Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk
aa r X i v : . [ m a t h . A P ] M a r MULTIPLE AHARONOV–BOHM EIGENVALUES: THE CASE OF THE FIRSTEIGENVALUE ON THE DISK
LAURA ABATANGELO
Abstract.
It is known that the first eigenvalue for Aharonov–Bohm operators with half-integer circu-lation in the unit disk is double if the potential’s pole is located at the origin. We prove that in fact itis simple as the pole a = 0. Introduction
In the present paper we are interested in the spectral properties of Schrödinger operators with Aharonov–Bohm vector potential (see e.g. [8, 27, 7]), acting on functions u : R → C , i.e.( i ∇ + A αa ) u := − ∆ u + 2 iA αa · ∇ u + | A αa | u, (1.1)where the vector potential is singular at the point a and takes the form A αa ( x , x ) = α (cid:18) − x − a ( x − a ) + ( x − a ) , x − a ( x − a ) + ( x − a ) (cid:19) . (1.2)We address here its eigenvalues in the unit disk in the special case when circulation α = .In order to pose the problem, we address here the general functional setting. If Ω ⊂ R is open,bounded and simply connected, for a ∈ Ω, we define the functional space H ,a (Ω , C ) as the completionof C ∞ c (Ω \ { a } , C ) with respect to the norm k u k H ,a (Ω , C ) := k∇ u k L (Ω , C ) + k u k L (Ω , C ) + (cid:13)(cid:13)(cid:13)(cid:13) u | x − a | (cid:13)(cid:13)(cid:13)(cid:13) L (Ω , C ) ! / . When the circulation of the vector potential is not an integer, i.e. α ∈ R \ Z , the latter norm is equivalentto the norm k u k H ,a (Ω , C ) = (cid:16) k ( i ∇ + A αa ) u k L (Ω , C ) + k u k L (Ω , C ) (cid:17) / , by the Hardy type inequality proved in [25] (see also [9] and [16, Lemma 3.1 and Remark 3.2]) Z D r ( a ) | ( i ∇ + A αa ) u | dx ≥ (cid:16) min j ∈ Z | j − α | (cid:17) Z D r ( a ) | u ( x ) | | x − a | dx, which holds for all r > a ∈ R and u ∈ H ,a ( D r ( a ) , C ). Here we denote as D r ( a ) the disk of center a and radius r .By a Poincaré type inequality, see e.g. [5, A.3], we can consider the equivalent norm on H ,a (Ω , C ) k u k H ,a (Ω , C ) := (cid:16) k ( i ∇ + A αa ) u k L (Ω , C ) (cid:17) / . We set the eigenvalue problem ( ( i ∇ + A αa ) ϕ = λϕ in Ω ϕ = 0 on ∂ Ω , (1.3) Date : September 28, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Magnetic Schrödinger operators, Aharonov–Bohm potential, Spectral theory, genericity. in a weak sense, that is λ ∈ C is an eigenvalue of problem (1.3) if there exists u ∈ H ,a (Ω , C ) \ { } (calledeigenfunction) such that Z Ω ( i ∇ + A αa ) u · ( i ∇ + A αa ) v dx = λ Z Ω uv dx for all v ∈ H ,a (Ω , C ) . From classical spectral theory, for every ( a, α ) ∈ Ω × R , the eigenvalue problem (1.3) admits a divergingsequence of real and positive eigenvalues { λ k ( a, α ) } k ≥ with finite multiplicity. These eigenvalues alsohave a variational characterization given by λ k ( a, α ) = min n sup u ∈ W k \{ } R Ω | ( i ∇ + A αa ) u | R Ω | u | : W k is a linear k -dim subspace of H ,a (Ω , C ) , o . (1.4)The paper [6] started the study of multiple eigenvalues of this operator with respect both to the positionof the pole a ∈ Ω and the circulation α ∈ (0 , D := { ( x , x ) ∈ R : x + x < } and the circulation α = , i.e. the problem ( ( i ∇ + A a ) ϕ = λϕ in Dϕ = 0 on ∂D. (1.5)Throughout the paper we will erase the index α , since it is fixed α = . Because of this choice, in viewof the correspondance between the magnetic problem and a real Laplacian problem on a double coveringmanifold (see [17, 30]), the operator (1.1) behaves as a real operator. As a consequence, the nodal setof the eigenfunctions of operator (1.1) (i.e. the set of points where they vanish) is made of curves andnot of isolated points as we could expect for complex valued functions. More specifically, the magneticeigenfunctions always have an odd number of nodal lines ending at the singular point a , and therefore atleast one.In particular, we are going to focus our attention on the first eigenvalue to problem (1.5) and to studyits multiplicity as the pole is moving from the origin around the disk. One can prove that this situationfulfills the assumptions of [6, Theorem 1.6], so that we know that the origin is locally the only point wherethe first eigenvalue is double. The main result of the paper is then the following Theorem 1.1.
Let λ ( a ) be the first eigenvalue of Problem (1.5) , i.e. λ ( a ) := λ ( a, ) . It is simple ifand only if a = 0 . We recall that the necessary condition is still known (see [11]). The new result is in fact the sufficientcondition. The proof relies essentially in two steps. Firstly, we observe that eigenvalue functions are radialfunctions. Thanks to the local analytic regularity of eigenvalues with respect to analytic perturbations ofthe problem, the double eigenvalue for a = 0 immediately splits in two locally analytic branches, whicha priori can be the same. We will show that in fact they are really different by means of their Taylorexpansion’s first terms. The first derivatives of the two branches at the origin can be computed in termsof the corresponding eigenfunctions’ asymptotic expansions in the spirit of [6]. This is the content ofSection 3.From a technical point of view, the disk gives us chances to compute eigenfunctions explicitly. Thiscan be done by reducing problem (1.5) to a suitable weighted Laplace eigenvalue problem on the doublecovering and thanks to a certain spectral equivalence between Problem 1.5 and suitable Laplace eigenvalueproblems with mixed boundary conditions (see Section 2). This is enough to prove that the first derivativesof the two aforementioned analytic branches computed at the origin are different, in particular withopposite sign, thus concluding Section 3.The proof is concluded in Section 4 thanks to the continuity and monotonicity of the two branches upto the boundary of the domain. HE FIRST AB EIGENVALUE ON THE DISK 3
Motivations.
The interest in Aharonov-Bohm operators with half-integer circulation α = ismotivated by the fact that nodal domains of their eigenfunctions are strongly related to spectral minimalpartitions of the Dirichlet Laplacian, i.e. partitions of the domain minimizing the largest of the firsteigenvalues on the components, in the special case when they present points of odd multiplicity (see[11]). We refer to papers [12, 13, 18, 17, 19, 20, 21, 22, 23] for details on the deep relation betweenbehavior of eigenfunctions, their nodal domains, and spectral minimal partitions. Related to this, theinvestigation carried out in [2, 3, 4, 14, 26, 29] highlighted a strong connection between nodal propertiesof eigenfunctions and asymptotic expansion of the function which maps the position of the pole a in thedomain to eigenvalues of the operator ( i ∇ + A a ) (see also [1, Section 3] for a brief overview).The interest in the case of disk comes from the seminal papers [19] and [10], where the so-called Mercedes Star Conjecture is introduced and discussed . Roughly speaking, the conjecture evokes that thespectral minimal 3-partition for the disk is in fact the
Mercedes Star partition (see [10, Figure 1]).For what concerns us, the disk gives us the opportunity to begin to tackle the interesting questionabout how rare multiple eigenvalues are with respect to the position of the pole globally in the domain.This is a first contribution to carry on the analysis started in [6]. On the other hand, the present paper isnot dealing directly with the aforementioned conjecture, but it presents arguments which may be usefultowards it. Finally, Theorem 1.1 validates numerical simulations presented in [10, Figure 1] for the firsteigenvalue. 2.
Explicit eigenfunctions and eigenvalues
The aim of this section is exploiting the symmetry of the disk in order to deduce peculiar featuresof eigenvalues to Problem (1.5). Firstly, we recall that the map a λ k ( a ) is a radial function for any k ∈ N \ { } .2.1. Eigenfunctions in the double covering.
In the papers [17, Lemma 3.3] and [30, Section 3] it isshown that in case of half-integer circulation the considered operator is equivalent to the standard Lapla-cian in the double covering. We then briefly recall some basic facts about Aharonov–Bohm operators.For any a ∈ R , we define θ a : R \ { a } → [0 , π ) the polar angle centered at a such that θ a ( a + r (cos t, sin t )) = t, for t ∈ [0 , π ) . (2.1)Thus, it results (see [16, 17, 6] for deeper explanations) that 2 A a is gauge equivalent to 0, as 2 A a = − ie − iθ a ∇ e iθ a = ∇ θ a . We introduce the following antilinear and antiunitary operator K a u = e iθ a u. which depends on the position of the pole a ∈ Ω through the angle θ a . It results that ( i ∇ + A a ) and K a commute. The restriction of the scalar product to L K a (Ω) := { u ∈ L (Ω , C ) : K a u = u } givesit the structure of a real Hilbert space and commutation implies that eigenspaces are stable under theaction of K a . Then we can find a basis of L K a (Ω) formed by K a -real eigenfunctions of ( i ∇ + A a ) . Beingallowed to consider K a -real eigenfunctions of ( i ∇ + A a ) allows to reduce the analysis to the real operator( i ∇ + A a ) L Ka (Ω) in the real space L K a (Ω). Definition 2.1. ( [14, Lemma 2.3] , [17, Lemma 3.3] ) Let Ω ⊂ R be an open simply connected andbounded set. Let a ∈ Ω be the pole of the operator. The double covering of Ω is the set ˜Ω := { y ∈ C : y + a ∈ Ω } . Lemma 2.2. ( [14, Lemma 2.3] ) Let θ denote the angle of the polar coordinates in R . If ϕ is a K -realeigenfunction of the problem (1.5) for a = 0 , then the function ψ ( y ) := e − iθ ( y ) ϕ ( y ) defined in ˜ D is real valued and it is a solution to the problem ( − ∆ ψ = 4 λ | y | ψ in ˜ Dψ = 0 on ∂ ˜ D. (2.2)The second basic special feature of the disk is stated in the following LAURA ABATANGELO
Lemma 2.3.
When a = 0 , the double covering of the unit disk D can be identified with the twofold unitdisk D .Proof. By Definition 2.1, the double covering of the unit disk D isΩ := { y ∈ C : y ∈ D } . If we identify C with R in the standard way and consider the polar coordinates ( x , x ) = ρ (cos θ, sin θ )we need that ( y , y ) = ρ (cos 2 θ, sin 2 θ ) ∈ D. Then, observing that y = x − x and y = 2 x x , a simple computation shows that y + y =( x + x ) < (cid:3) Thanks to Lemma 2.3, we are in position to have an explicit expression of eigenfunctions to Problem(1.5) by means of Bessel and trigonometric functions.
Lemma 2.4. If λ is an eigenvalue of the problem (2.2) , then it is double and its eigenfunctions takethe form ψ ( ρ cos θ, ρ sin θ ) = A J n/ ( p λ ρ ) cos( nθ ) + B J n/ ( p λ ρ ) sin( nθ ) , y = ( ρ cos θ, ρ sin θ ) ∈ ˜ D with A, B ∈ R and for some n ∈ N \ { } .Coming back to the original problem (1.5) on the original domain D , λ is a double eigenvalue of theproblem (1.5) and its eigenfunctions take the form ϕ ( r cos t, r sin t ) = e i t J n/ ( p λ r ) (cid:18) A cos (cid:0) n t (cid:1) + B sin (cid:0) n t (cid:1)(cid:19) x = ( r cos t, r sin t ) ∈ D. (2.3) Proof.
Standard separation of variables ψ ( ρ cos θ, ρ sin θ ) = u ( ρ ) v ( θ ) leads to v ( θ ) = C or v ( θ ) = A cos( nθ ) + B sin( nθ ) for n ∈ N being A, B, C ∈ R . The radial part produces a Bessel-type equation which reads ρ d udρ + ρ dudρ + (4 λ ρ − n ) u ( ρ ) = 0whose solutions are given by the so-called modified Bessel functions J n/ ( √ λ ρ ) or J − n/ ( √ λ ρ ) (forthe modified Bessel functions, see the book by Watson [33]). From the results in [16, 17] we know that theeigenfunction is regular at the origin, so its radial part will be given in terms of the only J n/ . Imposingthe boundary conditions at ρ = 1, we find J n/ ( √ λ ) = 0, which means that λ = α n/ ,k for some k ∈ N , where { α n/ ,k } k ∈ N denote the sequence of zeros of the Bessel function J n/ . This concludes the first partof the statement. By virtue of Lemma 2.2 the rest of the statement follows. (cid:3) Note that the case of the disk is covered by the paper [11]: the fact that every eigenvalue is double wasalready provided by [11, Proposition 5.3] in a more general context. Nevertheless, this is not the mainpoint we are interested in.We recall that there is a connection between the zeros of the Bessel functions (to this aim we refer to[33, Chapter XV]): in particular, the positive zeros of the Bessel function J n are interlaced with those ofthe Bessel function J n +12 and by Porter’s Theorem the positive zeros of J n are interlaced with those ofthe Bessel function J n +22 . Then, denoting z n ,k the k -th zero of the Bessel function J n , we have0 < z , < z , < z , < z , < z , < . . . Remark 2.5.
The first case is then ( n, k ) = (1 , and it corresponds to the double first eigenvalue forthe Aharonov–Bohm operator with half-integer circulation and pole at the origin.The second case is n = 3 and k = 1 , which produces the double third eigenvalue. HE FIRST AB EIGENVALUE ON THE DISK 5
Isospectrality and consequences on eigenvalues.
We introduce two auxiliary problems. Letus denote D + := { ( x , x ) ∈ D : x > } . Definition 2.6. ( [26] ) The two problems − ∆ u = λu in D + u = 0 on ∂D + \ ( t, × { } ∂u∂ν = 0 on ( t, × { } − ∆ u = λu in D + u = 0 on ∂D + \ [ − , t ) × { } ∂u∂ν = 0 on [ − , t ) × { } (2.4) are called Dirichlet–Neumann and
Neumann–Dirichlet eigenvalue problem for the Laplacian in the upperhalf-disk, respectively.
We recall the following result proved in [11] (see also [26, Proposition 5.3]).
Lemma 2.7. ( [11] ) Let a = ( t, for t ∈ [0 , . The set of the eigenvalues of Problem (1.5) { λ j ( t ) } j ≥ isthe union (counted with multiplicity) of the sequences { λ DNj ( t ) } j ≥ and { λ NDj ( t ) } j ≥ , being { λ DNj ( t ) } j ≥ and { λ NDj ( t ) } j ≥ the set of the eigenvalues of the Dirichlet–Neumann and Neumann–Dirichlet problems (2.4) respectively. By virtue of the latter Lemma 2.7 and the continuity result stated in [26] for Aharonov–Bohm eigen-values (see also [15, Section 10]), the following result holds true.
Lemma 2.8. ( [26] , [15] ) Fix k ∈ N \{ } and denote λ DNk ( t ) ( λ NDk ( t ) ) the k -th eigenvalue of the Dirichlet–Neumann problem in (2.4) (Neumann–Dirichlet problem, respectively). Then the maps t λ DNk ( t ) t λ NDk ( t ) are continuous in ( − , . We observe that in this case the standard Courant–Fisher characterization of eigenvalues establishes λ DNk ( t ) = min E ⊂H t subspace dim E = k max u ∈ E \{ } R Ω |∇ u | R Ω u , (2.5)where H t := (cid:26) u ∈ H (Ω) : u = 0 on ∂ Ω \ ( t, × { } and ∂u∂ν = 0 on ( t, × { } (cid:27) , analogously for λ NDk ( t ). Remark 2.9. By (2.5) , if − < t ≤ t < then H t ⊆ H t and then λ DNj ( t ) ≥ λ DNj ( t ) for any j ≥ , i.e. the function t λ DNj ( t ) is monotone non-decreasing for any j ≥ . As well, the function t λ NDj ( t ) is monotone non-increasing for any j ≥ .In the case of the disk, one can even see it by noting that λ DNj ( t ) = λ NDj ( − t ) because of the symmetryof the disk. Another consequence of Lemma 2.7 is the following result.
Lemma 2.10.
Let us consider the problems in (2.4) . For t = 1 we have λ DN (1) = λ ND (1) . We note the latter result can be proved by direct computation, in terms of Bessel-type functions, asin the proof of Lemma 2.4.Now, if a = ( t,
0) let us denote λ j ( t ) the j -th eigenvalue of the problem (1.5). By Lemma 2.7, symmetryof the disk and Remark 2.9 (non-increasing monotonicity of the map t λ ND ( t )), we have λ ( t ) = min (cid:8) λ DN ( t ) , λ ND ( t ) (cid:9) = min (cid:8) λ ND ( − t ) , λ ND ( t ) (cid:9) = λ ND ( t ) for any t ∈ [0 , . (2.6)We have as well λ ( t ) = min (cid:8) λ DN ( t ) , λ ND ( t ) (cid:9) = λ DN ( t ) for any t ∈ [0 , , (2.7)where the last equivalence follows from Lemma 2.8, Remark 2.9 and Lemma 2.10, recalling that λ ND (0) = λ DN (0) > λ DN (0) = λ ND (0). Indeed, if by contradiction there exists ¯ t ∈ (0 ,
1) such that λ ND (¯ t ) <λ DN (¯ t ), then Remark 2.9 implies λ ND (1) ≤ λ ND (¯ t ) < λ DN (¯ t ) ≤ λ DN (1) which denies Lemma 2.10. LAURA ABATANGELO Immediate splitting of the eigenvalue
The aim of this section is to show that as the pole is moved, then the double eigenvalue split and producetwo locally different analytic branches of eigenvalues. The first one is stricly monotone decreasing whereasthe second one is stricly monotone increasing in a small neighborhood of the origin, with respect to thedistance of the pole from the origin. In order to do this, we are going to exploit the results achieved inSection 2. In addition, by rotational symmetry, we will restrict ourselves to the case when the pole ismoving along x -axis.3.1. Analytic perturbation with respect to the pole.
As already pointed out in the Introduction(see also [6, Section 2], [26]), as the pole moves not only the operator changes, but also this producesdifferent variational settings: functional spaces depend on the position of the pole. In order to study themoving pole’s effect on eigenvalues, first of all we need to define a family of diffeomorphisms which allowus to set the eigenvalue problem on a fixed domain, in the spirit of [6, 26].We consider a particular case of the local perturbation introduced in [6, Subsection 5.1]. Let ξ ∈ C ∞ c ( R ) be a cut-off function such that0 ≤ ξ ≤ , ξ ≡ D / (0) , ξ ≡ R \ D / (0) , |∇ ξ | ≤
16 on R . (3.1)For a ∈ D / (0), we define the local transformation Φ a ∈ C ∞ ( R , R ) byΦ a ( x ) = x + aξ ( x ) . (3.2)Notice that Φ a (0) = a and that Φ ′ a is a perturbation of the identityΦ ′ a = I + a ⊗ ∇ ξ = a ∂ξ∂x a ∂ξ∂x a ∂ξ∂x a ∂ξ∂x ! , so that J a ( x ) := det(Φ ′ a ) = 1 + a ∂ξ∂x + a ∂ξ∂x = 1 + a · ∇ ξ. (3.3)Let R = 1 / a ∈ D R (0), Φ a is invertible, its inverse Φ − a is also C ∞ ( R , R ), see e.g. [28,Lemma 1]. Then, as in [6, Section 7], we define γ a : L (Ω , C ) → L (Ω , C ) by γ a ( u ) = p J a ( u ◦ Φ a ) , (3.4)where J a is defined in (3.3). Such a transformation γ a defines an isomorphism preserving the scalarproduct in L (Ω , C ). Moreover, since Φ a and √ J a are C ∞ , γ a defines an algebraic and topologicalisomorphism of H ,a (Ω , C ) in H , (Ω , C ) and inversely with γ − a , see [28, Lemma 2]. We notice that γ − a writes γ − a ( u ) = (cid:18)q J a ◦ Φ − a (cid:19) − ( u ◦ Φ − a ) . With a little abuse of notation we define the application γ a : ( H ,a (Ω , C )) ⋆ → ( H , (Ω , C )) ⋆ in sucha way that ( H , (Ω , C )) ⋆ h γ a ( f ) , v i H , (Ω , C ) = ( H , (Ω , C )) ⋆ h f, γ − a ( v ) i H ,a (Ω , C ) , (3.5)for any f ∈ ( H ,a (Ω , C )) ⋆ , and inversely for γ − a : ( H , (Ω , C )) ⋆ → ( H ,a (Ω , C )) ⋆ .We define the new operator G a : H , (Ω , C ) → ( H , (Ω , C )) ⋆ by the following relation G a ◦ γ a = γ a ◦ ( i ∇ + A a ) , (3.6)being γ a defined in (3.4) and (3.5). By [28, Lemma 3] the domain of definition of G a is given by γ a ( H ,a (Ω , C )), it coincides with H , (Ω , C ). Moreover, G a and ( i ∇ + A αa ) are spectrally equiva-lent , in particular they have the same eigenvalues with the same multiplicity and the map a G a is C ∞ ( D R (0) , BL ( H , (Ω , C ) , ( H , (Ω , C )) ⋆ ).Now, let us consider the special case a = ( a , x -axis.For simplicity, in the following we denote t := a and G t := G ( a , . HE FIRST AB EIGENVALUE ON THE DISK 7
Then, following the same argument in [26, Section 4], the family t G t is an analytic family of type (B)in the sense of Kato with respect to the variable t . In order to prove it, by definition (see [24, Chapter7, Section 4]) we need to show that the quadratic form g t associated to G t , defined as g t ( u ) = ( H , (Ω)) ⋆ h G t u, u i H , (Ω) , is an analytic family of type (a) in the sense of Kato , i.e. it fulfills the following two conditions:(i) the form domain is independent of t ;(ii) the form g t ( u ) is analytic with respect to the parameter t for any u in the form domain.The first assertion follows from (3.6) (see [6, Section 7.1]), whereas the second one follows from [6,Lemmas 5.1,5.2,7.1] possibly shrinking the interval ( − R, R ) where the parameter t is varying. TheKato-Rellich perturbation theory gives some information in the case when the considered eigenvalue isnot simple. Let λ be any double eigenvalue of G . Then there exist a family of 2 linearly independent L (Ω)-normalized eigenfunctions { u j ( t ) } j =1 , relative to the associated eigenvalue µ j ( t ) for j = 1 , t and such that for j = 1 , µ j (0) = λ and µ j ( t ) is an eigenvalueof the operator G t . We recall that G t has the same eigenvalues with the same multiplicity as operator( i ∇ + A ( t, ) . Note that the 2 functions t µ ( t ) , t µ ( t ) are not a priori necessarily distinct. TheFeynman-Hellmann formula (see [24, Chapter VII, Section 3]) then tells us that µ ′ j (0) = ( H , (Ω , C )) ⋆ h G ′ (0)[ t ] u j (0) , u j (0) i H , (Ω , C ) . (3.7)3.2. Computing the derivative at of the two branches. The aim of this subsection is showingthat the two ( a priori not necessarily different) analytic branches t µ j ( t ), j = 1 ,
2, have a differentderivative at t = 0. In order to do this, we refer to the paper [6]. In particular, for j = 1 , µ ′ j (0) = ( H , (Ω , C )) ⋆ h G ′ (0)[ t ] u j (0) , u j (0) i H , (Ω , C ) = π A j − B j ) (3.8)where A j , B j ∈ R are the coefficients in the expansion (2.3).What is left is detecting u j (0) for j = 1 ,
2. To this aim, we are going to exploit the symmetry propertyof the domain with respect to the x -axis. We refer to [11] and define the antiunitary antilinear operatorΣ : L ( D ) → L ( D ) Σ u := ¯ u ◦ σ, being σ ( x , x ) = ( x , − x ). We have that Σ and ( i ∇ + A ) commute (see [11, Section 5]), as well as Σand K . This means L K is stable under the action of Σ. Thus, if we write L K, Σ (Ω) := L K (Ω) ∩ ker(Σ − Id ) L K,a Σ (Ω) := L K (Ω) ∩ ker(Σ + Id ) , then we have the orthogonal decomposition L K (Ω) = L K, Σ (Ω) ⊕ L K,a Σ (Ω) . (3.9)We can therefore define the operators ( i ∇ + A ) and ( i ∇ + A ) a Σ , restrictions of ( i ∇ + A ) to L K, Σ (Ω)and L K,a Σ (Ω) respectively. The spectrum of ( i ∇ + A ) is the union (counted with multiplicities) of thespectra of ( i ∇ + A ) and ( i ∇ + A ) a Σ . Lemma 2.7 is then completed by the following result. Lemma 3.1. ( [11, Propositions 5.7 and 5.8] ) If u is a K -real Σ -invariant eigenfunction of ( i ∇ + A ) then the restriction to D + of e − i θ u is a real eigenfunction of the Dirichlet–Neumann problem in (2.4) .If u is a K -real a Σ -invariant eigenfunction of ( i ∇ + A ) then the restriction to D + of e − i θ u is areal eigenfunction of the Neumann–Dirichlet problem in (2.4) . Conversely, if v is an eigenfunction of theDirichlet–Neumann problem in D + , if ˜ v is the even extension of u in D , the function e i θ ˜ v is a ( K -real) Σ -invariant eigenfunction of ( i ∇ + A ) . If v is an eigenfunction of the Neumann–Dirichlet problem in D + , if ˜ v is the odd extension of u in D , the function e i θ ˜ v is a ( K -real) a Σ -invariant eigenfunction of ( i ∇ + A ) . LAURA ABATANGELO
In view of (3.4) and (3.6) we have that u (0) and u (0) are two K -real linearly independent eigen-functions of ( i ∇ + A ) . Therefore via (3.9), Lemma 3.1 and Lemma 2.4 u (0) is a Σ-invariant whereas u (0) is Σ-invariant. From Lemma 2.4, Remark 2.5 and the asymptotic expansion of the Bessel functions(see e.g. [33, Chapter 3]) there exist A, B ∈ R \ { } such that u ( r cos t, r sin t ) = e i t r / B sin t O ( r ) as r → + (3.10) u ( r (cos t, sin t )) = e i t r / A cos t O ( r ) as r → + . (3.11)Equations (3.7) and (3.8) immediately give µ ′ (0) = − π B < , (3.12) µ ′ (0) = π A > , (3.13)thus concluding the first step towards our main result.4. Conclusion
We are now in position to conclude the proof of our main result.
Proof of Theorem 1.1.
Thanks to rotational invariance of eigenvalues, it is sufficient to prove that if a = ( t,
0) and λ ( t ) is the first eigenvalue of the problem (1.5), which is double for t = 0, then λ ( t ) issimple for any t ∈ (0 , δ > µ ( t ) and µ ( t )are different for t ∈ ( − δ, δ ), since µ ′ (0) < µ ′ (0) > . (4.1)Moreover, we have that λ ( t ) = ( µ ( t ) for t ∈ ( − δ, µ ( t ) for t ∈ [0 , δ ) , (4.2)since µ j ( t ) are eigenvalues of the operator G t which is spectral equivalent to ( i ∇ + A a ) with a = ( t, t ∈ (0 , λ ( t ) < λ ( t ) for t ∈ (0 , (cid:3) tλ ND (0) = λ DN (0) λ DN ( t ) λ ND ( t ) λ ND ( t ) λ DN ( t ) b − b λ ND (0) = λ DN (0) λ ND ( t ) λ DN ( t ) Figure 1.
The double first Aharonov–Bohm eigenvalue λ (0) splits in two differentbranches of simple eigenvalues up to the boundary. HE FIRST AB EIGENVALUE ON THE DISK 9
Acknowledgements
The author would like to thank dr. Manon Nys for many discussions, as well as the anonymous refereeof the paper [6] who excited our interest about the theme.The author is partially supported by the project ERC Advanced Grant 2013 n. 339958: “ComplexPatterns for Strongly Interacting Dynamical Systems – COMPAT”, by the PRIN2015 grant “Varia-tional methods, with applications to problems in mathematical physics and geometry” and by the 2017-GNAMPA project “Stabilità e analisi spettrale per problemi alle derivate parziali”.
References [1] L. Abatangelo. Sharp asymptotics for the eigenvalue function of Aharonov–Bohm operators with a moving pole.
Rend.Sem. Mat. Univ. Politec. Torino Bruxelles-Torino Talks in PDE’s Turin, May 2–5, 2016 , Vol. 74, 2: 19–29, 2016.[2] L. Abatangelo and V. Felli. Sharp asymptotic estimates for eigenvalues of Aharonov–Bohm operators with varyingpoles.
Calc. Var. Partial Differential Equations , 54(4):3857–3903, 2015.[3] L. Abatangelo and V. Felli. On the leading term of the eigenvalue variation for Aharonov–Bohm operators with amoving pole.
SIAM J. Math. Anal. , 48(4):2843–2868, 2016.[4] L. Abatangelo, V. Felli, and C. Lena. On Aharonov–Bohm operators with two colliding poles.
Advanced Nonlin. Studies ,17:283–296, 2017.[5] L. Abatangelo, V. Felli, B. Noris, and M. Nys. Sharp boundary behavior of eigenvalues for Aharonov–Bohm operatorswith varying poles.
J. Funct. Anal. , 273: 2428–2487, 2017.[6] L. Abatangelo, M. Nys. On multiple eigenvalues for Aharonov–Bohm operators in planar domains.
Nonlinear Analysis ,2017, https://doi.org/10.1016/j.na.2017.11.010.Q3[7] R. Adami and A. Teta. On the Aharonov-Bohm Hamiltonian.
Lett. Math. Phys. , 43(1):43–53, 1998.[8] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quantum theory.
Phys. Rev. (2) , 115:485–491, 1959.[9] A. A. Balinsky. Hardy type inequalities for Aharonov-Bohm magnetic potentials with multiple singularities.
Math. Res.Lett.
10, no. 2-3:69–176, 2003.[10] V. Bonnaillie-Noël and B. Helffer. On spectral minimal partitions: the disk revisited.
Ann. Univ. Buchar. Math. Ser. ,4(LXII)(1):321–342, 2013.[11] V. Bonnaillie-Noël, B. Helffer, T. Hoffmann-Ostenhof, Aharonov-Bohm Hamiltonians, isospectrality and minimal par-titions.
J. Phys. A (18) , 42:185–203, 2009.[12] V. Bonnaillie-Noël, B. Helffer. Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on thesquare with application to minimal partitions.
Exp. Math. , 20 no. 3: 304–322, 2011.[13] V. Bonnaillie-Noël, C. Léna. Spectral minimal partitions of a sector.
Discrete Contin. Dyn. Syst. Ser. B , 19 no. 1:27–53, 2014.[14] V. Bonnaillie-Noël, B. Noris, M. Nys, and S. Terracini. On the eigenvalues of Aharonov–Bohm operators with varyingpoles.
Anal. PDE , 7(6):1365–1395, 2014.[15] M. Dauge, B. Helffer, Eigenvalues variation. II. Multidimensional problems.
J. Diff. Eq.
J. Eur. Math. Soc. (JEMS) , 13(1):119–174, 2011.[17] B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and M. P. Owen. Nodal sets for groundstates of Schrödingeroperators with zero magnetic field in non-simply connected domains.
Comm. Math. Phys. , 202(3):629–649, 1999.[18] B. Helffer. On spectral minimal partitions: a survey.
Milan J. Math. , 78, no. 2: 575–590, 2010.[19] B. Helffer, T. Hoffmann-Ostenhof. On minimal partitions: new properties and applications to the disk.
Spectrum anddynamics, CRM Proc. Lecture Notes , 52, Amer. Math. Soc., Providence, RI: 119–135, 2010.[20] B. Helffer, T. Hoffmann-Ostenhof. On a magnetic characterization of spectral minimal partitions.
J. Eur. Math. Soc.(JEMS) , 15, no. 6, 2081–2092, 2013.[21] B. Helffer, T. Hoffmann-Ostenhof, S. Terracini. Nodal domains and spectral minimal partitions.
Ann. Inst. H. PoincaréAnal. Non Linéaire , 26:101–138, 2009.[22] B. Helffer, T. Hoffmann-Ostenhof, S. Terracini. Nodal minimal partitions in dimension 3.
Discrete Contin. Dyn. Syst. ,28, no. 2: 617–635, 2010.[23] B. Helffer, T. Hoffmann-Ostenhof, S. Terracini. On spectral minimal partitions: the case of the sphere.
Around theresearch of Vladimir Maz’ya. III, Int. Math. Ser. (N. Y.) , 13, Springer, New York, 153–178, 2010.[24] T. Kato. Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics.
Springer-Verlag,Berlin , xxii+619 pp., 1995.[25] A. Laptev and T. Weidl. Hardy inequalities for magnetic Dirichlet forms. In
Mathematical results in quantum mechanics(Prague, 1998) , volume 108 of
Oper. Theory Adv. Appl. , pages 299–305. Birkhäuser, Basel, 1999.[26] C. Léna. Eigenvalues variations for Aharonov–Bohm operators.
J. Math. Phys. , 56(1):011502, 18, 2015.[27] M. Melgaard, E. M. Ouhabaz, and G. Rozenblum. Negative discrete spectrum of perturbed multivortex Aharonov–Bohm Hamiltonians.
Ann. Henri Poincaré , 5(5):979–1012, 2004.[28] A. M. Micheletti. Perturbazione dello spettro dell’operatore di Laplace, in relazione ad una variazione del campo.
Ann.Scuola Norm. Sup. Pisa (3) , 26:151–169, 1972. [29] B. Noris, M. Nys, and S. Terracini. On the Aharonov–Bohm operators with varying poles: the boundary behavior ofeigenvalues.
Comm. Math. Phys. , 339(3): 1101–1146, 2015.[30] B. Noris, S. Terracini. Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizingpartitions.
Indiana University Mathematics Journal (4) , 59: 1361–1403, 2010.[31] M. Teytel. How rare are multiple eigenvalues?
Communications on Pure and Applied Mathematics , 52: 917–934, 1999.[32] J. C. Saut, R. Temam. Generic properties of nonlinear boundary value problems.
Comm. Partial Differential Equations(3) , 4: 293–319, 1979.[33] G.N. Watson. A treatise on the theory of the Bessel functions.
Cambridge at the University press , 1944.
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