We consider Aharonov–Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.
The present paper is concerned with asymptotic estimates of theeigenvalue variation for magnetic Schrödinger operators with Aharonov–Bohm potentials.These special potentials generatelocalized magnetic fields, as they are produced byinfinitely long thin solenoids intersecting perpendicularly the planeat fixed points (poles), as the radius of the solenoids goes tozero and the magnetic flux remains constant.
The aim of the present paper is the investigation of eigenvalues of these operators as functions of the poles on the domain.This study was initiated by the set of papers [1, 2, 4, 10, 19],where a single point moving in the domain was considered, providing sharp asymptotics as it goes to an interior point or to a boundary point.On the other hand, to the best of our knowledge, the only paper considering different poles is , providing a continuity result for the eigenvalues and an improved regularity for simple eigenvalues as the poles are distinct and far from the boundary.
Additional motivations for the study of eigenvalue functions of these operators appear in the theory of spectral minimal partitions.We refer the interested reader to [7, 9, 14, 20] and references therein.
For , the Aharonov–Bohm magnetic potential with pole a and circulation is defined as
In this paper we consider potentials which are the sum of two different Aharonov–Bohm potentials whose singularities are located at two different points in the domain moving towards each other.For small, let and be the poles of the following Aharonov–Bohm potential:
Let Ω be an open, bounded and connected set in such that .We consider the Schrödinger operator
with homogeneous Dirichlet boundary conditions (see Section 3.1 for the notion of magnetic Hamiltonians) and its eigenvalues , counted with multiplicities.We denote by the eigenvalues of the Dirichlet Laplacian in Ω.As already mentioned, we know from  that for every ,
The main result of the present paper is a sharp asymptotic for the eigenvalue variation as the two poles and coalesce towards a point where the limit eigenfunction does not vanish.
A first result in this direction was given in , under a symmetry assumption on the domain.
Theorem 1.1 ([3, Theorem 1.13])
Let , .Let Ω be an open, bounded and connected set in satisfying and .Let be a simple eigenvalue of the Dirichlet Laplacian on Ω, and let be a -normalized eigenfunction associated to .Let be the order of vanishing of at 0, and let be such that the minimal slope of nodal lines of is equal to , so that
for some (see, e.g., ).Let us assume that (by symmetry of Ω, this forces , i.e, the -axis is the bisector of two nodal lines of ).
For small, let , , and let be the N-th eigenvalue for .Then
as , with being a positive constant depending only on k.
In the present paper we are able to remove, in the case (i.e., when the limit eigenfunction does not vanish at the collision point), the assumption on the symmetry of the domain, proving the following result.
Theorem 1.2 ([3, Theorem 1.17])
Let Ω be an open, bounded and connected set in such that .Let us assume that there exists such that the N-th eigenvalue of the Dirichlet Laplacian in Ω is simple.Let be a -normalized eigenfunction associated to .If , then
It is worthwhile mentioning that in  simple magnetic eigenvalues are proved to be analytic functions of the configuration of the poles, provided the limit configuration is made of interior distinct poles.A consequence of our result is that the latter assumption is even necessary, and simple eigenvalues are not analytic in a neighborhood of configurations of poles collapsing outside nodal lines of the limit eigenfunction.
The proof of Theorem 1.2 relies essentially onthe characterization of the magnetic eigenvalue asan eigenvalue of the Dirichlet Laplacian in Ω with a small set removed,in the flavor of  (see Section 3.2 below). In  only the case of symmetric domains was consideredand the magnetic problem was shown to be spectrallyequivalent to the eigenvalue problem for the Dirichlet Laplacian in the domain obtained by removing the segment joining the poles.In the general non-symmetric case, we can still derive a spectral equivalence with a Dirichlet problem in the domain obtained by removing from Ω the nodal lines of magnetic eigenfunctions close to the collision point.The general shape of this removed set (which is not necessarily a segment as in the symmetric case) creates some further difficulties.In particular, precise information about the diameter of such a set is needed in order to apply the following resultfrom .
Theorem 1.3 ([3, Theorem 1.7])
Let be a bounded connected open set containing 0.Let be a simple eigenvalue of the Dirichlet Laplacian in Ω, and let be a -normalized eigenfunction associated to such that .Let be a family of compact connected sets contained in Ω such that for every , there exists such that for every ( denoting the disk of radius r centered at 0).Then
where denotes the N-th eigenvalue of the DirichletLaplacian in .
In order to apply Theorem 1.3, a crucial intermediate step in the proof of Theorem 1.2 is the estimate of the diameter of nodal lines of magnetic eigenfunctions near the collision point.More precisely, we prove that when a is sufficiently small, locally near 0 suitable (magnetic-real) eigenfunctions have a nodal set consisting in a single regular curve connecting and .If denotes the diameter of such a curve, we obtain that
see Section 4.
The paper is organized as follows.In Section 2 we obtain some preliminary upper bounds for the eigenvalue variation , testing the Rayleigh quotient for eigenvalues with proper test functions constructed by suitable manipulation of limit eigenfunctions.In Section 3 we prove that, as the two poles of the operator (1.2) move towards each other colliding at 0, is equal to the N-th eigenvalue of the Laplacian in Ω with a small piece of nodal line of the magnetic eigenfunction removed.Combining the upper estimates of Section 2 with Theorem 1.3, in Section 4 we succeed in estimating the diameter of the removed small set as in (1.4); we then conclude the proof of Theorem 1.2 by combining (1.4) and Theorem 1.3.
2 Estimates from Above
We denote by the closure of with respect to the norm
We observe that, by Poincaré and diamagneticinequalities together with the Hardy type inequality proved in , with continuous inclusion. In orderto estimate from above the eigenvalue , we recall thewell-known Courant–Fisher minimax characterization:
Let .For every , there exists a continuous radial cut-off function such that , which also has the following properties:
for all ,
if , and if ,
The function is continuous and locally in ,with .The function is supported in the disk of radius centered at 0.We therefore have
which proves (iv).We have if or , and
if . From this we directly obtain identity (iii).∎
For all , there exists a smooth function satisfying
where is the segment in defined by .Furthermore, for every , .
See [3, Lemma 3.1].∎
The first step in the proof of Theorem 1.2 is the following upper bound for the eigenvalue .
For every ,
The proof of Proposition 2.3 is based on estimates from above of the Rayleigh quotient for computed at some proper test functions constructed by suitable manipulation of limit eigenfunctions.To this end, let us consider, for each , a real eigenfunction of with homogeneous Dirichlet boundary conditionsassociated with , with .Furthermore, we choose these eigenfunctions so that
For and small enough, we set
We have that .Lemma 2.1 and the Dominated Convergence Theorem imply that tends to in when .This implies, in particular, that the functions are linearly independent for a small enough.
Hence, for small enough, is anN-dimensional subspace of , so that, in view of (2.1),
Hence, the conclusion follows from Lemma 2.1 (iv).∎
where, for all , denotes the disk of center and radius r.
Let us fix j and k in (possibly equal).In , we have that
and, since ,
An integration by parts on the last term of (2.8) gives
After cancellations, we get
where is the quadratic form defined as
with being the Kronecker delta.
To estimate the largest eigenvalue of the quadratic form , we will use the following technical lemma.
For every , let us consider the quadratic form
with such that .Assume that there exist real numbers and such that
The result is contained in [1, Lemma 6.1], hence we omit the proof.∎
Since , we can write
Since , from Lemma 2.1 (iv), it follows that
Since , we have that as . Then Lemma 2.1 (iii) implies that
For all , we have that
and hence, since and in view of Lemma 2.1,
3 Gauge Invariance, Nodal Sets and Reduction to the Dirichlet–Laplacian
In the following, by a path γ we mean a piecewise map , with being a closed interval.It follows from the definition of (see (1.1))that for any closed path γ (i.e., ),
where (resp. )) is the winding number of γ around (resp. ).
3.1 Gauge Invariance
Let us give some results concerning the gauge invariance of our operators.In view of applying them to several different situations, we give statements valid for a magnetic Hamiltonian in an open and connected domain D, without restricting ourselves to the Aharonov–Bohm case.
In the following, the term vector potential (in an openconnected domain D)stands for a smooth real vector field . Inorder to define the quantum mechanical Hamiltonian for a particle inD, under the action of the magnetic field derived from the vectorpotential , we first consider the differential operator
acting on smooth functions compactly supported in D.Using integration by parts (Green’s formula), one can easily see that P is symmetric and positive.This is formally the desired Hamiltonian, but to obtain a self-adjoint Schrödinger operator, we have to specify the boundary conditions on , which we choose to be Dirichlet boundary conditions everywhere.More specifically, our Hamiltonian is the Friedrichs extension of the differential operator P.We denote it by , and we call it the magnetic Hamiltonian on D associated with .
We observe that the Aharonov–Bohm operator ,introduced in (1.2) with the poles and in Ω, can be defined as the magnetic Hamiltonian on , where , and that the spectrum of consists of the eigenvalues defined by (2.1).The space is the form domain of .
We call gauge function a smooth complex valued function such that .To any gauge function ψ, we associate a gauge transformation acting as , with
where .We notice that, since , is a real vector field.Two magnetic potentials are said to be gauge equivalent if one can be obtained from the other by a gauge transformation (this is an equivalence relation).
The following result is a consequence of [17, Theorem 1.2].
If and are two gauge equivalent vector potentials, then the operators and are unitarily equivalent.
The equivalence between two vector potentials (which is equivalent to the fact that their difference is gauge-equivalent to 0) can be determined using the following criterion.
Let be a vector potential in D.This is gauge equivalent to 0 if and only if
for every closed path γ contained in D.
Let us first prove the direct implication. We assume that is gauge equivalent to 0, that is to say that there exists a gauge function ψ such that
Fix a closed path and consider the mapping from I to , where .By the lifting property, there exists a piecewise function such that for all .This implies that
This implies that
Since γ is a closed path, , and therefore
Let us now consider the reverse implication.We define a gauge function ψ in the following way.We fix an (arbitrary) point .Let us show that for , the quantity
does not depend on the choice of the path γ from toX. Indeed, let and be two such paths, and let be the closed path obtained by going from to Xalong and then from X to along . On theone hand, we have
On the other hand, if (3.2) holds, we have
This implies that
By the connectedness of D, there exists a path from to X for any (we can even choose it piecewise linear).We can therefore define, without ambiguity, a function by
It is immediate from the definition that and that ψ is smooth, with
It is therefore a gauge function sending to 0.∎
Lemma 3.3 can be used to define a set of eigenfunctions for having special nice properties, as was done in [13, Section 3] for the Neumann boundary condition.It is analogous to the set of real eigenfunctions for the usual Dirichlet–Laplacian.To define it, we will construct a conjugation, that is, an antilinear antiunitary operator, which commutes with .To simplify the notation, we denote by and by H in the rest of this section.
We now define the antilinear antiunitary operator K by
For all ,
The above formula and the fact that K is antilinear and antiunitary,imply that for all u and v in ,
where denotes the standard scalar product on the complex Hilbert space .By density, we conclude that
We say that a function is magnetic-realwhen .
Let us denote by the set of magnetic-real functions in . The restriction of the scalar product to gives it the structure of a real Hilbert space.The commutation relation implies that is stable under the action of H.We denote by the restriction of H to .There exists an orthonormal basis of formed by eigenfunctions of .Such a basis can be seen as a basis of magnetic-real eigenfunctions of the operator H in the complex Hilbert space .
Let us now fix an eigenfunction u of (or, equivalently, a magnetic-real eigenfunction of H). We define its nodal set as the closure in of the zero-set .Let us describe the local structure of . In the sequel, by a regular curve or regular arc we mean a curve admitting a parametrization for some .
The set has the following properties:
is, locally in , a regular curve, exceptpossibly at a finite number of singular points .
For , in the neighborhood of , consists of an even number of regular half-curves meeting at with equal angles (so that can be seen as a cross-point).
In the neighborhood of (resp. ), consists of an odd number of regular half-curves meeting at (resp. ) with equal angles (in particular this means that and are always contained in ).
The proof is essentially contained in [20, Theorem 1.5] (see also );for the sake of completeness we present a sketch of it.Let the eigenfunction u be associated with the eigenvalue λ, so that .Let be a point in .For , we denote by the open disk .Let us show that we can find small enough and a local gauge transformation such that and is a real-valued function in .Indeed, let us define, as before, a gauge function ψ such that .For small enough, we can define a smooth function such that for all , by taking
with a determination of the argument in .A direct computation shows that for ,
The gauge transformation on associated with φ therefore sends to 0.Furthermore, since u is K-real, we have in , and therefore .The real-valued function satisfies ,and, since on ,we have that .Parts (i) and (ii) of Theorem 3.6 then follow from classical results on the nodal set of Laplacian eigenfunctions (see, for instance,[15, Theorem 2.1] and [20, Theorem 4.2]).
To prove part (iii) of Theorem 3.6, we use the regularity result of  for the Dirichlet problem associated with a one-pole Aharonov–Bohm operator.Indeed, let be small enough so that and .By this choice of ε, on D, with f asmooth function, so that the domain D and the magnetic potential, restricted to D, satisfy the hypotheses of[20, Theorem 1.5]. The function u is a solution of the Dirichlet problem
with . A direct application of [20, Theorem 1.5] gives property (iii) around .We can obtain property (iii) around by exchanging the role of and .∎
3.2 Reduction to the Dirichlet–Laplacian
Our aim in this subsection is to show that, as the two poles of the operator (1.2) coalesce into a point at which does not vanish, is equal to the N-th eigenvalue of the Laplacian in Ω with a small subset concentrating at 0 removed.
Let us assume that there exists such that the N-theigenvalue of the Dirichlet Laplacian in Ω issimple. Let be a -normalized eigenfunction associated and assume that .Then, for all sufficiently small, there exists a compact connected set such that
and concentrates around 0 as , i.e., for any , there exists such that if , then .
We will divide the proof into two lemmas.
Let be such that and for all . Let .We denote by the closed ring
There exists such that if and u is a magnetic-real eigenfunction associated with , then u does not vanish in.
Let us assume, by contradiction, that there exists a sequence such that for all , admits an eigenfunction whichvanishes somewhere in .Let us denotes by a zero of in .
According to [18, Section III], we can assume, up to extraction and a suitable normalization of , that in .Since H is a uniformly regular elliptic operator in a neighborhood of , converges to uniformly on .Furthermore, up to one additional extraction, we can assume that .This implies that , contradicting the fact that for all .∎
For all such that and for all ,there exists such that if and is a magnetic-real eigenfunction associated with ,then consists in a single regular curve connecting and .
By the continuity of (see ), we have that
Let us choose such that
where is the 1-st eigenvalue of the Laplacian in the unit disk .According to Lemma 3.8, there exists such that if , then any eigenfunction associated to does not vanish in the closed ring .
Let us assume that and , and let be an eigenfunction associated with .The proof relies on a topological analysis of , inspired by previous work on nodal sets and minimal partitions (see [8, Section 6] and references therein).Lemma 3.8 implies that is compactly included in .Theorem 3.6 implies that consists of a finite number of regular arcs connecting a finite number of singular points.In other words, is a regular planar graph.Let us denote by V the set of vertices of , by the number of its connected components and by μ the number of its faces.By face, we mean a connected component of .There is always one unbounded face, so .Furthermore, for all , we denote by the degree of the vertex w, that is to say the number of half-curves ending at w.Let us note that, according to Theorem 3.6, both and belong to V and have an odd degree, and any other vertex can only have an even degree.These quantities are related through Euler’s formula for planar graphs:
For this classical formula, see, for instance, [6, Theorems 1.1 and 9.5].Note that this reference treats the case of a connected graph.The generalization used here is easily obtained by linking the connected components of the graph with edges, in order to go back to the connected case.
Let us show by contradiction that .If , there exists a bounded face of the graph , which is a nodal domain of entirely contained in .Let us call it ω.We denote by the k-th eigenvalue of the operator in ω, with homogeneous Dirichlet boundary condition on .Since ω is a nodal domain, for some depending on a, we have that
By the diamagnetic inequality,
where is the 1-st eigenvalue of the Dirichlet Laplacian in ω.By domain monotonicity,
Hence, we obtain
thus contradicting (3.4).We conclude that .
Going back to Euler’s formula (3.5), we obtain
We are now in position to prove Theorem 3.7.
Proof of Theorem 3.7.
From Lemma 3.9, it follows that for a sufficiently small, there exists a curve in connecting and and (in view of Lemma 3.8) concentrating at 0, where is a magnetic-real eigenfunction associated with .
Let us write .Since is contained in , we have that there exists (depending on a) such that
where denotes the -th eigenvalue of .
Let us consider a closed path γ in .By the definition of , γ does not meet , which means that is contained in a connected component of . Since the function is constant on all connected components of , we have that .According to (3.1), this implies that
We observe that stays bounded as .Indeed if, by contradiction, along some sequence , by (3.9), we should have
thus contradicting (3.3).
Then, for any sequence , there exists a subsequence such that for some k.Since is integer-valued we have that necessarily for j sufficiently large.Hence, (3.9) yields .It is well known (see, e.g., [11, Theorem 1.2]) that as ; hence, taking into account (1.3), we conclude that .Moreover, since the limit of does not depend on the subsequence and isinteger-valued, we conclude that for all a sufficientlysmall, so that (3.9) becomes
and the proof is complete.∎
4 Proof of Theorem 1.2
We are in position to complete the proof of Theorem 1.2.
Proof of Theorem 1.2.
For small, let be as in Theorem 3.7.We denote as
the diameter of .From Theorem 1.3, it follows that
Hence, in view of Theorem 3.7,
On the other hand, since , we have that ,so that and
for every , and then, letting , weobtain that
Dedicated to Professor Ireneo Peral on the occasion of his 70th birthday
Funding source: European Research Council
Award Identifier / Grant number: 339958
Funding source: Ministero dell’Istruzione, dell’Università e della Ricerca
Award Identifier / Grant number: 201274FYK7_008
Funding statement: The authors have been partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems –COMPAT”, funded by the European Research Council.V. Felli is partially supported by PRIN-2012-grant n. 201274FYK7_008: “Variational and perturbative aspects of nonlinear differential problems”, funded by the Ministero dell’Istruzione, dell’Università e della Ricerca.
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