Abstract. We study the behavior of certain eigenvalues for magnetic Aharonov-Bohm op- erators with half-integer circulation and Dirichlet boundary conditions in a planar domain.

AHARONOV-BOHM OPERATORS WITH A MOVING POLE

LAURA ABATANGELO AND VERONICA FELLI

Abstract. We study the behavior of certain eigenvalues for magnetic Aharonov-Bohm op- erators with half-integer circulation and Dirichlet boundary conditions in a planar domain.

We analyze the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and determining its exact coefficients.

1. Introduction

This paper is concerned with the behavior of certain eigenvalues of magnetic Aharonov- Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain.

A remarkable mathematical motivation for the study of Aharonov-Bohm operators with half-integer circulation can be found in the deep relation between nodal domains of eigen- functions of such operators and spectral minimal partitions of the Dirichlet Laplacian, i.e.

partitions of the domain minimizing the largest of the first eigenvalues on the components.

From [10] it is known that the boundary of the optimal partition is the union of a finite num- ber of regular arcs; moreover, if the number of half lines meeting at each intersection point is even, then the partition components are nodal domains of an eigenfunction of the Dirichlet Laplacian. On the other hand, partitions with points of odd multiplicity can be obtained as nodal domains by minimizing a certain eigenvalue of an Aharonov-Bohm Hamiltonian with respect to the number and the position of poles, as suggested in [2, 3, 4, 14] and confirmed by the magnetic characterization of minimal partitions given in [8]. The properties of the map associating eigenvalues of magnetic operators to the position of poles and its connection between its critical points and nodal properties of eigenfunctions was further investigated in [5, 12, 13]. The present paper completes the analysis started in [1] and aims at giving the sharp asymptotic expansion for the eigenvalue variation with respect to moving poles.

For every a = (a ₁ , a ₂ ) ∈ R ² , the Aharonov-Bohm vector potential with pole a and circula- tion 1/2 is defined as

A a (x 1 , x 2 ) = 1 2

 −(x ₂ − a ₂ )

(x 1 − a ₁ ) ² + (x 2 − a ₂ ) ² , x 1 − a ₁

(x 1 − a ₁ ) ² + (x 2 − a ₂ ) ²



, (x 1 , x 2 ) ∈ R ² \ {a}.

Date: Revised version, March 17, 2016.

The authors have been partially supported by 2014 INdAM-GNAMPA research project “Stabilit` a spettrale e analisi asintotica per problemi singolarmente perturbati”, 2015 INdAM-GNAMPA research project “Opera- tori di Schr¨ odinger con potenziali elettromagnetici singolari: stabilit` a spettrale e stime di decadimento”, and by ERC Advanced Grant 2013 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”.

2010 Mathematics Subject Classification. 35J10, 35P20, 35Q40, 35Q60, 35J75.

Keywords. Magnetic Schr¨ odinger operators, Aharonov-Bohm potential, asymptotics of eigenvalues, blow-up analysis.

1

Let Ω ⊂ R ² be a bounded, open and simply connected domain. For every a ∈ Ω, we consider the eigenvalue problem

(E a )

( (i∇ + A a ) ² u = λu, in Ω,

u = 0, on ∂Ω,

in the weak sense (15).

From classical spectral theory (see e.g. [6, Chapter 6]), the eigenvalue problem (E _a ) admits a sequence of real diverging eigenvalues (repeated according to their finite multiplicity) λ ^a ₁ ≤ λ ^a ₂ ≤ · · · ≤ λ ^a _j ≤ . . . . We are concerned with the behavior of the function a 7→ λ ^a _j in a neighborhood of a fixed point b ∈ Ω; without loss of generality, we can consider b = 0 ∈ Ω.

Let us assume that there exists n ₀ ≥ 1 such that

(1) λ ⁰ _n

is simple,

and denote

λ ₀ = λ ⁰ _n

and, for any a ∈ Ω,

λ _a = λ ^a _n

.

In [12, Theorem 1.3] it is proved that, for all j ≥ 1 such that λ ⁰ _j is simple, the function a 7→ λ ^a _j is analytic in a neighborhood of 0. In particular, under assumption (1), a 7→ λ _a is continuous and, if a → 0, then

(2) λ _a → λ ₀ .

Let ϕ ₀ ∈ H ₀ ^1,0 (Ω, C)\{0} (see section 2 for the definition of the functional space H ₀ ^1,0 (Ω, C)) be a L ² (Ω, C)-normalized eigenfunction of problem (E 0 ) associated to the eigenvalue λ 0 = λ ⁰ _n

, i.e. satisfying

(3)



 

 

(i∇ + A ₀ ) ² ϕ ₀ = λ ₀ ϕ ₀ , in Ω,

ϕ ₀ = 0, on ∂Ω,

R

Ω |ϕ ₀ (x)| ² dx = 1.

From [7, Theorem 1.3] (see also [14, Theorem 1.5] and [1, Proposition 2.1]) it is known that (4) ϕ ₀ has at 0 a zero of order k

2 for some odd k ∈ N, and that there exist β 1 , β 2 ∈ C such that (β 1 , β 2 ) 6= (0, 0) and

(5) r ^−k/2 ϕ 0 (r(cos t, sin t)) → e ⁱ

 β 1 cos

 k 2 t



+ β 2 sin

 k 2 t

 

in C ^1,τ ([0, 2π], C)

as r → 0 ⁺ for any τ ∈ (0, 1). We recall that, by [9] (see also [5, Lemma 2.3]), the function e ⁻ⁱ

ϕ ₀ (r(cos t, sin t)) is a (complex) multiple of a real-valued function; therefore (5) implies that the function γ(t) = β 1 cos ^k ₂ t
+ β ₂ sin ^k ₂ t
is real-valued up to a complex multiplicative constant and then either β 1 = γ(0) = 0 or ^β _β

1

= ± ^γ(π) _γ(0) is real. Then ϕ 0 has exactly k nodal lines meeting at 0 and dividing the whole angle into k equal parts; these nodal lines are tangent to the k half-lines 

t, tan(α ₀ + j ^2π _k )t
: t > 0 , j = 0, 1, . . . , k − 1, where

(6) α 0 =

( 2

k arccot − ^β _β

1

, if β ₁ 6= 0,

0, if β ₁ = 0.

At a deeper study, the rate of convergence of λ _a to λ ₀ is strictly related to the number of nodal lines of ϕ 0 ending at 0. First results in this direction are proved in [5], in which the authors provide some estimates for the rate of convergence (2). A significant improvement of these studies is obtained in [1], where sharp asymptotic behavior of eigenvalues is provided as the pole is approaching an internal zero of an eigenfunction ϕ ₀ of the limiting problem (3) along the half-line tangent to any nodal line of ϕ 0 ; more precisely, under assumption (1) and being k as in (4), in [1, Theorem 1.2] it is proved that the limit

(7) lim

|a|→0

λ 0 − λ _a

|a| ^k is finite and strictly positive as a → 0 tangentially to a nodal line.

The above positive limit can be expressed in terms of the value m _k defined as follows. Let s ₀ be the positive half-axis s 0 = [0, +∞)×{0}. For every odd natural number k, the function (8) ψ _k (r cos t, r sin t) = r ^k/2 sin  k

2 t



, r ≥ 0, t ∈ [0, 2π],

is the unique (up to a multiplicative constant) function which is harmonic on R ² \ s ₀ , homo- geneous of degree k/2 and vanishing on s ₀ . Let s := {(x ₁ , x ₂ ) ∈ R ² : x ₂ = 0 and x ₁ ≥ 1}, R ² + = {(x ₁ , x ₂ ) ∈ R ² : x ₂ > 0)}, and denote as D s ^1,2 (R ² + ) the completion of C _c ^∞ (R ² ₊ \ s) under the norm ( R

R

|∇u| ² dx) ^1/2 . By standard minimization methods, the functional J _k : D _s ^1,2 (R ² + ) → R, J _k (u) = 1

2 Z

R

|∇u(x)| ² dx − Z

∂R

\s

u(x 1 , 0) ∂ψ k

∂x ₂ (x 1 , 0) dx 1 , achieves its minimum over the whole space D ^1,2 s (R ² + ) at some function w k ∈ D _s ^1,2 (R ² + ), i.e.

there exists w _k ∈ D ^1,2 _s (R ² + ) such that

(9) m _k = min

u∈D

(R

)

J k (u) = J k (w k ).

We notice that

(10) m _k = J _k (w _k ) = − 1 2

Z

R

|∇w _k (x)| ² dx = − 1 2

Z 1 0

∂ ₊ ψ _k

∂x ₂ (x ₁ , 0) w _k (x ₁ , 0) dx ₁ < 0, where, for all x 1 > 0, ^∂ _∂x

^ψ

2

(x 1 , 0) = lim _t→0

^ψ

^(x

^,t)−ψ _t

^(x

^,0) = ^k ₂ x

k 2

−1

1 . In [1] it is proved that the limit in (7) is equal to

(11) C 0 = −4(|β 1 | ² + |β 2 | ² ) m _k with (β ₁ , β ₂ ) 6= (0, 0) being as in (5).

From [1, Theorem 1.2] we can easily deduce that, under assumption (1) and being k as in (4), the Taylor polynomials of the function a 7→ λ ₀ − λ _a with center 0 and degree strictly smaller than k vanish.

Lemma 1.1. Let Ω ⊂ R ² be a bounded, open and simply connected domain such that 0 ∈ Ω and let n ₀ ≥ 1 be such that the n ₀ -th eigenvalue λ ₀ = λ ⁰ _n

of (i∇ + A ₀ ) ² on Ω is simple with associated eigenfunctions having in 0 a zero of order k/2 with k ∈ N odd. For a ∈ Ω let λ a = λ ^a _n

be the n 0 -th eigenvalue of (i∇ + A a ) ² on Ω. Then

(12) λ ₀ − λ _a = P (a) + o(|a| ^k ), as |a| → 0 ⁺ ,

0 α

a r

Figure 1. a = |a|(cos α, sin α) approaches 0 along the direction determined by the angle α.

for some homogeneous polynomial P 6≡ 0 of degree k

(13) P (a) = P (a ₁ , a ₂ ) =

k

X

j=0

c _j a ^k−j ₁ a ^j ₂ .

The main result of the present paper is the determination of the exact value of all coefficients of the polynomial P (and hence the sharp asymptotic behavior of λ _a − λ ₀ as a → 0 along any direction, see Figure 1).

Theorem 1.2. Under the same assumptions of Lemma 1.1, let α ∈ [0, 2π). Then λ ₀ − λ _a

|a| ^k → C ₀ cos k(α − α ₀ )

as a → 0 with a = |a|(cos α, sin α), where α 0 is defined in (6) and C 0 in (11).

Remark 1.3. By Theorem 1.2 it follows that the polynomial (13) of Lemma 1.1 is given by P (|a|(cos α, sin α)) = C ₀ |a| ^k cos(k(α − α ₀ )).

Hence

P (a ₁ , a ₂ ) = C ₀ Re e ^−ikα

(a ₁ + i a ₂ ) ^k
, thus yielding ∆P = 0, i.e. the polynomial P in (12)-(13) is harmonic.

The proof of Theorem 1.2 is based on a combination of estimates from above and below of the Rayleigh quotient associated to the eigenvalue problem with a fine blow-up analysis for scaled eigenfunctions

(14) ϕ _a (|a|x)

|a| ^k/2 ,

which gives a sharp characterization of upper and lower bounds for eigenvalues. Differently from the blow-up analysis performed in [1] for poles moving tangentially to nodal lines, in the general case of poles moving along any direction we cannot explicitly construct the limit profile of the family of scaled functions (14). Such a difficulty is overcome by studying the dependence of the limit profile on the position of the pole and the symmetry/periodicity properties of its Fourier coefficient with respect to a basis of eigenvectors of an associated angular problem:

such symmetry and periodicity turn into some symmetry and periodicity invariances of the

polynomial P . A complete classification of homogeneous k-degree polynomials with such

periodicity/symmetry invariances then allows us to determine explicitly the polynomial P .

The paper is organized as follows. Section 2 is devoted to recalling some known facts and introducing some notation. In section 3 we prove sharp asymptotics for λ 0 − λ _a in dependence on the angle α. In section 4 we describe some symmetry properties of the sharp asymptotics, which allow us to prove Theorem 1.2 in section 5.

2. Preliminaries

In this section we present some preliminaries as needed in the forthcoming argument. For every a ∈ Ω, we introduce the functional space H ^1,a (Ω, C) as the completion of

{u ∈ H ¹ (Ω, C) ∩ C ^∞ (Ω, C) : u vanishes in a neighborhood of a}

with respect to the norm kuk _H

(Ω,C) = 

k∇uk ² _L

(Ω,C

) + kuk ² _L

(Ω,C) + _|x−a| ^u

2 L

(Ω,C)

 1/2

,

which, in view of the Hardy type inequality proved in [11] (see also [7, Lemma 3.1 and Remark 3.2]), is equivalent to the norm

 k(i∇ + A _a )uk ² _L

(Ω,C

) + kuk ² _L

(Ω,C)

 1/2

.

We denote as H ₀ ^1,a (Ω, C) the space obtained as the completion of C c ^∞ (Ω \ {a}, C) with respect to the norm k · k _H

_(Ω,C) .

For every a ∈ Ω, we say that λ ∈ R is an eigenvalue of problem (E a ) in a weak sense if there exists u ∈ H ₀ ^1,a (Ω, C) \ {0} (called an eigenfunction) such that

(15)

Z

Ω

(i∇u + A a u) · (i∇v + A a v) dx = λ Z

Ω

uv dx for all v ∈ H ₀ ^1,a (Ω, C).

2.1. Change of coordinates. Up to a change of coordinates (a rotation), it is not restrictive to assume in (5) that

(16) β ₁ = 0,

see [1, Remark 2.2]. Under condition (16), we have that α ₀ = 0 and one nodal line of ϕ ₀ is tangent the x 1 -axis.

2.2. Polar eigenfunctions. The limit function in (5) is an eigenfunction of the operator Lψ = −ψ ⁰⁰ + iψ ⁰ + 1

4 ψ acting on 2π-periodic functions. The eigenvalues of L are  _j

4 : j ∈ N, j is odd ; moreover each eigenvalue ^j ₄

has multiplicity 2 and the functions

(17) ψ ₁ ^j (t) = e ⁱ

√ π cos

 j 2 t



, ψ ^j ₂ (t) = e ⁱ

√ π sin

 j 2 t



form an L ² ((0, 2π), C)-orthonormal basis of the eigenspace associated to the eigenvalue ^j ₄

.

2.3. Angles and approximating eigenfunctions. As in [5], for every α ∈ [0, 2π) and b = (b 1 , b 2 ) = |b|(cos α, sin α) ∈ R ² \ {0}, we define

(18) θ _b : R ² \ {b} → [α, α + 2π) and θ ₀ ^b : R ² \ {0} → [α, α + 2π) such that

θ _b (b + r(cos t, sin t)) = t and θ ₀ ^b (r(cos t, sin t)) = t, for all r > 0 and t ∈ [α, α + 2π).

E.g. if b ₁ > 0 and b ₂ > 0 the functions θ _b and θ ^b ₀ are given by

θ b (x 1 , x 2 ) =



 

 

 



 

 

 



arctan ^x _x

^−b

1

−b

, if x ₁ > b ₁ , x ₂ ≥ ^b _b

1

x ₁ ,

π

2 , if x 1 = b 1 , x 2 > b 2 , π + arctan ^x _x

^−b

1

−b

, if x 1 < b 1 ,

3

2 π, if x 1 = b 1 , x 2 < b 2 , 2π + arctan ^x _x

^−b

1

−b

, if x 1 > b 1 , x 2 < ^b _b

1

x 1 , (19)

θ ₀ ^b (x 1 , x 2 ) =



 

 

 



 

 

 



arctan ^x _x

1

, if x ₁ > 0, x ₂ ≥ ^b _b

1

x ₁ ,

π

2 , if x 1 = 0, x 2 > 0, π + arctan ^x _x

1

, if x ₁ < 0,

3

2 π, if x 1 = 0, x 2 < 0, 2π + arctan ^x _x

1

, if x ₁ > 0, x ₂ < ^b _b

1

x ₁ . We notice that θ _b and θ ₀ ^b are regular except on the half-lines

(20) s _b := tb : t ≥ 1 , s ^b ₀ := t b : t ≥ 0 ,

respectively, whereas the difference θ ^b ₀ − θ _b is regular except for the segment {tb : t ∈ [0, 1]}

from 0 to b.

We also define

θ ₀ : R ² \ {0} → [0, 2π) as

(21) θ 0 (x 1 , x 2 ) =



 

 

 



 

 

 



arctan ^x _x

1

, if x ₁ > 0, x ₂ ≥ 0,

π

2 , if x 1 = 0, x 2 > 0, π + arctan ^x _x

1

, if x ₁ < 0,

3

2 π, if x 1 = 0, x 2 < 0, 2π + arctan ^x _x

1

, if x ₁ > 0, x ₂ < 0,

so that θ ₀ (cos t, sin t) = θ ₀ ⁰ (cos t, sin t) = t for all t ∈ [0, 2π) and θ ₀ is regular except for the half-axis {(x 1 , 0) : x 1 ≥ 0}. We notice that

(22) (θ ^b ₀ − θ ₀ )(r cos t, r sin t) =

( 0, if t ∈ [α, 2π), 2π, if t ∈ [0, α).

Let us now consider a suitable family of eigenfunctions relative to the approximating eigen- value λ _a . For all a ∈ Ω, let ϕ _a ∈ H ₀ ^1,a (Ω, C) \ {0} be an eigenfunction of problem (E a ) associated to the eigenvalue λ a , i.e. solving

(23)

(

(i∇ + A a ) ² ϕ a = λ a ϕ a , in Ω,

ϕ a = 0, on ∂Ω,

such that (24)

Z

Ω

|ϕ _a (x)| ² dx = 1 and Z

Ω

e

^(θ

^−θ

^)(x) ϕ a (x)ϕ 0 (x) dx is a positive real number, where ϕ ₀ is as in (3). From (1), (3), (23), (24), and standard elliptic estimates, it follows that ϕ _a → ϕ ₀ in H ¹ (Ω, C) and in C _loc ² (Ω \ {0}, C) and

(25) (i∇ + A a )ϕ a → (i∇ + A ₀ )ϕ 0 in L ² (Ω, C).

2.4. Limit profile in dependence on α. A key role in the proof of our main result is played by a suitable magnetic-harmonic function in R ² , which will turn out to be the limit of blowed-up sequences of eigenfunctions with poles approaching 0 along the half-line starting from 0 with slope tan α.

For every p ∈ R ² , we denote by D ^1,2 p (R ² , C) the completion of C c ^∞ (R ^N \{0}, C) with respect to the magnetic Dirichlet norm

kuk _D

p

(R

,C) :=

 Z

R

 (i∇ + A p )u(x)  

2 dx

 1/2

.

We recall from [11] that functions in D ^1,2 p (R ² , C) satisfy the Hardy type inequality Z

R

|(i∇ + A _p )u| ² dx ≥ 1 4

Z

R

|u(x)| ²

|x − p| ² dx;

furthermore (see also [7, Lemma 3.1 and Remark 3.2]) the inequality Z

D

(p)

|(i∇ + A _p )u| ² dx ≥ 1 4

Z

D

(p)

|u(x)| ²

|x − p| ² dx,

holds for all r > 0 and u ∈ H ^1,p (D r (p), C), where D r (p) denotes the disk of center p and radius r.

If a = |a|(cos α, sin α) → 0 with α ∈ [0, 2π), i.e. if a → 0 along the line of slope α, all the functions of the blowed-up family (14) are singular at the same point p = (cos α, sin α). We will prove in section 3 that the family (14) converges to the limit profile Ψ _p described in the following proposition.

Proposition 2.1. Let α ∈ [0, 2π) and p = (cos α, sin α). There exists a unique function Ψ _p ∈ H _loc ^1,p (R ² , C) such that

(26) (i∇ + A _p ) ² Ψ _p = 0 in R ² in a weak H ^1,p -sense, and

(27)

Z

R

\D

 (i∇ + A _p )(Ψ _p − e

^(θ

^−θ

⁾ e

^θ

ψ _k )  

2 dx < +∞, for any r > 1, where D r = D r (0).

Proof. Let η be a smooth cut-off function such that η ≡ 0 in D 1 and η ≡ 1 in R ² \ D _R for some R > 1. We observe that

F = (∆η)e

^(θ

^−θ

⁾ e

^θ

ψ k − 2i∇η · (i∇ + A _p )



e

^(θ

^−θ

⁾ e

^θ

ψ k



= −(i∇ + A _p ) ² 

ηe

^(θ

^−θ

⁾ e

^θ

ψ _k 

∈ D _p ^1,2 (R ² , C)
?

.

Hence, via Lax-Milgram’s Theorem there exists a unique solution g ∈ D ^1,2 p (R ² , C) to the problem

(i∇ + A p ) ² g = F , in D _p ^1,2 (R ² , C)
?

. The function Ψ _p = g + ηe

^(θ

^−θ

⁾ e

^θ

ψ _k satisfies (26) and (27).

To prove uniqueness, it is enough to observe that, if two functions Ψ ¹ _p , Ψ ² _p ∈ H _loc ^1,p (R ² , C) satisfy (26) and (27), then their difference Ψ ¹ _p − Ψ ² _p belongs to the space D p ^1,2 (R ² , C) (see [1, Proposition 4.3]); since (i∇ + A _p ) ² (Ψ ¹ _p − Ψ ² _p ) = 0 in (D ^1,2 p (R ² , C)) ^? , we conclude that

necessarily Ψ ¹ _p − Ψ ² _p ≡ 0. 

Remark 2.2. We observe that from [7, Theorem 1.5] it follows easily that Ψ p − e

^(θ

^−θ

⁾ e

^θ

ψ k = O(|x| ^−1/2 ), as |x| → +∞.

3. Sharp asymptotics for λ 0 − λ _a in dependence on α

In this section we prove some estimates of the eigenvalue variation λ ₀ − λ _a , by evaluating the Rayleigh quotient at suitable test functions obtained by manipulation of eigenfunctions.

Although this procedure follows the scheme of [1], it presents some additional difficulties requiring nontrivial adaptions, mainly due to the fact that the limit profile of Proposition 2.1 cannot be explicitly constructed as in [1]. We describe below this procedure, referring to [1]

for details of arguments already developed there and instead highlighting the differences with [1] and the difficulties in the adaption to the general case of poles moving along a generic direction.

For all 1 ≤ j ≤ n ₀ and a ∈ Ω, let ϕ ^a _j ∈ H ₀ ^1,a (Ω, C) \ {0} be an eigenfunction of problem (E _a ) associated to the eigenvalue λ ^a _j , i.e. solving

(28)

(

(i∇ + A a ) ² ϕ ^a _j = λ ^a _j ϕ ^a _j , in Ω,

ϕ ^a _j = 0, on ∂Ω,

such that (29)

Z

Ω

|ϕ ^a _j (x)| ² dx = 1 and Z

Ω

ϕ ^a _j (x)ϕ ^a <sub>`</sub> (x) dx = 0 if j 6= `.

For j = n ₀ , we choose

(30) ϕ ^a _n

= ϕ _a ,

with ϕ _a as in (23)–(24).

3.1. Estimate from below of λ 0 − λ _a . Letting p = (cos α, sin α), we define the function w _R as the unique solution to the minimization problem

Z

D

|(i∇ + A _p )w R (x)| ² dx

= min

Z

D

|(i∇ + A _p )u(x)| ² dx : u ∈ H ^1,p (D R , C), u = e

^(θ

^−θ

⁾ e

^θ

ψ k on ∂D R

 , which then solves

(31)

( (i∇ + A _p ) ² w _R = 0, in D _R ,

w _R = e

^(θ

^−θ

⁾ e

^θ

ψ _k , on ∂D _R .

Arguing as in [1, Subsection 6.2], we can prove the following lemma.

Lemma 3.1. For α ∈ [0, 2π) and a = |a|(cos α, sin α) ∈ Ω, let λ _a ∈ R and ϕ a ∈ H ₀ ^1,a (Ω, C) solve (23)-(24) and λ ₀ ∈ R and ϕ 0 ∈ H ₀ ^1,0 (Ω, C) solve (3). If (1) and (4) hold and (16) is satisfied, then, for all R > ˜ R and a = |a|(cos α, sin α) ∈ Ω,

λ 0 − λ _a

|a| ^k ≥ g _R (a) where

|a|→0 lim g _R (a) = i|β ₂ | ² ˜ κ _R , with

(32) ˜ κ _R =

Z

∂D



e ⁻

^θ

e

^(θ

^−θ

⁾ (i∇ + A _p )w _R · ν − (i∇)ψ _k · ν  ψ _k ds being p = (cos α, sin α) and ψ _k as in (8).

Proof. The proof follows exactly as in [1, Lemma 6.6], so we omit it.  For any R > 1 let us introduce the following Fourier-type coefficient

(33) υ _R (r) :=

Z 2π 0

e ⁻

^θ

(r cos t,r sin t) w _R (r cos t, r sin t)e

^θ

(r cos t,r sin t) ψ ^k ₂ (t) dt, r ∈ [1, R], with ψ ^k ₂ defined in (17). Due to jumps of the phases appearing in (33) on s p (see (20)), the derivation of the equation satisfied by the Fourier-type coefficient υ _R is more delicate than in [1]; hence we give the details in Lemma 3.2 below.

Lemma 3.2. For any R > 1 the function υ _R defined in (33) satisfies

(34) r ^−k/2 υ _R (r)
0

= c _R

r ^1+k , in (1, R), for some c _R ∈ C.

Proof. To prove (34) it is enough to show that Z R

1



− υ ⁰⁰ _R − 1

r υ ⁰ _R + k ² 4r ² υ _R



rη(r) dr = 0, for all η ∈ C _c ^∞ (1, R).

By (31), it is easy to see that the function u(x) := e ⁻

^θ

^(x) w R (x) is harmonic in D R \ s _p , where s _p is defined in (20). Let us consider an arbitrary function η(r) ∈ C _c ^∞ (1, R) and the function

g(t) := 1

√ π e

^(θ

^−θ

)(cos t,sin t) sin( ^k ₂ t) = (− ^{√ 1}

π sin( ^k ₂ t) t ∈ [0, α)

√ 1

π sin( ^k ₂ t) t ∈ [α, 2π).

Testing equation −∆u = 0 with v(r cos t, r sin t) = η(r)g(t) in D _R \ s _p , integrating by parts and observing that both v and ∇u jump across s p , we obtain that

0 = Z R

1

 Z 2π 0



r∂ r u(r cos t, r sin t)η ⁰ (r)g(t) + η(r)

r g ⁰ (t)∂ t u(r cos t, r sin t)

 dt

 dr

= − Z R

1

η(r)

 Z 2π 0



∂ _r u(r cos t, r sin t) + r∂ _rr ² u(r cos t, r sin t)  g(t) dt

 dr +

Z R 1

η(r) r

 Z 2π 0

∂ _t u(r cos t, r sin t)g ⁰ (t) dt

 dr

= − Z R

1



η(r)υ _R ⁰ (r) + rη(r)υ _R ⁰⁰ (r)

 dr +

Z R 1

η(r) r

 Z 2π 0

∂ t u(r cos t, r sin t)g ⁰ (t) dt

 dr.

A further integration by parts yields Z _2π

0

∂ _t u(r cos t, r sin t)g ⁰ (t) dt = − Z _2π

0

u(r cos t, r sin t)g ⁰⁰ (t) dt

+ g ₋ ⁰ (2π)u(r cos(2π ⁻ ), r sin(2π ⁻ )) − g ⁰ ₊ (α)u(r cos(α ⁺ ), r sin(α ⁺ )) + g ₋ ⁰ (α)u(r cos(α ⁻ ), r sin(α ⁻ )) − g ₊ ⁰ (0)u(r cos(0 ⁺ ), r sin(0 ⁺ ))

= − Z 2π

0

u(r cos t, r sin t)g ⁰⁰ (t) dt = k ² 4

Z 2π 0

u(r cos t, r sin t)g(t) dt = k ² 4 υ _R (r) in view of the fact that g ⁰ ₊ (0) = g ⁰ ₋ (2π), g ₊ ⁰ (α) = −g ₋ ⁰ (α), and

lim

t→α

u(r cos(t), r sin(t)) = − lim

t→α

u(r cos(t), r sin(t)).

The conclusion then follows. 

For α ∈ [0, 2π) and p = (cos α, sin α), let us define the following Fourier-type coefficient of the limit profile Ψ p

(35) ξ _p (r) :=

Z 2π 0

e ⁻

^θ

(r cos t,r sin t) Ψ _p (r cos t, r sin t)e

^θ

(r cos t,r sin t) ψ ^k ₂ (t) dt, r ≥ 1.

Lemma 3.3. Let ˜ κ _R be as in (32). Then

R→+∞ lim κ ˜ R = ik √ π( √

π − ξ p (1)), where ξ _p (r) is defined in (35).

Proof. The proof follows from integration of (34) arguing as in the proof of [1, Lemma 6.7].  Combining the results of Lemmas 3.1 and 3.3 we have that

(36) λ 0 − λ _a ≥ |a| ^k k|β 2 | ² √ π



ξ p (1) − √

π + o(1)



as a = |a|p → 0.

3.2. Blow-up analysis and Rayleigh quotient for λ ₀ . Differently from the lower bound for λ 0 − λ _a , the upper bound of the eigenvalue variation presents significant new difficulties with respect to the case of poles moving along nodal lines of ϕ ₀ treated in [1, Subsection 6.1].

Indeed, when the direction along which a → 0 is not a nodal line of ϕ 0 the value (ξ p (1) − √ π) can have any sign (and vanish along some directions); this does not allow deriving the exact asymptotic behavior of the normalization term in the blow-up analysis from estimates of the Rayleigh quotient from above and below as done in [1]. On the other hand, from [1] we can derive Lemma 1.1 and hence obtain a control on the size of the eigenvalue variation along any direction.

The proof of Lemma 1.1 is based on the following result (see also [5, Lemma 6.6]).

Lemma 3.4. Let Q(x 1 , x 2 ) = P h

j=0 c j x ^j ₁ x ^h−j ₂ be a homogeneous polynomial in two variables x ₁ , x ₂ of degree at most h ∈ N. If there exist ¯ θ ∈ [0, 2π) and an odd natural number k such that k > h and

(37) Q 

cos ¯ θ + j ^2π _k
, sin ¯ θ + j ^2π _k


= 0, for all j = 0, 1, . . . , k − 1, then Q ≡ 0.

Proof. Up to a rotation, it is not restrictive to assume that ¯ θ = 0. If x 1 6= 0, we can write Q as

Q(x ₁ , x ₂ ) = x ^h ₁ Q ˜ ^x _x

1

, where ˜ Q(t) =

h

X

j=0

c _j t ^h−j .

Since k is odd, we have that cos j ^2π _k
6= 0 for all j = 0, 1, . . . , k − 1. Then, from assumption (37) it follows that ˜ Q tan j ^2π _k

= 0 for all j = 0, 1, . . . , k − 1. Since k is odd, we also have that tan j ^2π _k
6= tan ` <sup>2π</sup> <sub>k</sub>
for all j, l ∈ {0, 1, . . . , k − 1} with j 6= `. Hence ˜ Q has k distinct zeros. Since ˜ Q is a polynomial of degree at most h and h < k, from the Fundamental Theorem of Algebra we conclude ˜ Q ≡ 0, i.e. c _j = 0 for all j = 0, 1, . . . , k − 1. Hence Q ≡ 0.  Proof of Lemma 1.1. Since the function a = (a 1 , a 2 ) 7→ λ 0 − λ _a is C ^∞ in a neighborhood of 0 (see [5, Theorem 1.3]), it admits a Taylor expansion up to order k of the form

λ ₀ − λ _a =

k

X

j=1

P _j (a ₁ , a ₂ ) + o(|a| ^k ), as |a| → 0,

where, for every j = 1, . . . , k, P _j (a ₁ , a ₂ ) is either identically zero or a homogeneous polynomial in the two variables a 1 , a 2 of degree j. From [1, Theorem 1.2] (see also (7)) we have that, for every ` < k,

P `



cos α 0 + j ^2π _k
, sin α ₀ + j ^2π _k


= 0, for all j = 0, 1, . . . , k − 1,

where α 0 is as in (6) (i.e. α 0 + j ^2π _k , with j = 0, 1, . . . , k − 1, identify the directions of the k half-lines tangent to the nodal lines of the eigenfunctions associated to λ ₀ ). The conclusion

follows directly from Lemma 3.4. 

From the expansion (12)–(13) in Lemma 1.1 it follows that

(38) |λ _a − λ ₀ | = O(|a| ^k )

as |a| → 0 along any direction. Exploiting (38) we can perform a sharp blow-up analysis prior

to the estimate from above of the eigenvalue variation λ ₀ − λ _a .

Let α ∈ [0, 2π) and p = (cos α, sin α). Arguing as in [1] we can prove that, for every δ ∈ (0, 1/4), there exist r _δ , K _δ > 0 such that, for all R ≥ K _δ ,

(39) the family of functions  ˜ ϕ a : a = |a|p, |a| < ^r _R

is bounded in H ^1,p (D R , C) where

(40) ϕ ˜ a (x) := ϕ a (|a|x)

pH _a,δ , and

H a,δ := 1 K _δ |a|

Z

∂D

|ϕ _a | ² ds.

Furthermore, from [1, Estimates (113) and (114)] we have that (41) H _a,δ ≥ C _δ |a| ^k+2δ , if |a| < r _δ

K _δ , for some C δ > 0 independent of a, and

(42) H _a,δ = O(|a| ^1−2δ ) as |a| → 0.

We observe that ˜ ϕ _a weakly solves

(43) (i∇ + A p ) ² ϕ ˜ a = |a| ² λ a ϕ ˜ a , in _|a| ¹ Ω = {x ∈ R ² : |a|x ∈ Ω}, and

(44) 1

K _δ Z

∂D

| ˜ ϕ a | ² ds = 1.

Let R > 2. For |a| sufficiently small we define the functions v _j,R,a as follows:

(45) v _j,R,a =

( v ^ext _j,R,a , in Ω \ D _R|a| ,

v ^int _j,R,a , in D _R|a| , j = 1, . . . , n ₀ , where

v _j,R,a ^ext := e

^(θ

^−θ

⁾ ϕ ^a _j in Ω \ D _R|a| ,

with ϕ ^a _j as in (28)–(30) and θ _a , θ ₀ ^a as in (18) (notice that e

^(θ

^−θ

⁾ is smooth in Ω \ D _R|a| ), so that it solves

( (i∇ + A 0 ) ² v ^ext _j,R,a = λ ^a _j v ^ext _j,R,a , in Ω \ D _R|a| , v _j,R,a ^ext = e

^(θ

^−θ

⁾ ϕ ^a _j on ∂(Ω \ D _R|a| ), whereas v _j,R,a ^int is the unique solution to the problem

( (i∇ + A 0 ) ² v ^int _j,R,a = 0, in D _R|a| , v _j,R,a ^int = e

^(θ

^−θ

⁾ ϕ ^a _j , on ∂D _R|a| . It is easy to verify that dim span{v _1,R,a , . . . , v _n

_,R,a }
= n ₀ .

For all R > 2 and a = |a|p ∈ Ω with |a| small, we define

(46) Z _a ^R (x) := v ^int _n

0

,R,a (|a|x)

pH _a,δ .

Arguing as in [1, Lemma 6.2] we can prove that, as a consequence of (39) and the Dirichlet principle,

(47) the family of functions Z _a ^R : a = |a|p, |a| < ^r _R

is bounded in H ^1,0 (D R , C).

We also define

W a (x) := ϕ 0 (|a|x)

|a| ^k/2 . (48)

As in [1], under assumption (16) and letting k as in (4), from [7, Theorem 1.3 and Lemma 6.1] we have that

(49) W a → √ β 2

π e

^θ

ψ _k as |a| → 0 in H ^1,0 (D R , C) for every R > 1, with β 2 as in (5).

Theorem 3.5. For every R > 2, kv _n

_,R,a − ϕ ₀ k _H

0

(Ω,C) = O

 pH _a,δ 

as a = |a|p → 0.

Proof. Let R > 2. We first notice that v n

,R,a → ϕ ₀ in H ₀ ^1,0 (Ω, C) as |a| → 0 ⁺ . Indeed Z

Ω

 (i∇ + A 0 )(v n

,R,a − ϕ ₀ )  

2 dx = Z

Ω

|e

^(θ

^−θ

⁾ (i∇ + A a )ϕ a − (i∇ + A ₀ )ϕ 0 | ² dx +

Z

D

 pH _a,δ (i∇ + A ₀ )Z _a ^R − |a| ^k/2 (i∇ + A ₀ )W _a   

2

dx

− Z

D

 pH a,δ e

^(θ

^−θ

⁾ (i∇ + A p ) ˜ ϕ a − |a| ^k/2 (i∇ + A 0 )W a

  

2

dx = o(1) in view of (25), (39), (47), (49) and (42).

From [1, Lemma 7.1] the function

F : C × H 0 ^1,0 (Ω, C) −→ R × R × (H _0,R ^1,0 (Ω, C)) ^? (λ, ϕ) 7−→ 

kϕk ²

H

(Ω,C) − λ ₀ , Im R

Ω ϕϕ ₀ dx
, (i∇ + A ₀ ) ² ϕ − λϕ  is Fr´ echet-differentiable at (λ 0 , ϕ 0 ) and its Fr´ echet-differential dF (λ 0 , ϕ 0 ) is invertible. In the above definition, (H _0,R ^1,0 (Ω, C)) ^? is the real dual space of H _0,R ^1,0 (Ω, C) = H ₀ ^1,0 (Ω, C), which is here meant as a vector space over R endowed with the norm

kuk _H

(Ω,C) =

 Z

Ω

 (i∇ + A ₀ )u  

2 dx

 1/2

. Therefore

|λ _a − λ ₀ | + kv _n

_,R,a − ϕ ₀ k _H

(Ω,C)

≤ k(dF (λ ₀ , ϕ ₀ )) ⁻¹ k

L(R×R×(H

(Ω,C))

,C×H

(Ω,C)) kF (λ _a , v _n

_,R,a )k

R×R×(H

(Ω))

(1 + o(1)) as |a| → 0 ⁺ . To prove the theorem it is then enough to estimate the norm of

F (λ _a , v _n

_,R,a ) = (α _a , β _a , w _a )

=



kv _n

_,R,a k ²

H

(Ω,C) − λ ₀ , Im R

Ω v n

,R,a ϕ 0 dx
, (i∇ + A ₀ ) ² v n

,R,a − λ _a v n

,R,a



in R × R × (H _0,R ^1,0 (Ω)) ^? . The estimates of β _a and w _a can be performed as in [1, Proof of Theorem 7.2] obtaining that

β _a = o pH _a,δ

and kw _a k _(H

0,R

(Ω,C))

= O pH _a,δ
,

as a = |a|p → 0. As far as α _a is concerned, differently from [1], the estimate of [1, Proposition 6.10] which, in the case α = 0, implied that |λ a − λ ₀ | = O(H _a,δ ), is not available after preliminary estimates of the Rayleigh quotient for generic values of α since (ξ _p (1) − √

π) can have any sign. This difficulty can be overcome by observing that (41) and (38) imply that

|λ _a − λ ₀ | = O(|a|

^−δ pH _a,δ ) and then

|λ _a − λ ₀ | = o(pH _a,δ ), as a = |a|p → 0. Then, from (39) and (47), we obtain that

α _a = Z

D

|(i∇ + A ₀ )v _n ^int

_,R,a | ² dx − Z

D

|(i∇ + A _a )ϕ _a | ² dx

!

+ (λ _a − λ ₀ )

= H _a,δ

Z

D

|(i∇ + A ₀ )Z _a ^R | ² dx − Z

D

|(i∇ + A _p ) ˜ ϕ a | ² dx



+ (λ a − λ ₀ )

= o(pH _a,δ ),

as a = |a|p → 0, thus concluding the proof. 

Via Theorem 3.5 and a change of variables, it follows that, letting α ∈ [0, 2π), p = (cos α, sin α), and R > 1,

(50) Z

1

|a|

Ω

\D

(i∇ + A _p ) 

˜

ϕ _a (x) − e

^(θ

^−θ

^{) |a|} √

H

W _a     

2

dx = O(1), as a = |a|p → 0.

Theorem 3.6. For α ∈ [0, 2π), p = (cos α, sin α) and a = |a|p ∈ Ω, let ϕ _a ∈ H ₀ ^1,a (Ω, C) solve (23-24) and ϕ ₀ ∈ H ₀ ^1,0 (Ω, C) be a solution to (3) satisfying (1), (4), and (16). Let ˜ ϕ _a and K δ be as in (40) and Ψ p be as in Proposition 2.1. Then

(51) lim

|a|→0

|a| ^k/2 pH _a,δ = 1

|β ₂ |

s K δ

R

∂D

|Ψ _p | ² ds and

(52) ϕ ˜ _a → β ₂

|β ₂ |

s K _δ

R

∂D

|Ψ _p | ² ds Ψ _p as a = |a|p → 0, in H ^1,p (D _R , C) for every R > 1, almost everywhere and in C _loc ² (R ² \ {p}, C).

Proof. Step 1. We first prove that for every sequence a n = |a n |p with |a _n | → 0, there exist Φ ∈ H ˜ _loc ^1,p (R ² , C), ˜ Φ 6≡ 0, and a subsequence a _n

such that ˜ ϕ _a

→ ˜ Φ in H ^1,p (D _R , C) for every R > 1, almost everywhere and in C _loc ² (R ² \ {p}, C) and ˜ Φ weakly solves

(53) (i∇ + A _p ) ² Φ = 0, ˜ in R ² .

To prove it, we observe that from (39) it follows that, for every sequence a n = |a n |p with

|a _n | → 0, by a diagonal process there exists ˜ Φ ∈ H _loc ^1,p (R ² , C), and a subsequence a n

such

that ˜ ϕ _a

* ˜ Φ weakly in H ^1,p (D _R , C) for every R > 1 and almost everywhere. ˜ Φ 6≡ 0

since _K ¹

δ

R

∂D

| ˜ Φ| ² ds = 1 thanks to (44) and the compactness of the trace embedding H ^1,p (D K

, C) ,→ L ² (∂D K

, C).

Passing to the limit in (43), we have that ˜ Φ weakly solves (53), whereas, arguing as in the proof of [1, Theorem 8.1], we can prove that the convergence of the subsequence ˜ ϕ _a

to ˜ Φ is actually strong in H ^1,p (D R , C) for every R > 1. The convergence in C _loc ² (R ² \ {p}, C) follows easily from classical elliptic estimates.

Step 2. We claim that, for every sequence a n = |a n |p with |a _n | → 0, there exists a subse- quence a _n

such that

`→+∞ lim

|a _n

| ^k/2 q H a

,δ

is finite and strictly positive.

To prove the claim, we argue by contradiction, assuming that

(i) either there exists a sequence a n = |a n |p with |a _n | → 0 such that lim _n→+∞ √ ^|a

^|

H

= 0 (ii) or there exists a sequence a n = |a n |p with |a _n | → 0 such that lim _n→+∞ √ ^|a

^|

H

= +∞.

If (i) holds, then, by step 1, along a subsequence, ˜ ϕ a

→ ˜ Φ in H ^1,p (D R , C) for every R > 1, for some ˜ Φ 6≡ 0 weakly solving (53). Then from (49), passing to the limit in (50) we would obtain that

Z

R

\D

|(i∇ + A _p ) ˜ Φ(x)| ² dx < +∞,

contradicting the fact that ˜ Φ 6≡ 0 is a non trivial weak solution to (53) (and so cannot have finite energy otherwise by testing the equation we would get that ˜ Φ ≡ 0, see [1, Proof of Proposition 4.3]). Hence case (i) cannot occur.

If (ii) holds, then from (50) we would have, for all R > 2

|a| ^k H _a,δ

Z

D

\D

(i∇ + A _p )  √

H

|a|

ϕ ˜ _a (x) − e

^(θ

^−θ

⁾ W _a     

2

dx = O(1), as a = |a|p → 0, and hence, in view of (49) and (39), passing to the limit along the sequence we would obtain that

|a _n | ^k H _a

_,δ

 Z

D

\D

(i∇ + A p )



e

^(θ

^−θ

⁾ β 2 e

^θ

ψ k

    

2

dx + o(1)



= |a _n | ^k H _a

_,δ



|β ₂ | ² Z

D

\D

|∇ψ _k | ² dx + o(1)



= O(1), as n → +∞,

which is not possible if lim _`→+∞ ^|a

^|

√

H

= +∞ as in case (ii), since R

D

\D

|∇ψ _k | ² dx > 0.

Hence also case (ii) cannot occur and the claim of step 2 is proved.

Step 3. From steps 1 and 2, it follows that, for every sequence a n = (|a n |, 0) = |a _n |p with |a n | → 0, there exist c ∈ (0, +∞), ˜ Φ ∈ H _loc ^1,p (R ² , C) weakly solving (53), ˜ Φ 6≡ 0, and a subsequence a _n

such that lim _`→+∞ ^|a

^|

√

H

= c and ˜ ϕ _a

→ ˜ Φ in H ^1,p (D _R , C) for every

R > 1 and in C _loc ² (R ² \ {p}, C). Passing to the limit along a n

in (50) and recalling (49), we obtain that, for every R > 2,

Z

R

\D

(i∇ + A p )  ˜ Φ(x) − cβ 2 e

^(θ

^−θ

⁾ e

^θ

ψ _k

    

2

dx < +∞.

Hence from Proposition 2.1 we conclude that necessarily

(54) Φ = cβ ˜ ₂ Ψ _p .

Since _K ¹

δ

R

∂D

| ˜ Φ| ² ds = 1, from (54) and the fact that c is a positive real number, it follows that c = _|β ¹

2

|

K

R

∂DKδ

|Ψ

|

ds

1/2

. Hence we have that

˜

ϕ _a

→ β ₂

|β ₂ |

r K

R

∂DKδ

|Ψ

|

ds Ψ _p in H ^1,p (D _R , C) for every R > 1 and in C _loc ² (R ² \ {p}, C), and

|a _n

| ^k/2 q H _a

_,δ

→ 1

|β ₂ |

s K δ

R

∂D

|Ψ _p | ² ds .

Since the above limits depend neither on the sequence {a n } _n nor on the subsequence {a n

} _` , we conclude that the above convergences hold as |a| → 0 ⁺ , thus proving (51) and (52).  Remark 3.7. Combining (51) and (52) we deduce that

ϕ _a (|a|x)

|a| ^k/2 → β ₂ Ψ _p as a = |a|p → 0,

in H ^1,p (D _R , C) for every R > 1 and in C _loc ² (R ² \ {p}, C). Furthermore, arguing as in [1, Lemma 8.3], from Theorem 3.6 we can deduce that, letting Z _a ^R as is (46),

Z _a ^R → β ₂

|β ₂ |

s K _δ

R

∂D

|Ψ _p | ² ds z R as a = |a|p → 0, in H ^1,0 (D _R , C) for every R > 2, where z R is the unique solution to

( (i∇ + A ₀ ) ² z _R = 0, in D _R , z R = e

^(θ

^−θ

⁾ Ψ p , on ∂D R .

Thanks to the convergences of blow-up sequences established in Theorem 3.6 and Remark 3.7, we can now follow closely the arguments of [1, Subsection 6.1, Lemma 9.1] thus obtaining the following upper bound for the difference λ ₀ − λ _a .

Lemma 3.8. For α ∈ [0, 2π) and a = |a|(cos α, sin α) ∈ Ω, let λ a ∈ R and ϕ a ∈ H ₀ ^1,a (Ω, C) solve (23-24) and λ 0 ∈ R and ϕ 0 ∈ H ₀ ^1,0 (Ω, C) solve (3). If (1) and (4) hold and (16) is satisfied, then, for a = |a|(cos α, sin α) and p = (cos α, sin α),

lim sup

|a|→0

λ 0 − λ _a

|a| ^k ≤ |β ₂ | ² k √

π(ξ p (1) − √ π), with ξ _p (r) defined in (35).

Collecting (36) and Lemma 3.8 we can state the following result.

Proposition 3.9. For α ∈ [0, 2π) and a = |a|(cos α, sin α) ∈ Ω, let ϕ _a ∈ H ₀ ^1,a (Ω, C) and λ _a ∈ R solve (23-24) and λ 0 ∈ R and ϕ 0 ∈ H ₀ ^1,0 (Ω, C) solve (3). If (1) and (4) hold and (16) is satisfied, then, for a = |a|(cos α, sin α),

|a|→0 lim

λ 0 − λ _a

|a| ^k = |β 2 | ² k √ πf (α), where

(55) f : [0, 2π) → R, f (α) = (ξ _p (1) − √

π), p = (cos α, sin α), with ξ _p (r) defined in (35).

4. Properties of f (α)

To prove our main result, we are going to investigate two suitable symmetry properties of the function f (α). Let us define two transformations R 1 , R 2 acting on a general point

x = (x 1 , x 2 ) = (r cos t, r sin t), r > 0, t ∈ [0, 2π), as

(56) R ₁ (x) = R ₁ (x ₁ , x ₂ ) = M _k x ₁ x ₂



, M _k = cos ^2π _k − sin ^2π _k sin ^2π _k cos ^2π _k

!

i.e.

R ₁ (r cos t, r sin t) = 

r cos(t + ^2π _k ), r sin(t + ^2π _k )  , and

(57) R ₂ (x) = R 2 (x 1 , x 2 ) = (x 1 , −x 2 ), i.e.

R ₂ (r cos t, r sin t) = (r cos(2π − t), r sin(2π − t)),

The transformation R ₁ is a rotation of ^2π _k and R ₂ is a reflexion through the x ₁ -axis.

We would like to study how the coefficient ξ p (1) (see (35)) changes when the above trasfor- mations act on p. In particular, we are going to prove that such a quantity ξ _p (1) is invariant under the transformations R 1 , R 2 .

In order to obtain such an invariance, we first study the relation between the limit profiles Ψ _p (R _j (x)) and Ψ _R

j

(p) (x), j = 1, 2.

Lemma 4.1. For p = (cos α, sin α), α ∈ [0, 2π), let Ψ p be the limit profile introduced in Proposition 2.1 and let R ₁ , R ₂ be the transformations introduced in (56) and (57). Then

(58) Ψ _R

1

(p) = −e ⁻ⁱ

Ψ p ◦ R ₁
and

(59) Ψ _R

_(p) = −e ^iθ

Ψ _p ◦ R ₂
.

Proof. In order to prove (58), we observe that, by direct calculations, A p ◦ R ₁
(x) = A _R

1

(p) (x)M _k ⁻¹ , (60)

e

^(θ

^◦R

⁾ (ψ _k ◦ R ₁ ) = −e ⁱ

e

^θ

ψ _k , (61)

e

^(θ

^◦R

^)(x) ∇ψ _k (R 1 (x)) = −e ⁱ

e

^θ

^(x) ∇ψ _k (x)M _k ⁻¹ .

(62)

Furthermore

θ _p (R ₁ (x)) = ( θ _R

1

(p) (x) + ^2π _k , if α ∈  _2π

k , 2π
, θ _R

1

(p) (x) + ^2π _k − 2π, if α ∈ 0, ^2π _k
, and

θ ₀ ^p (R 1 (x)) =





 θ ^R

−1 1

(p)

0 (x) + ^2π _k , if α ∈  _2π

k , 2π
, θ ^R

−1 1

(p)

0 (x) + ^2π _k − 2π, if α ∈ 0, ^2π _k
, so that

(63) θ _R

1

(p) − θ ^R

−1 1

(p)

0 = θ p ◦ R ₁ − θ ^p ₀ ◦ R ₁ .

Let us denote e Ψ p (y) = Ψ p (R 1 (y)). By direct calculations we have that, since Ψ p weakly solves the equation (i∇ + A _p ) ² Ψ _p = 0, the function e Ψ _p solves (i∇ + (A _p ◦ R ₁ )M _k ) ² Ψ e _p = 0 and hence, in view of (60),

(64) (i∇ + A _R

1

(p) ) ² Ψ e p = 0, in R ² in a weak H ^1,R

^(p) -sense.

Passing to the limit in (50) and taking into account (49) and Theorem 3.6, we obtain that, for all R > 1,

(65) Z

R

\D

(i∇ + A _p ) 

Ψ _p − e

^(θ

^−θ

^+θ

⁾ ψ _k     

2

dx

= Z

R

\D

(i∇ + A _p )Ψ _p − e

^(θ

^−θ

^+θ

⁾ i∇ψ _k    

2

dx < +∞.

By the change of variable x = R ₁ (y) in the above integral, using (60), (62), and (63) we obtain that

Z

R

\D

(i∇ + A _p )Ψ _p (x) − e

^(θ

^−θ

^+θ

^)(x) i∇ψ _k (x)    

2

dx (66)

= Z

R

\D

(i∇ + A _R

1

(p) ) e Ψ _p (y) + e

^π e

i 2

θ

R−1 1 (p)

−θ

−1 1 (p)

0

+θ

(y)

i∇ψ _k (y)    

2

dy

= Z

R

\D

(i∇ + A _R

1

(p) ) − e ⁻

^π Ψ e p
(y) − e

i 2

θ

R−1 1 (p)

−θ

−1 1 (p)

0

+θ

(y)

i∇ψ k (y)    

2

dy < +∞.

From (64), (66) and Proposition 2.1 we conclude that

−e ⁻

^π Ψ e _p = Ψ _R

1

(p)

thus proving (58).

To prove (59), we first observe that direct calculations yield A _R

_(p) ◦ R ₂
M ⁻¹ = −A _p , where M = 1 0

0 −1

 , (67)

ψ k ◦ R ₂ = ψ k , ∇ψ _k (R 2 (x)) = ∇ψ k (x)M ⁻¹ .

(68)