7 Numerical illustration

= 0.

Then we saw in Step 1 that cos

ϑ˜+2πl k +ε_l

6= 0 and sin

ϑ˜+2πl⁰ k +ε_l⁰

6= 0

for every integer l⁰ ∈ [0, k−1] different from l. By rewriting Ph+1 in the following form

P_h+1(ϑ) = sinϑP_h(ϑ) +β_h+1cos^h+1ϑ, withP_h as in (6.24), we deduce both that β_h+1 = 0 and that

P_h

ϑ˜+2πl⁰ k +ε_l⁰

= 0

for every l⁰ ∈ [0, k−1] different from l. These are k−1 conditions for a polynome of orderh≤(k−3)/2, so the induction hypothesis impliesP_h≡0 and in turnP_h+1≡0.

End of the proof of Theorem 1.7. Take any a ∈ Ω sufficiently close to 0, then by Lemma 6.4

λ^a−λ⁰=

H

X

h=1

|a|^hPh(ϑ(a)) +o(|a|^H).

By combining Lemmas 6.5 and 6.6 we obtain that P_h ≡ 0 for every h ∈ [1,(k−1)/2], therefore|λ^a−λ⁰| ≤C|a|^(k+1)/2 for some constantC indepen-dent ofa.

7 Numerical illustration

Let us now illustrate some results of this paper using the Finite Element Library [25] with isoparametricP6 lagrangian elements. We will restrict our attention to the case of half-integer circulationα= 1/2.

The numerical method we used here was presented in details in [5].

Given a domain Ω and a point a ∈ Ω, to compute the eigenvalues λ^a_j of

the Aharonov-Bohm operator (i∇+Aa)² on Ω, we compute those of the Dirichlet Laplacian on the double covering Ω^R_a of Ω\ {a}, denoted by µ^R_j . This spectrum of the Laplacian on Ω^R_a is decomposed in two disjoint parts:

• the spectrum of the Dirichlet Laplacian on Ω,λ_j,

• the spectrum of the magnetic Schrödinger operator (i∇+Aa)²,λ^a_j. Thus we have

{µ^R_j )_j≥1 ={λ^a_j}_j≥1^G{λ_j}_j≥1.

Therefore by computing the spectrum of the Dirichlet Laplacian on Ω and, for every a∈ Ω, that on the double covering Ω^R_a, we deduce the spectrum of the Aharonov-Bohm operator (i∇+Aa)² on Ω. This method avoids to deal with the singularity of the magnetic potential and furthermore allows to work with real valued functions. We have just to compute the spectrum of the Dirichlet Laplacian, which is quite standard. The only effort to be done is to mesh a double covering domain.

Let us now present the computations for the angular sector of aperture π/4:

Σ_π/4 =

(x1, x2)∈R², x1 >0, |x₂|< x1tanπ

8, x²₁+x²₂ <1

. An analysis of the spectral minimal partitions of angular sectors can be found in [8]. By symmetry, it is enough to compute the spectrum fora in the half-domain. We take a discretization grid of step 1/N withN = 100 or N = 1000:

a∈Π_N :=

(m N, n

N

,0< m < N,0< |n|

m <tanπ

8,m²+n² N² <1

) . Figure 2 gives the first nine eigenvaluesλ^a_j fora∈Π₁₀₀. In these figures, the angular sector is represented by a dark thick line. Outside the angular sector are represented the eigenvaluesλ_j of the Dirichlet Laplacian on Σ_π/4 (which do not depend ona). We observe the convergence proved in Theorem 1.1:

∀j≥1, λ^a_j →λ_j as a→∂Σ_π/4.

In Figure 3, we provide the 3-D representation of Figures 2(a) and 2(b).

Let us now deal more accurately with the singular points on the sym-metry axis. Numerically, we take a discretization step equal to 1/1000 and consider a ∈ {(m/1000,0),1 ≤ m ≤ 1000}. Figure 4 gives the first nine

60 can identify the pointsabelonging to the symmetry axis such thatλ^a_j is not simple. If we look for example at the first and second eigenvalues, we see that they are not simple respectively for one and three values of a on the symmetry axis. At such values, the function a7→λ^a_j, j= 1,2, is not differ-entiable, as can be seen in Figure 3. Figure 3 illustrates Theorem 1.3 for a domain with a piecewise C^∞ boundary: we see that the functiona 7→ λ^a_j, j= 1,2, is regular except at the points where the eigenvalueλ^a_j is not simple.

Going back to Figure 4, we see that the only critical points of λ^a_j which correspond to simple eigenvalues are inflexion points. As an example, we have analyzed the inflexion points for λ^a₃, λ^a₄, λ^a₅ when a = (a1,0) with

(a)a7→λ^a₁,a∈Π100 (b)a7→λ^a₂,a∈Π100

Figure 3: 3-D representation of Figures 2(a) and 2(b).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 50 100 150 200 250 300 350 400 450

a₍₃₎ a₍₄₎ a₍₅₎

Figure 4: a7→λ^a_j,a∈(₁₀₀₀^m ,0),0< m <1000 , 1≤j≤9.

a1 ∈ (0.6,0.7), a1 ∈(0.75,0.85) and a1 ∈ (0.45,0.55) respectively. We will denote these points bya_(j),j= 3,4,5. Figure 5 gives the nodal lines for three different pointsa= (a₁,0) on the symmetry axis y = 0 with a₁ = 0.6,0.63 and 0.65. This illustrate the emergence of a triple point when the pole is moved along the liney= 0. In Figure 6, we have plotted the nodal lines of the eigenfunctions ϕ^a_j^(j) associated with λ^a_j^(j), j = 3,4,5. We observe that

eachϕ^a_j^(j) has a zero of order 3/2 ata_(j). Correspondingly, the derivative of λ^a_j ata_(j) vanishes in Figure 4, thus illustrating Theorem 1.7. In the three examples proposed here, also the second derivative ofλ^a_j vanishes ata_(j).

(a)λ^a₃,a= (0.60,0)'a(3) (b) λ^a₃,a= (0.63,0)'a(3) (c) λ^a₃,a= (0.65,0)'a(3)

Figure 5: Nodal lines of an eigenfunction associated with λ^a₃, a = (a1,0), a₁ = 0.6,0.63,0.65.

(a)λ^a₃,a= (0.63,0)'a(3) (b) λ^a₄,a= (0.79,0)'a(4) (c)λ^a₅,a= (0.48,0)'a(5)

Figure 6: Nodal lines of an eigenfunction associated withλ^a_j^(j),j= 3,4,5.

Let us now move a little the singular point around a_(j). We use a dis-cretization step of 1/1000. Figure 7 represents the behavior ofλ^a_j foraclose to a_(j). It indicates that these points are degenerated saddle points. The behavior of the functiona7→ λ^a_j, j = 3,4,5, around a_(j) is quite similar to that of the function (t, x)7→t(t²−x²) around the origin (0,0).

We remark that computing the first twelve eigenvalues of (i∇+Aa)² on Σ_π/4, we have never found an eigenfunction for which five or more nodal lines end at a singular pointa.

As we have already remarked, all the local maxima and minima ofλ^a_j in Figure 4 correspond to non-simple eigenvalues. Plotting the nodal lines of the corresponding eigenfunctions, we have found that they all have a zero

0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68

of order 1/2 at a, i.e. one nodal line ending at a. Nonetheless, this is not a general fact: in performing the same analysis in the case Ω is a square [0,1]×[0,1], we have found that the third and fourth eigenfunctions have a zero of order 3/2 at the center a= (¹₂,¹₂), see Figure 10, which is in this case a maximum of a 7→ λ^a₃ and a minimum of a 7→ λ^a₄, see Figures 8, 9.

We observe in Figure 8 that the first and second derivatives ofλ^a₃ and ofλ^a₄ seem to vanish at the centera= (¹₂,¹₂).

Figure 8: a7→λ^a_j for aalong the perpendicular bisector or the diagonal of a square

(a)λ^a3,a= (¹₂,¹₂) (b) λ^a4,a= (¹₂,¹₂)

Figure 10: Nodal lines of an eigenfunction associated with λ^a_j, j = 3,4, a= (¹₂,¹₂).

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bonnaillie@math.cnrs.fr

IRMAR, ENS Rennes, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz (France)

benedettanoris@gmail.com

INdAM-COFUND Marie Curie Fellow

Laboratoire de Mathématiques, Université de Versailles-St Quentin, 45 avenue des États-Unis, 78035 Versailles cedex (France)

manonys@gmail.com

Fonds national de la Recherche scientifique-FNRS

Département de Mathématiques, Université Libre de Bruxelles (ULB), Boulevard du triomphe, B-1050 Bruxelles (Belgium)

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano (Italy)

susanna.terracini@unito.it

Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 20123 Torino (Italy)