈ T

^∗

( T

²

) 7→ |v|

²

+ V (x)

with T

^∗

( T

²

) the cotangent bundle of T

²

. Similarily, would it be possible to find an effective

“classical” equation describing the semiclassical limit in the case of a singular potential V = ξ or B = ξ ? See the book [12] from Helffer for details on the semiclassical limit and the work [6]

by Dumaz and Labbé for a very precise description of the asymptotics in one dimension for the Anderson Hamiltonian.

Organisation of the paper

In the first section, we construct the domain and prove density in L

²

. We compare the graph norm and the natural norms of the domain which gives closedness of the operator. We also give an explicit form comparison between the random magnetic Laplacian H and the Laplacian ∆ . In the second section, we show that the operator is symmetric as a weak limit of the regularised operator.

The form comparison of H and ∆ with the Babuška-Lax-Milgram Theorem gives self-adjointness.

Finally, we show that H is the resolvent-limit of the regularised operator H

ε

in Proposition 2.4 and compare the spectrum of H and ∆ . In particular, this implies an almost sure Weyl-type law for the random magnetic Laplacian. The third Section deals with the construction of the enhanced potential A built from the noise ξ through a renormalisation procedure. We gather in Appendix A all the results we need for the paracontrolled calculus and refer to [18] for the details. In particular, P and Π respectively denotes the paraproduct and the resonent term such that

f g = P

f

g + P

g

f + Π(f, g) and P e is a paraproduct intertwinned with P via the relation

∆ ◦ P e = P ◦ ∆

and satisfies the same continuity properties as P . We also denote

∆

⁻¹

:= − Z

1

0

e

^t∆

dt

an inverse for the Laplacian up to the regularising operator e

^∆

.

1 – Definition of the operator

In this section, we first construct the domain and show that its natural norms are equivalent to the graph norm of H . In particular, this guarantees the closedness of the operator. Finally, we compare the respective forms associated to H and L .

1.1 – Construction of the domain

Fix α ∈ (

²₃

, 1) and let A be an enhanced magnetic potential

A = (A, A

²

) ∈ X

^α

:= C

^α−1

( T

²

, R

²

) × C

^2α−2

( T

²

, R ) with its natural norm

kAk

X^α

:= kAk

_C^α−1

+ kA

²

k

_C^2α−2

.

For A ∈ L

^∞

, the term A

²

can be interpreted as A·A while it is not defined if A is only a distribution.

It is enhanced in the sense that one does not have a natural interpretation of A · A for singular potential, this is specified by the additional data A

²

. Section 3 is devoted to the particular case of magnetic white noise where A

²

is constructed through a probabilistic renormalisation procedure.

Thus we refer as noise-dependent a quantity that depends on this enhanced potential A . For any regular function u ∈ C

^∞

( T

²

) , we have

2iA · ∇u + A

²

u = P

_∇u

2iA + P

u

A

²

+ (])

where (]) ∈ C

^∞

( T

²

) with P

u

2iA ∈ H

^α−1

and P

∇u

A

²

∈ H

^2α−2

. Assuming Hu ∈ L

²

yields

−∆u = Hu − 2iA · ∇u + A

²

u ∈ H

^2α−2

since 2α−2 < α−1 hence u is expected to belong to H

^2α

by elliptic regularity theory. For u ∈ H

^2α

, we have

2iA · ∇u + A

²

u = 2iP

_∇u

A + 2iP

_A

∇u + 2iΠ(∇u, A) + P

_u

A

²

+ P

_A2

u + Π(u, A

²

)

= (α − 1) + (3α − 2) + (2α − 2) + (4α − 2)

= P

u

A

²

+ P

_∇u

2iA + (3α − 2)

where (β ) denotes a term of formal regularity H

^β

for any β ∈ R. Following the paracontrolled calculus approach, we want to consider a paracontrolled function of the form

u = P e

u

X

1

+ e P

_∇u

X

2

+ u

^]

with u

^]

a smoother remainder such that Hu ∈ L

²

. Thus we take

X

₁

:= ∆

⁻¹

(A

²

) and X

₂

:= ∆

⁻¹

(2iA) and define the domain of H as follows.

Definition. We define the set D

A

of functions paracontrolled by A as D

A

:= {u ∈ L

²

; u − P e

_u

X

₁

− P e

_∇u

X

₂

∈ H

²

}.

The domain is defined as

D

A

= Φ

⁻¹

(H

²

) with

Φ(u) := u − e P

u

X

1

− P e

∇u

X

2

.

However the domain could be anything from trivial to dense in L

²

a priori. For s ∈ (0, 1) , we introduce the map Φ

^s

as

Φ

^s

:

D

A

→ H

²

u 7→ u − P e

^s_u

X

1

− P e

^s_∇u

X

2

with e P

^s

the paraproduct truncated at scale s ; see Appendix A for the definition. In particular, the map

Φ

^s

: H

^β

→ H

^β

is a perturbation of the identity for any β ∈ [0, 2α) invertible for s small enough, we denote its inverse Γ . Since P e

v

− P e

^s_v

X is a smooth function, the domain is also given by D

A

= (Φ

^s

)

⁻¹

(H

²

) = Γ(H

²

).

The reader should keep in mind that Γ implicitely depends on s , we do not keep it in the notation to lighten this work. This parametrisation of the domain will be crucial to prove that the domain is dense in L

²

and to study H . In particular, sharp bounds on the eigenvalues of H are needed to get a Weyl-type law for H . To do so, we need to keep a careful track of the different constants.

The reader interested only in the construction of the operator and its self-adjointess can skip these computations. Obtaining sharp bounds requires explicit constants with respect to the parameter s and the regularity exponent in the paracontrolled calculus. We recall the needed bounds in Appendix A, see [18] for details and proofs. For β ∈ [0, 2α) , let

s

β

(A) :=

β

^∗

mkAk

X^α

_2α−β⁴

where β

^∗

= 1 − β if β ∈ [0, 1) and β

^∗

= 2α − β if β ∈ [1, 2α) and m > 0 is a universal constant, see Appendix A. The following Proposition gives regularity estimates for Φ

^s

and Γ .

Proposition 1.1. Let β ∈ [0, 2α) and s ∈ (0, 1). We have

kΦ

^s

(u) − uk

_Hβ

≤ m s

^2α−β4

β

^∗

kAk

_Xα

kuk

_Hβ

.

In particular, s < s

β

(A) implies that the map Φ

^s

is invertible and its inverse Γ verifies the bound

kΓu

^]

k

_Hβ

≤ 1 1 − m

^s

2α−β 4

β^∗

kAk

_X^α

ku

^]

k

_Hβ

.

Proof : If β < 1 , the bounds on Φ

^s

follow directly from

ke P

^s_u

X

1

+ e P

^s_∇u

X

2

k

_Hβ

≤ m s

^2α−β4

1 − β kuk

L²

kX

1

k

_C^2α

+ m s

^α+1−β4

1 − β k∇uk

_H^β−1

kX

2

k

_Hα+1

≤ m s

^2α−β4

1 − β kuk

_Hβ

kX

₁

k

_C2α

+ kX

₂

k

_Cα+1

≤ m s

^2α−β4

1 − β kuk

_Hβ

kAk

_X^α

. For β ∈ [1, 2α) , we have

ke P

^s_u

X

₁

+ P e

^s_∇u

X

₂

k

_Hβ

≤ m s

^2α−β4

2α − β kuk

_L2

kX

₁

k

_Cα

+ m s

^α+1−β2

α + 1 − β k∇uk

_L2

kX

₂

k

_Hα+1

≤ m s

^2α−β4

2α − β kuk

_H1

kX

1

k

_C2α

+ kX

2

k

_Cα+1

≤ m s

^2α−β4

2α − β kuk

_H1

kAk

X^α

. The result for Γ follows since 2α − β > 0 .

We also consider the associated maps Φ

^s_ε

and Γ

ε

for a regularisation A

ε

of the enhanced poten- tial. It is defined as

Φ

^s_ε

(u) := u − e P

u

X

₁^(ε)

− e P

∇u

X

₂^(ε)

and

Γ

ε

u

^]

= P e

_Γεu]

X

₁^(ε)

+ P e

_∇Γεu]

X

₂^(ε)

+ u

^]

where

X

₁^(ε)

:= ∆

⁻¹

(A

²_ε

) and ∆X

₂^(ε)

:= 2iA

_ε

.

These satisfy the same continuity properties as Φ

^s

and Γ with bounds uniform with respect to ε .

Moreover, we have the following approximation Lemma holds.

Lemma 1.2. Let β ∈ [0, 2α) and s ∈ (0, 1). If s ≤ s

β

(A), we have

k Id − ΓΓ

⁻¹_ε

k

_L2→H^β

.

^A,s

kA − A

ε

k

X^α

. In particular, this implies the convergence of Γ

ε

to Γ with the bound kΓ − Γ

_ε

k

_Hβ→H^β

.

^A,s

kA − A

_ε

k

X^α

.

Proof : Given any u ∈ H

^β

, we have u = ΓΓ

⁻¹

(u) = Γ(u − e P

^s_u

X

1

− e P

^s_∇u

X

2

) . We get ku − ΓΓ

⁻¹_ε

(u)k

_Hβ

=

Γ u − P e

^s_u

X

1

− P e

^s_∇u

X

2

− Γ u − e P

^s_u

X

₁^(ε)

− e P

^s_∇u

X

₂^(ε)

H^β

= Γ

e P

^s_u

X

₁^(ε)

− X

₁

+ e P

^s_∇u

X

₂^(ε)

− X

₂

_Hβ

.

^A,s

P e

^s_u

X

₁^(ε)

− X

1

+ P e

^s_∇u

X

₂^(ε)

− X

2

H^β

.

^A,s

kA

ε

− Ak

_X2α

kuk

_L2

since s < s

_β

(A) implies the continuity of Γ : H

^β

→ H

^β

and X

_i^(ε)

− X

_i

depends linearly on A

_ε

− A for i ∈ {1, 2} . The result on Γ − Γ

_ε

follows from the bound on Γ

_ε

uniform with respect to ε .

This allows to prove density of the domain.

Corollary 1.3. The domain D

A

is dense in H

^β

for every β ∈ [0, 2α).

Proof : Given f ∈ H

²

, Γ(g

ε

) ∈ D

A

where g

ε

= Γ

⁻¹_ε

f ∈ H

²

thus we can conclude with the previous Lemma that

ε→0

lim kf − Γ(g

ε

)k

_Hβ

= 0.

The density of H

²

in H

^β

completes the proof.

1.2 – First properties of H

Since H is formally given by

H = −∆ + 2iA · ∇ + A · A,

we are able to define (H, D

A

) as an unbounded operator in L

²

associated to the enhanced potential A .

Definition 1.4. We define H : D

_A

⊂ L

²

→ L

²

as

Hu := −∆u

^]

+ R(u) where u

^]

= Φ(u) and

R(u) := P

2iA

∇u + Π(∇u, 2iA) + P

_A2

u + Π(u, A

²

).

The definition of H is independant of the parameter s ∈ (0, 1) . It is a very usefull tool to get differents bounds on the operator with the different representations

Hu = −∆u

^]_s

+ R(u) + Ψ

^s

(u) where u

^]_s

:= Φ

^s

(u) and

Ψ

^s

(u) := −∆ e P

_u

− P e

^s_u

X

₁

− ∆ P e

_∇u

− P e

^s_∇u

X

₂

∈ C

^∞

( T

²

).

For example, we can compare the graph norm of H

kuk

²_H

:= kuk

²_L2

+ kHuk

²_L2

and the natural norms of the domain

kuk

²_DA

:= kuk

²_L2

+ kΦ

^s

(u)k

²_H2

with the following Proposition provided s is small. Let β :=

¹₂

(

⁴₃

+ 2α) and δ > 0 . For s ∈ (0, 1) such that s < s

β

(A) , we introduce the constant

m

²_δ

(A, s) := ks

^α−22

kAk

_X^α

+ kδ

^−2−ββ





kAk

_X^α

1 − m

^s

2α−β 4

β^∗

kAk

_Xα





2 2−β

1 + s

^α2

kAk

_X^α

with k > 0 a large enough constant depending. In particular, m

²_δ

(A, s) diverges as s goes to 0 or s

_β

(A) or as δ goes to 0 .

Proposition 1.5. Let u ∈ D

_A

and s ∈ (0, 1) such that s < s

_β

(A). Then for any δ > 0, we have (1 − δ)ku

^]_s

k

_H²

≤ kHuk

L²

+ m

²_δ

(A, s)kuk

L²

and

kHuk

L²

≤ (1 + δ)ku

^]_s

k

_H²

+ m

²_δ

(A, s)kuk

L²

with u

^]_s

= Φ

^s

(u).

Proof : Recall that for any s ∈ (0, 1) , the operator is given by Hu = −∆u

^]_s

+ R(u) + Ψ

^s

(u) thus we need to bound R and Ψ

^s

. For u ∈ D

A

, we have

kP

2iA

∇u + Π(∇u, 2iA)k

L²

. k2iAk

_C^α−1

kuk

_Hβ

kP

_A2

u + Π(u, A

²

)k

_L2

. kA

²

k

_C2α−2

kuk

_Hβ

hence

kR(u)k

_L2

. kAk

_X^α

kuk

_Hβ

. We also have

kΨ

^s

(u)k

_L2

. k(e P

u

− e P

^s_u

)X

1

+ (e P

_∇u

− e P

^s_∇u

)X

2

k

_H2

. s

^α−22

kAk

_X^α

kuk

_L2

. For s < s

β

(A) , Proposition 1.1 gives

kuk

_Hβ

≤ 1 1 − m

^s

2α−β 4

β^∗

kAk

_Xα

ku

^]_s

k

_Hβ

thus we get

kHu + ∆u

^]_s

k

L²

. kAk

_Xα

1 − m

^s

2α−β 4

β^∗

kAk

_X^α

ku

^]_s

k

_Hβ

+ s

^α−22

kAk

X^α

kuk

L²

.

Since 0 < β < 2 , we have for any t > 0

ku

^]_s

k

_Hβ

.

Z

t 0

(−t

⁰

∆)e

^t0∆

u

^]_s

dt

⁰

t

⁰

_Hβ

+ e

^t∆

u

^]_s

_Hβ

. t

^2−β2

ku

^]_s

k

_H2

+ t

^−β2

ku

^]_s

k

_L2

. t

^2−β2

ku

^]_s

k

_H²

+ t

^−β2

1 + s

^2α4

kAk

X^α

kuk

L²

.

For any δ > 0 , take

t =





δ 1 − m

^s

2α−β 4

β^∗

kAk

_X^α

kkAk

_Xα





2 2−β

with k the constant from the previous inequality. This yields

kHu + ∆u

^]_s

k

_L2

. m

²_δ

(A, s)kuk

_L2

+ δku

^]_s

k

_H2

. and completes the proof.

Remark : In comparison to the work [18] on the Anderson Hamiltonian

u 7→ −∆u + uξ

where the space white noise can be interpreted as an electric potential, one needs s small for these bounds to hold here. In fact, one could perform the same kind of expansion with

∆X

₂

= P

_∇X1

2iA

at the price of a nonlinear dependance of X

₂

with respect to A in order to bypass the smallness condition on s. This would change the different bounds one get for Φ

^s

and Γ but still yield a self- adjoint operator that is the limit of the regularised H

_ε

. Theorem XIII .26 from [21] guarantees that the different choice of construction coincide provided that H is self-adjoint.

In particular, this implies that (H, D

A

) is a closed operator in L

²

. Proposition 1.6. The operator H is closed on its domain D

A

.

Proof : Let (u

n

)

n≥0

⊂ D

A

be a sequence such that

u

n

→ u in L

²

and Hu

n

→ v in L

²

. Proposition 1.5 gives that Φ

^s

(u

_n

)

n≥0

is a Cauchy sequence in H

²

hence converges to u

^]_s

∈ H

²

for s < s

β

(A) . Since Φ

^s

: L

²

→ L

²

is continuous, we have Φ

^s

(u) = u

^]_s

hence u ∈ D

A

. Finally, we have

kHu − vk

L²

≤ kHu − Hu

_n

k

L²

+ kHu

n

− vk

L²

.

A

ku

^]_s

− Φ

^s

(u

_n

)k

_H2

+ ku − u

_n

k

_L2

+ kHu

_n

− vk

_L2

hence Hu = v and H is closed on D

A

.

We conclud this Section by computing the Hölder regularity of the functions in the domain.

Corollary. We have

D

A

⊂ C

^1−κ

for any κ > 0.

Proof : The Besov embedding in two dimensions implies H

²

, → B

¹_∞,2

, → L

^∞

and Φ

^s

: L

^∞

→ L

^∞

is also invertible for s small enough hence D

A

= (Φ

^s

)

⁻¹

(H

²

) ⊂ L

^∞

. First for u ∈ D

A

, we have

kuk

_C^α

. kuk

L^∞

kX

1

k

_C^α

+ k∇uk

_C⁻¹

kX

2

k

_Cα+1

+ ku

^]

k

_C^α

.

^A

kuk

L^∞

+ ku

^]

k

_H²

.

Finally, this gives

kuk

_C^1−κ

. kuk

L^∞

kX

1

k

_C^1−κ

+ k∇uk

_C^α−1

kX

2

k

_C^2−α+κ

+ ku

^]

k

_C^1−κ

.

A

kuk

_L^∞

+ kuk

_Cα

+ ku

^]

k

_H2

and the proof is complete.

1.3 – Form comparison between H and ∆

We have proven in Theorem 1.5 that Hu can be seen as a small perturbation of −∆u

^]

in norm.

Here, we prove a similar statement in the quadratic form setting. Let η :=

^α₄

and δ > 0 . For s ∈ (0, 1) such that s < s

_1−η

(A) , define

m

¹_δ

(A, s) := (1 + s

^α−22

)kAk

_X^α

+ δ

^−1−ηη

kAk

X^α

(1 − η

⁻¹

s

^2α+η−14

kAk

_Xα

)

²

!

¹_η

(1 + s

^α2

kAk

_X^α

)

with k > 0 a large enough constant. In particular, m

¹_δ

(A, s) diverges as s goes to 0 or s

_1−η

(A) or as δ goes to 0 .

Proposition 1.7. Let u ∈ D

A

and s ∈ (0, 1) such that s < s

₁₋^α

4

(A). For any δ > 0, we have (1 − δ)h∇u

^]_s

, ∇u

^]_s

i ≤ hu, Hui + m

¹_δ

(A, s)kuk

²_L2

and

(1 − δ)h∇u

^]_s

, ∇u

^]_s

i ≤ hu, H

ε

ui + m

¹_δ

(A, s)kuk

²_L2

where u

^]_s

= Φ

^s

(u).

Proof : For u ∈ D

A

, recall that

Hu = −∆u

^]_s

+ R(u) + Ψ

^s

(u) where u

^]_s

= Φ

^s

(u) ∈ H

²

. We have

u, −∆u

^]_s

=

e P

^s_u

X

1

, −∆u

^]_s

+

P e

^s_∇u

X

2

, −∆u

^]_s

+

u

^]_s

, −∆u

^]_s

= −

P

^s_u

∆X

1

, u

^]_s

−

P

^s_∇u

∆X

2

, u

^]_s

+

∇u

^]_s

, ∇u

^]_s

thus

hu, Hui = −

P

^s_u

∆X

₁

, u

^]_s

−

P

^s_∇u

∆X

₂

, u

^]_s

+

∇u

^]_s

, ∇u

^]_s

+

u, R(u) +

u, Ψ

^s

(u) .

For η ≤

^α₂

, we have

P

^s_u

∆X

₁

, u

^]_s

. kP

^s_u

∆X

₁

k

_H2α−2

ku

^]_s

k

_H1−η

. kAk

_Xα

kuk

_L2

ku

^]_s

k

_H1−η

,

P

^s_∇u

∆X

2

, u

^]_s

. kP

^s_∇u

∆X

2

k

_Hα−1−η

ku

^]_s

k

_H1−η

. kAk

_X^α

kuk

_H1−η

ku

^]_s

k

_H1−η

,

u, P

_2iA

∇u

. kuk

_H1−η

kP

_2iA

∇uk

_Hα−1−η

. kAk

_Xα

kuk

²_H1−η

,

u, P

A²

u + Π(u, A

²

)

. kuk

L²

kP

A²

u + Π(u, A

²

)k

L²

. kAk

X^α

kuk

L²

kuk

_H^1−η

,

u, Ψ

^s

(u)

. kuk

L²

k(P

u

− P

^s_u

)LX

1

+ (P

∇u

− P

^s_∇u

)LX

2

k

L²

. s

^α−22

kAk

X^α

kuk

²_L2

. The only term that is not a priori controlled is

u, Π(2iA, ∇u) since the resonant term is singular if we only suppose that u ∈ H

¹

; this is where the almost duality property comes into play. We have

u, Π(2iA, ∇u)

=

P

u

2iA, ∇u

+ F(u, 2iA, ∇u)

with the corrector F(u, 2iA, ∇u) controlled if u ∈ H

^1−η

with η <

^α₂

. The paraproduct is not singular however one can not use better regularity than L

²

for u thus we use an integration by part to get

P

u

2iA, ∇u

= −

div (P

u

2iA), u

= −

P

u

div (2iA), u +

B(u, 2iA), u with

B a, (b

1

, b

2

)

:= div P

a

(b

1

, b

2

)

− P

a

div (b

1

, b

2

), see Appendix A for continuity estimates on B . We have

u, Π(2iA, ∇u) .

B(u, 2iA), u

+ |F(u, 2iA, ∇u)| . kAk

_Xα

kuk

²_H1−η

since div (A) = 0 . Since s < s

_1−η

(A) , we get

hu, Hui − h∇u

^]_s

, ∇u

^]_s

i

. (1 + s

^α−22

)kAk

_Xα

kuk

²_L2

+ kAk

_Xα

1 − η

⁻¹

s

^2α+η−14

kAk

_Xα

²

ku

^]_s

k

²_H1−η

.

To complete the proof, one only has to interpolate the H

^1−η

norm of u

^]_s

between its H

¹

norm and its L

²

norm which is controlled by the L

²

norm of u , as in the proof of Proposition 1.5. Since 0 < 1 − η < 1 , we have for any t > 0

ku

^]_s

k

_H1−η

.

Z

t 0

(−t

⁰

∆)e

^t0∆

u

^]_s

dt

⁰

t

⁰

H^1−η

+ e

^t∆

u

^]_s

H^1−η

. t

^η2

ku

^]_s

k

_H1

+ t

^−1−η2

ku

^]_s

k

_L2

. t

^η2

ku

^]_s

k

_H1

+ t

^−1−η2

1 + s

^2α4

kAk

X^α

kuk

L²

.

For any δ > 0 , take

t = δ 1 − η

⁻¹

s

^2α+η−14

kAk

X^α

2

kkAk

_X^α

!

η²

with k the constant from the previous inequality. This yields hu, Hui − h∇u

^]_s

, ∇u

^]_s

i

. m

²_δ

(A, s)kuk

_L2

+ δku

^]_s

k

_H1

. and completes the proof.

2 – Self-adjointness and spectrum

In this section, we prove that H is self-adjoint with pure point spectrum. It is symmetric since H Γ is the limit in norm of the regularised H

ε

Γ

ε

as proved in Section 2.1. Hence it is enough to prove that

H + k : D

A

→ L

²

is surjective for some k ∈ R, this is the content of Section 2.2. In Section 2.3, we prove that H

ε

converges to H in the stronger resolvent sense. Finally, we give in Section 2.4 bounds for the eigenvalues of H using the different representation

H = −∆Φ

^s

+ R + Ψ

^s

parametrised by s ∈ (0, 1) from the eigenvalues of −∆ . In particular, it implies a Weyl-type law for H .

2.1 – The operator is symmetric

To prove that H is symmetric, we use the regularised operator H

ε

. Recall that (H

ε

, H

²

) is self- adjoint and that Φ

^s_ε

: H

²

→ H

²

is continuous. In some sense, the operator H should be the limit of

H

ε

:= −∆ + 2iA

ε

· ∇ + A

²_ε

as ε goes to 0 with A

ε

:= (A

ε

, A

²_ε

) a smooth approximation of A in X

^α

. Since D(H

ε

) = H

²

, one can not compare directly the operators. However given any u ∈ L

²

, we have

u = Γ ◦ Φ

^s

(u) = lim

ε→0

Γ

ε

◦ Φ

^s

(u).

Thus for u ∈ D

A

, the approximation u

ε

:= Γ

ε

◦ Φ

^s

(u) belongs to H

²

and one can consider the difference

kHu − H

_ε

u

_ε

k

L²

= k(H Γ − H

_ε

Γ

_ε

)u

^]

k

L²

with u

^]

:= Φ

^s

(u) . The following Proposition assures the convergence of H

_ε

Γ

_ε

to H Γ provided s < s

β

(A) where β =

¹₂

(

⁴₃

+ 2α) .

Proposition 2.1. Let u ∈ D

A

and s ∈ (0, 1) such that s < s

_β

(A). Then kHu − H

_ε

u

_ε

k

L²

.

^A,s

ku

^]_s

k

_H²

kA − A

_ε

k

X^α

with u

^]_s

= Φ

^s

(u) and u

_ε

:= Γ

_ε

u

^]_s

. In particular, this implies that H

_ε

Γ

_ε

converges to HΓ in norm

as ε goes to 0 as operators from H

²

to L

²

.

Proof : We have

H

ε

u

ε

= −∆u

^]_s

+ R

ε

(u

ε

) + Ψ

^s_ε

(u

ε

)

where R

_ε

and Ψ

^s_ε

are defined as R and Ψ

^s

with A

_ε

instead of A . Since

⁴₃

< β < 2α , we have kR(u)−R

ε

(u

ε

)k

L²

≤ kR(u − u

ε

)k

L²

+ k(R − R

ε

)(u

ε

)k

L²

.

^s,A

ku − u

ε

k

_Hβ

+ kA − A

ε

k

_Xα

ku

ε

k

_Hβ

and

kΨ

^s

(u) − Ψ

^s_ε

(u

_ε

)k

L²

.

s,A

ku − u

_ε

k

L²

+ kA − A

_ε

k

X^α

kuk

L²

and the proof is complete since s < s

β

(A) implies

kuk

_Hβ

.

s,A

ku

^]_s

k

_Hβ

.

The symmetry of H immediately follows.

Corollary. The operator H is symmetric.

Proof : Let u, v ∈ D

A

and consider u

^]

:= Φ

^s

(u) and v

^]

:= Φ

^s

(v) for s < s

β

(A) . Since H

ε

is a symmetric operator, we have

hHu, vi = lim

ε→0

hH

ε

Γ

_ε

u

^]_s

, Γ

_ε

v

_s^]

i = lim

ε→0

hΓ

ε

u

^]_s

, H

_ε

Γ

_ε

v

^]_s

i = hu, Hvi using that H

_ε

Γ

_ε

converges to H Γ and Γ

_ε

to Γ in norm convergence.

2.2 – The operator is self-adjoint

In this Section, we prove that (H, D

A

) is self-adjoint. Being closed and symmetric, it is enough to prove that

(H + k)u = v

admits a solution for some k ∈ R, see Theorem X .1 in [22]. This is done using the Babuška-Lax- Milgram Theorem, see [2], and Theorem 1.7 which implies that H is almost surely bounded below for any δ ∈ (0, 1) and s small enough.

Proposition 2.2. Let δ ∈ (0, 1) and s ∈ (0, 1) such that s < s

₁₋^α

4

(A). For k > m

¹_δ

(A, s), the operators H + k and H

ε

+ k are invertibles as unbounded operator in L

²

. Moreover the operators

H + k

−1

: L

²

→ D

A

H

_ε

+ k

−1

: L

²

→ H

²

are bounded.

Proof : Since s < s

₁₋^α

4

(A) and k > m

¹_δ

(A, s) , Proposition 1.7 gives k − m

¹_δ

(A, s)

kuk

²_L2

<

(H + k)u, u for u ∈ D

A

. Considering the norm

kuk

²_DA

= kuk

²_L2

+ ku

^]_s

k

²_H2

on D

A

, this yields a weakly coercive operator using Proposition 1.5 in the sense that kuk

DA

.

A

k(H + k)uk

L²

= sup

kvk_L2=1

(H + k)u, v

for any u ∈ D

A

. Moreover, the bilinear map

B : D

_A

× L

²

→ R (u, v) 7→

(H + k)u, v

is continuous since Proposition 1.5 implies

|B(u, v)| ≤ k(H + k)uk

L²

kvk

L²

.

^A

kuk

D_A

kvk

L²

for u ∈ D

A

and v ∈ L

²

. The last condition we need is that for any v ∈ L

²

\{0} , we have sup

kukDA=1

|B(u, v)| > 0.

Let assume that there exists v ∈ L

²

such that B(u, v) = 0 for all u ∈ D

A

. Then

∀u ∈ D

A

, hu, vi

_DA,D^∗

A

= 0.

hence v = 0 as an element of D

^∗_A

. By density of D

_A

in L

²

, this implies v = 0 in L

²

hence the property we want. By the Theorem of Babuška-Lax-Milgram, for any f ∈ L

²

there exists a unique u ∈ D

A

such that

∀v ∈ L

²

, B(u, v) = hf, vi.

Moreover, we have kuk

_DA

.

^A

kf k

_L2

hence the result for (H + k)

⁻¹

. The same argument works for H

ε

+ k since Peroposition 1.7 also holds for H

ε

with bounds uniform in ε .

As explained before, this immediatly implies that H is self-adjoint. Moreover, the resolvent is a compact operator from L

²

to itself since D

A

⊂ H

^β

for any β ∈ [0, 2α) hence it has pure point spectrum.

Corollary 2.3. The operator H is self-adjoint with discret spectrum λ

_n

(A)

n≥1

which is a nonde- creasing diverging sequence without accumulation points. Moreover, we have

L

²

= M

n≥1

Ker H − λ

n

(A)

with each kernel being of finite dimension. We finally have the min-max principle λ

n

(A) = inf

D

sup

u∈D;kuk_L2=1

hHu, ui

where D is any n-dimensional subspace of D

A

that can also be written as λ

n

(A) = sup

v₁,...,vn−1∈L²

inf

u∈Vect(v1,...,v_n−1)^⊥

kukL2 =1

hHu, ui.

2.3 – H is the resolvent-limit of H

_ε

Since the intersection of the domains of H and H

_ε

is trivial, the natural convergence of H

_ε

to H is in the resolvent sense, this is the following Proposition. In particular, this result explains why our operator H is natural since the regularised operator satisfies

i∂

₁

+ A

^(ε)₁

²

+ i∂

₂

+ A

^(ε)₂

²

= H + c

_ε

+ o

_ε→0

(1) in the norm resolvent sense.

Proposition 2.4. Let δ > 0 and s ∈ (0, 1) such that s < s

β

(A). Then for any constant k > m

¹_δ

(A, s), we have

k(H + k)

⁻¹

− (H

ε

+ k)

⁻¹

k

_L2→L²

.

^A,s

kA − A

ε

k

_X^α

.

Proof : Let v ∈ L

²

. Since H + k : D

A

→ L

²

is invertible, there exists u ∈ D

A

such that v = (H + k)u

thus

k(H + k)

⁻¹

v − (H

_ε

+ k)

⁻¹

vk

L²

= ku − (H

_ε

+ k)

⁻¹

(H + k)uk

L²

.

We introduce u

ε

:= Γ

ε

Φ

^s

(u) which converges to u in L

²

and we have

ku − (H

ε

+ k)

⁻¹

(H + k)uk

L²

≤ ku − u

ε

k

L²

+ ku

ε

− (H

ε

+ k)

⁻¹

(H + k)uk

L²

. Since Lemma 1.2 gives

ku − u

_ε

k

L²

.

A,s

kA − A

_ε

k

X^α

, we only have to bound the second term. We have

ku

ε

− (H

ε

+ k)

⁻¹

(H + k)uk

_L2

= k(H

ε

+ k)

⁻¹

(H

ε

+ k)u

ε

− (H + k)u k

_L2

. k(H

ε

+ k)u

ε

− (H + k)uk

L²

. kH

ε

u

ε

− Huk

_L2

+ kku

ε

− uk

_L2

using Proposition 2.1. In the end, we have

k(H + k)

⁻¹

v − (H

ε

+ k)

⁻¹

vk

L²

. ku

^]_s

k

_H²

kA − A

ε

k

X^α

hence the result since (H + k)

⁻¹

: L

²

→ D

A

is continuous.

2.4 – Comparison between the spectrum of H and ∆

The following Proposition provides sharp bounds for the eigenvalues of H from the eigenvalues of

∆ , we denote by (λ

_n

)

_n≥1

the non-decreasing positive sequence of the eigenvalues of −∆ since it corresponds to the case A = 0 . For δ ∈ (0, 1) and s ∈ (0, 1) , introduce the constant

m

⁺_δ

(A, s) := (1 + δ)(1 + ms

^α2

kAk

_Xα

).

For s < s

0

(A) , we also introduce

m

⁻_δ

(A, s) := 1 − δ 1 − ms

^α2

kAk

X^α

.

Recall β =

¹₂

(

⁴₃

+ 2α) . Write a, b ≤ c to mean that we have both a ≤ c and b ≤ c . Proposition 2.5. Let δ ∈ (0, 1) and s ∈ (0, 1) such that s < s

β

(A) ∧ s

₁₋^α

4

(A). Given any n ∈ Z

⁺

, we have

λ

n

(A), λ

n

(A

ε

) ≤ m

⁺_δ

(A, s)λ

n

+ 2 + 2ms

^α2

kAk

_X^α

+ m

²_δ

(A, s).

If moreover s < s

0

(A), we have

λ

n

(A), λ

n

(A

ε

) ≥ m

⁻_δ

(A, s)λ

n

− m

¹_δ

(A, s).

Proof : Let u

^]₁

, . . . , u

^]_n

∈ H

²

be an orthonormal family of eigenfunctions of −∆ associated to λ

1

, . . . , λ

n

and consider

u

i

:= Γu

^]_i

∈ D

A

for 1 ≤ i ≤ n . Since Γ is invertible, the family (u

1

, . . . , u

n

) is free thus the min-max representation of λ

_n

(A) yields

λ

_n

(A) ≤ sup

u∈Vect(u1,...,un)

kukL2 =1

hHu, ui.

Given any normalised u ∈ Vect (u

1

, . . . , u

n

) , we have

hHu, ui ≤ kHuk

L²

≤ (1 + δ)ku

^]_s

k

_H²

+ m

²_δ

(A, s) for u

^]_s

= Φ

^s

(u) using Proposition 1.5 and s < s

_β

(A) . Moreover

ku

^]_s

k

_H2

≤ (1 + λ

_n

)ku

^]_s

k

_L2

≤ (1 + λ

_n

)

1 + ms

^α2

kAk

_Xα

hence the upper bound

λ

_n

(A) ≤ m

⁺_δ

(A, s)λ

_n

+ 2 + 2ms

^α2

kAk

X^α

+ m

²_δ

(A, s).

For the lower bound, we use the min-max representation of λ

n

(A) under the form λ

_n

(A) = sup

v1,...,vn−1∈L²

inf

u∈Vect(v1,...,vn−1)^⊥

kukL2 =1

hHu, ui.

Introducing

F := Vect (u

m

; m ≥ n),

we have that F

^⊥

is a subspace of L

²

of finite dimension n − 1 thus there exists a orthogonal family (v

₁

, . . . , v

_n−1

) such that F

^⊥

= Vect (v

₁

, . . . , v

_n−1

) . Since F is a closed subspace of L

²

as an intersection of hyperplans, we have F = Vect (v

1

, . . . , v

_n−1

)

^⊥

hence

λ

n

(A) ≥ inf

u∈F

kukL2 =1

hHu, ui.

Let u ∈ F with kuk

L²

= 1 . Using Proposition 1.7, we have

hHu, ui ≥ (1 − δ)h∇u

^]_s

, ∇u

^]_s

i − m

¹_δ

(A, s)

≥ (1 − δ)hu

^]_s

, −∆u

^]_s

i − m

¹_δ

(A, s)

≥ (1 − δ)λ

n

ku

^]_s

k

²_L2

− m

¹_δ

(A, s).

Finally using Proposition 1.1 for s < s

₁₋^α

4

(A) , we get hHu, ui ≥ 1 − δ

1 − ms

^α2

kAk

X^α

λ

n

− m

¹_δ

(A, s)

and the proof is complete.

In particular, taking

s = δ

mkAk

X^α

_α²

gives the simpler bounds

λ

_n

− m

¹_δ

(A) ≤ λ

_n

(A) ≤ (1 + δ)

²

λ

_n

+ m

²_δ

(A)

for any δ small enough. This is sharp enough to get an almost sure Weyl-type law from the Weyl law for the Laplacian ∆ , we denote by N (λ) the number of eigenvalues of −∆ lower than λ ∈ R.

Corollary 2.6. We have

λ→∞

lim λ

⁻¹

|{n ≥ 0; λ

n

(A) ≤ λ}| = π

Proof : The lower and upper bounds on the eigenvalues give

N

λ − m

²_δ

(A) 1 + δ

≤

{n ≥ 0; λ

n

(A) ≤ λ}

≤ N λ + m

¹_δ

(A) hence the proof is complete using the result for the Laplacian.

3 – Stochastic renormalisation of the singular potential

The enhanced potential

A := (A, A

²

) ∈ X

^α

= C

^α−1

× C

^2α−2

would be defined with A

²

:= A · A for regular enough potential, for example A ∈ L

^∞

is enough. For arbitrary distributions A ∈ C

^α−1

, this does not make sense since the product of two distributions in Hölder spaces can be defined only if the sum of their regularity exponents is positive. The magnetic Laplacian with white noise B = ξ as magnetic field corresponds to this framework since the associated magnetic potential

A := ∇

^⊥

ϕ