Spectral flow for pair compatible equipartitions
SSPECTRAL FLOW FOR PAIR COMPATIBLEEQUIPARTITIONS
BERNARD HELFFER AND MIKAEL PERSSON SUNDQVIST
Abstract.
We show that a recent spectral flow approach proposed byBerkolaiko–Cox–Marzuola for analyzing the nodal deficiency of the nodalpartition associated to an eigenfunction can be extended to more generalpartitions. To be more precise, we work with spectral equipartitionsthat satisfy a pair compatible condition. Nodal partitions and spectralminimal partitions are examples of such partitions.Along the way, we discuss different approaches to the Dirichlet-to-Neumann operators: via Aharonov–Bohm operators, via a double cover-ing argument, and via a slitting of the domain. Introduction
Main goals.
We consider the Dirichlet Laplacian L Ω = − ∆ in abounded domain Ω ⊂ R (and subdomains of Ω), where ∂ Ω is assumedto be piecewise differentiable.We would like to analyze the relations between the nodal domains ofthe real-valued eigenfunctions of this Laplacian and the partitions D of Ωby k open sets { D i } ki =1 , which are spectral equipartitions in the sense thatin each D i ’s the ground state energy λ ( D i ) of the Dirichlet realization ofthe Laplacian L D i in D i is the same. In addition we will consider spectralequipartitions which satisfy a pair compatibility condition (PCC) for any pairof neighbouring D i ’s, i.e. for any pair of neighbors D i , D j in, there is a linearcombination of the ground states in D i and D j which is an eigenfunction ofthe Dirichlet problem in Int( D i ∪ D j ).Nodal partitions and minimal partitions are typical examples of thesePCC-equipartitions but a difficult question is to recognize which PCC-equipartitions are minimal. This problem has been solved in the bipartitecase (which corresponds to the Courant sharp situation) but the problemremains open in the general case.Our main goal is to extend the construction and analysis of spectral flowand Dirichlet-to-Neumann operators, which was done for nodal partitionsin [7], to spectral equipartitions that satisfy the PCC. We describe brieflythe construction for nodal domains first:1.2. The spectral flow construction by Berkolaiko–Cox–Marzuola.
We describe shortly the result in [7] that we want to generalize, togetherwith their construction.
Mathematics Subject Classification.
Key words and phrases.
Spectral flow, Nodal deficiency, Dirichlet-to-Neumann operators,Aharonov–Bohm Hamiltonians. a r X i v : . [ m a t h . SP ] S e p BERNARD HELFFER AND MIKAEL PERSSON SUNDQVIST
Let Ω ⊂ R and let λ ∗ be some eigenvalue of the Dirichlet Laplacian L Ω , with corresponding eigenfunction ϕ ∗ . We denote by Γ the nodal set of ϕ ∗ inside Ω, i.e. Γ = { x ∈ Ω : ϕ ∗ ( x ) = 0 } , and introduce also the setsΩ ± = { x ∈ Ω : ± ϕ ∗ ( x ) > } , so that ϕ ∗ is positive in Ω + and negativein Ω − . Also, let k ∗ be the label of the eigenvalue λ ∗ if it is simple and theminimal label if λ ∗ is degenerate. Also, let µ ( ϕ ∗ ) denote the number ofnodal domains of ϕ ∗ , i.e. the number of connected components of the set { x ∈ Ω : ϕ ∗ ( x ) (cid:54) = 0 } .To state the main result of [7], we need to introduce Dirichlet-to-Neumannoperators. We only do this at an intuitive level at this point, and refer thereader to [3] for more details. Assume that E ⊂ R is a bounded domain,and that λ is not in the spectrum of L E . Given a sufficiently regular function g on ∂E , let u be the unique solution to (cid:40) − ∆ u = λu in E,u = g on ∂E. Then the Dirichlet-to-Neumann operator DN E ( λ ) : L ( ∂E ) → L ( ∂E ) isdefined by DN E ( λ ) g := ∂u∂ν , where ν is a unit normal vector pointing out of E . For λ in the spectrum of L E one has to be more careful and work in the orthogonal complement of afinite-dimensional subspace of L ( ∂E ). Again, the reader is referred to [3,Section 2] for more details. Theorem 1.1 ([7]) . If ε > is sufficiently small, then k ∗ − µ ( ϕ ∗ ) = 1 − dim ker( L Ω − λ ∗ ) + Mor (cid:0) DN Ω + ( λ ∗ + ε ) + DN Ω − ( λ ∗ + ε ) (cid:1) , (1.1) where Mor counts the number of negative eigenvalues of an operator (theso-called Morse index of the operator).Remark . The number k ∗ − µ ( ϕ ∗ ) in the left-hand side above is non-negative due to Courant’s nodal theorem. It is usually called the nodaldeficiency of the eigenfunction ϕ ∗ (see for example [7]). If λ ∗ is a simpleeigenvalue of L Ω then the right-hand side above is non-negative, and anindependent argument for Courant’s theorem is provided.It turns out, that to characterize the negative eigenvalues of the sumDN Ω + ( λ ∗ + ε ) + DN Ω − ( λ ∗ + ε ) it is fruitful to study the family of operators L Ω ,σ , 0 ≤ σ < + ∞ , induced by the bilinear form B σ ( u, v ) = (cid:90) Ω ∇ u · ∇ v dx + σ (cid:90) Γ u v ds, u, v ∈ H (Ω) . Also, let L Ω , + ∞ be the Laplacian in Ω with Dirichlet boundary conditionsimposed on ∂ Ω ∪ Γ. Indeed, if we denote by { λ k ( σ ) } + ∞ k =1 the set of eigenvaluesof L Ω ,σ , in increasing order, then Berkolaiko–Cox–Marzuola shows that if ε > − σ is an eigenvalue of DN Ω + ( λ ∗ + ε )+DN Ω − ( λ ∗ + ε )if, and only if, λ ∗ + ε = λ k ( σ ) for some k ∈ N .They also show that each analytic branch of the eigenvalues (not to beconfused with the family { λ k ( σ ) } ) is increasing with σ . In fact, either it PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 3 starts for σ = 0 with an eigenvalue of L Ω , + ∞ , and then it will be constantas σ increases, or the eigenvalue will increase strictly with σ . Moreover, as σ → + ∞ , the eigenvalues λ k ( σ ) converges to the eigenvalues of L Ω , + ∞ .Due to the construction, the eigenvalue λ ∗ is in fact the lowest eigenvalueof L Ω , + ∞ , with multiplicity µ ( ϕ ∗ ). Thus,lim σ → + ∞ λ k ( σ ) (cid:40) = λ ∗ , if 1 ≤ k ≤ µ ( ϕ ∗ ) ,> λ ∗ , if k > µ ( ϕ ∗ ) . By the definition of k ∗ , the operator L Ω , = L Ω has exactly k ∗ − − L Ω − λ ∗ ) eigenvalues less than, or equal to λ ∗ , and so exactly k ∗ − − L Ω − λ ∗ ) − µ ( ϕ ∗ ) of them will pass λ ∗ + ε , where ε > Examples: Equipartitions of the unit circle
Even though we will consider domains in R , we start by doing somecalculations for the unit circle. We assume that N is odd (even N correspondto the nodal case) and consider N -equipartitions D (see Figure 2.1 for thecases N = 3 and N = 5), k ( D ) = N of the unit circle and the angular part of the Laplacian, − d dθ , with Dirichletconditions at each sub-dividing point. Each interval have length Θ = 2 π/N ,and the smallest eigenvalue—the energy of the partition—is given by Λ( D ) =( N/ . (a) (b) Figure 2.1.
The unit circle with (a) the 3-partition and (b)the 5-partition.The corresponding magnetic operator on the circle is given by T = − (cid:16) ddθ − i π (cid:17) , and its spectrum consist of eigenvalues (cid:8)(cid:0) n − (cid:1) (cid:9) + ∞ n =1 , each with multiplicitytwo, dim ker (cid:104) T − (cid:16) n − (cid:17) (cid:105) = 2 . In particular, the minimal label (cid:96) ( D ) of the eigenvalue Λ( D ) = ( N/ isgiven by (cid:96) ( D ) = N. BERNARD HELFFER AND MIKAEL PERSSON SUNDQVIST
We are going to test the formula (cid:96) ( D ) − k ( D ) = 1 − dim ker (cid:0) T − Λ( D ) (cid:1) + T ( ε, D ) , (2.1)where T ( ε, D ) denotes the number of negative eigenvalues of a Dirichlet-to-Neumann operator, discussed below. In fact, since we just saw that (cid:96) ( D ) = N , k ( D ) = N , dim ker (cid:0) T − Λ( D ) (cid:1) = 2, we need to check that T ( ε, D ) = 1 . This is similar to the setting for Quantum graphs. In [21] the number ofnegative eigenvalues of a Dirichlet-to-Neumann operator of a graph Laplaciancorresponding to energy λ is calculated as a difference between the numberof eigenvalues of the corresponding Neumann and Dirichlet graph laplaciansless than λ . But graphs with loops are excluded.First we compute the Dirichlet-to-Neumann operator and the associated2 × M λ which associates with the solution u of − d dθ u = λu , u (0) = u , u (Θ) = u , the pair ( v , v ) = ( − u (cid:48) (0) , u (cid:48) (Θ)) . This leads to (cid:20) v v (cid:21) = M λ (cid:20) u u (cid:21) , where M λ is the matrix M λ = √ λ cot( √ λ Θ) − √ λ sin( √ λ Θ) − √ λ sin( √ λ Θ) √ λ cot( √ λ Θ) = (cid:20) α ( λ ) β ( λ ) β ( λ ) α ( λ ) (cid:21) , and α ( λ ) and β ( λ ) are defined via the equation above. We continue in thesame way along the circle. With ( u k , u k +1 ) = ( u ( k Θ) , u (( k + 1)Θ)) and( v k , v k +1 ) = ( − u (cid:48) ( k Θ) , u (cid:48) (( k + 1)Θ)), we find that (cid:20) v k v k +1 (cid:21) = M λ (cid:20) u k u k +1 (cid:21) , ≤ k ≤ N − . But when we come to ( u N , v N ) we have walked around the circle, and areback at the point we started. We replace ( u N , v N ) by ( − u , − v ). Thus, wefind that the N × N matrix M λ , that associates with ( u , u , . . . , u N − ) the N -tuple ( v , v , . . . , v N − ), is given by M λ := 12 α ( λ ) β ( λ ) 0 0 · · · − β ( λ ) β ( λ ) 2 α ( λ ) β ( λ ) 0 · · · β ( λ ) 2 α ( λ ) β ( λ ) · · ·
00 0 β ( λ ) 2 α ( λ ) · · · ...... ... ... . . . . . . β ( λ ) − β ( λ ) 0 0 · · · β ( λ ) 2 α ( λ ) . Thus, M λ has α on the main diagonal, β/ − β/ , N ) and ( N, M λ are given by µ k = α − β cos(2 kπ/N ) , k = 0 , . . . , N − . (2.2) PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 5
Hence the lowest one is µ = α ( λ ) − β ( λ ), and this eigenvalue is negative if √ λ = N/ ε , with ε > µ . After divison by β ( λ ), we have to analyze the sign of δ := − cos(2 π √ λ/N ) − cos(2 π/N ) . If we take √ λ = N/ ε , we have δ ( ε ) = cos(2 πε/N ) − cos(2 π/N ) > ε > √ λ = N/ ε , with ε > M λ has exactly 1 negative eigenvalue. This means that Formula (2.1)is indeed true. Remark . A more general situation in one dimension, corresponding toan interval, is analyzed in [5].3.
Equipartitions: Notation and definitions
In this section, we describe in which framework we will generalize theresults of [7].3.1.
Equipartitions, nodal partitions, and minimal partitions.
Weconsider a bounded connected open set Ω in R . A k -partition of Ω is afamily D = { D i } ki =1 of mutually disjoint, connected, open sets in Ω suchthat Ω = ∪ ki =1 D i . We denote by O k (Ω) the set of k -partitions of Ω. If D = { D i } ki =1 ∈ O k (Ω) and the eigenvalues λ ( D i ) of the Dirichlet Laplacianin D i are equal for 1 ≤ i ≤ k , we say that the partition D is a spectralequipartition . This is the type of partitions we will work on. We give twoexamples of how such partitions occur.We denote by { λ j (Ω) } + ∞ j =1 the increasing sequence of eigenvalues of theDirichlet Laplacian in Ω and by { u j } + ∞ j =1 some associated orthonormal basis ofreal-valued eigenfunctions. The ground state u can be chosen to be strictlypositive in Ω, but the other eigenfunctions { u j } j ≥ must have zerosets.For a function u ∈ C (Ω), we define the zero set of u as N ( u ) = { x ∈ Ω (cid:12)(cid:12) u ( x ) = 0 } , and call the components of Ω \ N ( u ) the nodal domains of u . Such a partitionof Ω is called a nodal partition , and we denote the number of nodal domainsof u by µ ( u ). These µ ( u ) nodal domains define a k -partition of Ω, with k = µ ( u ).Since an eigenfunction u j , restricted to each nodal domain satisfy theeigenvalue equation − ∆ u j = λ j u j together with the Dirichlet boundary con-dition, it follows that each nodal partition is indeed a spectral equipartition.By the Courant nodal theorem, µ ( u j ) ≤ j . We also say that the pair ( λ j , u j )is Courant sharp if µ ( u j ) = j .For any integer k ≥
1, and for D in O k (Ω), we introduce the energy Λ( D ) of the partition D , Λ( D ) = max i λ ( D i ) . Then we define L k (Ω) = inf D∈ O k Λ( D ) . BERNARD HELFFER AND MIKAEL PERSSON SUNDQVIST and call
D ∈ O k a minimal spectral k -partition if L k (Ω) = Λ( D ).If k = 2, it is rather well known (see [26] or [20]) that L (Ω) = λ (Ω) andthat the associated minimal 2-partition is a nodal partition, consisting ofthe nodal domains of some eigenfunction corresponding to second eigenvalue λ (Ω). In general, every minimal spectral partition is an equipartition(see [30]).3.2. Regularity assumptions on partitions.
Attached to a partition D ,we associate a closed set in Ω, which is called the boundary set of the partition: N ( D ) = ∪ i ( ∂D i ∩ Ω) . N ( D ) plays the role of the nodal set (in the case of a nodal partition).Further, we call a partition D regular if its associated boundary set N ( D )is a regular closed set in Ω. In general, a closed set K ⊂ Ω is said to be regular closed in Ω if(i) Except for finitely many distict critical points { x (cid:96) } ⊂ K ∩ Ω, the set K is locally diffeomorphic to a regular curve. In the neighborhood ofeach critical point x (cid:96) the set K consists of a union of ν (cid:96) ≥ x (cid:96) .(ii) The set K ∩ ∂ Ω consists of a (possibly empty) finite sets of boundarypoints { z m } . Moreover, in a neighborhood of each boundary point z m , the set K is a union of ρ m distinct smooth half-curves with oneend at z m .(iii) The set K has the equal angle meeting property . By this we meanthat the half-curves meet with equal angle at each critical point of K , as well as at the boundary (together with the tangent to theboundary).Nodal sets are regular [9] and in [30] it is proven that minimal partitionsare regular (modulo a set of capacity 0).For our discussion we need a weaker version of regularity which is onlyexpressed on the “boundary set”. The first and second items remain as inthe previous definition, but (iii) is changed. Indeed, we say that the closedset K ⊂ Ω is weakly regular if (i) and (ii) above hold, and further if(iv) The set K has the transversal meeting property . By this we mean thefollowing: The set K ∩ ∂ Ω consists of a (possibly empty) finite set ofboundary points { z m } . Moreover K is near each boundary point z m the union of ρ m smooth half-curves (with distinct tangent vectorsat z m ) which hit z m transversally to the boundary ∂ Ω. Finally, ateach critical point of K in the interior of Ω, the half-curves meet ina transversal way (i.e. no cusps).3.3. Odd and even points.
Given a partition D of Ω, we denote by X odd ( D ) the set of odd critical points, i.e. points x (cid:96) for which ν (cid:96) is odd.When ∂ Ω has one exterior boundary and m interior boundaries (correspond-ing to m holes), we should also consider the property (see [29]) that an oddnumber of lines arrives at some component of the interior boundary (thinkof the hole as a point). It seems that the assumption that there was onlyone boundary component was implicitly done in the litterature, or at leastwe should distinguish between the odd interior boundaries and the even PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 7 𝑥 𝑧 𝑧 𝑧 𝑧 𝑧 (a) 𝑥 𝑧 𝑧 𝑧 𝑧 𝑧 (b) Figure 3.1.
Partitions of a set Ω with three holes. In bothcases ν = 5, ρ = ρ = ρ = ρ = 1 and ρ = 3. (a) A regularpartition. Note that the angles between the curves meeting at x are 2 π/ z and the boundary is π/
4. At z , z , z and z the curvesmeet the boundary under a right angle. (b) This partitionis weakly regular. The curves meet the boundary and thecritical point transversally, but not necessarily under equalangles.boundaries. This would play a role in the definition of the Aharonov–Bohmoperator or in the construction of the double covering.We define by ∂ Ω odd ( D ) the union of the interior components of ∂ Ω forwhich an odd number of lines of N ( D ) arrive. In other words, we will speakof odd holes when we are in this case and ∂ Ω odd ( D ) corresponds to the unionof the boundaries of the odd holes. In Figure 3.2 we have marked ∂ Ω odd ( D )in bold.3.4. Pair compatibility condition.
Given an partition D = { D i } of Ω,we say that D i and D j are neighbors , which we write D i ∼ D j , if the set D ij := Int( D i ∪ D j ) \ ∂ Ω is connected. We associate with D a graph G ( D )by associating with each D i a vertex and to each pair D i ∼ D j an edge. Werecall that a graph is said to be bipartite if its vertices can be colored by twocolors so that all pairs of neighbors have different colors. We say that D is admissible if the associated graph G ( D ) is bipartite. Nodal partitions arealways admissible, since the eigenfunction changes sign when going from onenodal domain to a neighbor nodal domain.We turn to a compatibility condition between neighbors in a partition,developed in [26]. Let D = { D i } ki =1 be a regular equipartition of energyΛ( D ). Given two neighbors D i and D j , Λ( D ) is the groundstate energy ofboth L D i and L D j . There is, however, in general no way to construct afunction u ij in the domain of L D ij such that u ij = c i u i in D i and u ij = c j u j in D j . For this to be possible, it must hold that the normal derivatives of u i and u j are proportional on ∂D i ∩ ∂D j . BERNARD HELFFER AND MIKAEL PERSSON SUNDQVIST 𝑥 𝑧 𝑧 𝑧 𝑧 𝑧 (a) (b) Figure 3.2. (a) The partition D of Ω from Figure 3.1(a),here with the set ∂ Ω odd ( D ) in bold. (b) The graph G ( D )associated with the partition D . Note that it is bipartite.We say that the regular partition D = { D i } ki =1 satisfies the pair com-patibility condition , (for short PCC), if, for some λ ∈ R , and for any pair( i, j ) such that D i ∼ D j , there is an eigenfunction u ij (cid:54)≡ L D ij such that L D ij u ij = λu ij , and where the nodal set of u ij is given by ∂D i ∩ ∂D j . Werefer to Figure 4.1 for some 5-partitions of the square that satisfy the PCC.Nodal partitions and spectral minimal partitions satisfy the PCC. Remark . In the case of bipartite equipartitions which are “generic”(i.e. whose boundary set has no critical points and satisfies transversalityconditions at the boundary) necessary and sufficient conditions to have(PCC) are given by Berkolaiko–Kuchment–Smilansky in [8]. They also givea formula for the nodal deficiency using the Morse index of some functional.3.5.
Admissible k -partitions and Courant sharp eigenvalues. It hasbeen proved by Conti–Terracini–Verzini [18, 19, 20] and Helffer–T. Hoffmann-Ostenhof–Terracini [30], that, for any k ∈ N , there exists a minimal regular k -partition. Other proofs of a somewhat weaker version of this statementhave been given by Bucur–Buttazzo–Henrot [16], Caffarelli–F.H. Lin [17].It is also proven (see [26], [30]) that if the graph of a minimal partition isbipartite, then this partition is nodal. A natural question was to determinehow general the previous situation is. Surprisingly this only occurs in theCourant sharp situation.For any integer k ≥
1, we denote by L k (Ω) the smallest eigenvalue of L Ω ,whose eigenspace contains an eigenfunction with k nodal domains. We set L k (Ω) = + ∞ , if there are no eigenfunction with k nodal domains. In general,one can show that λ k (Ω) ≤ L k (Ω) ≤ L k (Ω) . The following result gives the full picture of the equality cases:
PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 9
Theorem 3.2 ([30]) . Suppose that Ω ⊂ R is smooth and that k ∈ N . If L k (Ω) = L k (Ω) or L k (Ω) = λ k (Ω) then λ k (Ω) = L k (Ω) = L k (Ω) , and one can find a Courant sharp eigenpair ( λ k , u k ) . The Aharonov–Bohm approach
The Aharonov–Bohm operator.
Let Ω ⊂ R be a bounded con-nected domain. We recall some definitions and results about the Aharonov–Bohm (AB) Hamiltonian with poles at a finite number of points. Theseresults were initially motivated by the work of Berger–Rubinstein [6] andfurther developed in [1, 29, 11, 10]. We begin with the case of one pole.4.1.1. Simply connected Ω , one AB pole. We assume that there is one ABpole X is located at X = ( x , y ) ∈ Ω and introduce the magnetic vectorpotential A X ( x, y ) = ( A X ( x, y ) , A X ( x, y )) = Φ2 π (cid:16) − y − y r , x − x r (cid:17) . Here Φ is the intensity of the AB magnetic field, and r denotes the Euclideandistance between ( x, y ) and ( x , y ). We know that in this case the magneticfield vanishes identically in the punctured domain˙Ω X = Ω \ { X } . We introduce the magnetic gradient ∇ A X as ∇ A X = ∇ − i A X , and considerthe self-adjoint AB Hamiltonian T A X = − ( ∇ A X ) . This operator is definedas the Friedrichs extension associated with the quadratic form C + ∞ ( ˙Ω X ) (cid:51) u (cid:55)→ (cid:90) Ω (cid:12)(cid:12) ∇ A X u (cid:12)(cid:12) dx. We introduce next the multi-valued complex argument function ϕ X ( x, y ) = arg (cid:0) x − x + i( y − y ) (cid:1) . This function satisfies A X = Φ2 π ∇ ϕ X . This implies that with the flux conditionΦ2 π = 12one has − A X = A X − ∇ ϕ X , and that multiplication with the function e iϕ X , uni-valued in ˙Ω X , is a gaugetransformation intertwining T A X and T − A X .The anti-linear operator K X : L (Ω) → L (Ω), defined by u (cid:55)→ K X u = exp(i ϕ X )¯ u becomes a conjugation operator. In particular ( K X ) is the identity operator, (cid:104) K X u, K X v (cid:105) = (cid:104) u, v (cid:105) , and K X T A X = T A X K X . We say that a function u is K X -real, if it satisfies K X u = u. Then theoperator T A X is preserving the K X - real functions. In the same way oneproves that the Dirichlet Laplacian admits an orthonormal basis of realvalued eigenfunctions or one restricts this Laplacian to the vector spaceover R of the real-valued L functions, one can construct for T A X a basisof K X -real eigenfunctions or, alternately, consider the restriction of the ABHamiltonian to the vector space over R L K X ( ˙Ω X ) = { u ∈ L ( ˙Ω X ) , K X u = u } . Simply connected Ω , several AB poles. We can extend our constructionof an Aharonov–Bohm Hamiltonian in the case of a configuration with (cid:96) distinct points X = { X j } (cid:96)j =1 in Ω (putting a flux Φ = π at each of thesepoints). We can just take as magnetic potential A X = (cid:96) (cid:88) j =1 A X j . The corresponding AB Hamiltonian T A X is again defined as the Friedrichsextension, this time via the natural quadratic form in C + ∞ ( ˙Ω X ), where˙Ω X = Ω \ X .We can also construct (see [29]) the anti-linear operator K X , where ϕ X isreplaced by a multivalued functionΦ X = (cid:96) (cid:88) j =1 ϕ X j which satisfies ∇ Φ X = 2 A X . Moreoverexp(iΦ X ) = (cid:96) (cid:89) j =1 exp(i ϕ X j )is uni-valued and belongs to C ∞ ( ˙Ω X ). As in the case of one AB pole, wecan consider the (real) subspace of the K X -real functions in L K X ( ˙Ω X ), andour operator as an unbounded selfadjoint operator in L K X ( ˙Ω X ).4.1.3. Non-simply connected Ω . If Ω is not simply connected, we also acceptsome of the Aharonov–Bohm fluxes to be placed in holes in the boundedcomponents of the complement of Ω. If Ω has m holes ω , . . . , ω m , we allow m (cid:48) “odd” poles X (cid:48) i (with 0 ≤ m (cid:48) ≤ m and X (cid:48) i ∈ ω i for i ∈ { , . . . , m (cid:48) } ).We call odd holes the holes for which we have introduced the poles X (cid:48) j . Inthis case, we can reproduce the same construction. We call the remainingholes in Ω even .At the end, for (cid:98) X := X ∪ X (cid:48) , we have constructed an AB Hamiltonianin ˙Ω X associated with a magnetic potential A (cid:98) X with poles at X and halfrenormalized flux created by the magnetic potential in each odd hole, theflux created in the even hole being 0. Similarly, we can as before use the“conjugation” operator K (cid:98) X .It was shown in [29, 1] (inspired by the previous work [6]) that the nodalset of such a K (cid:98) X -real eigenfunction has the same structure as the nodal setof a real-valued eigenfunction of the Dirichlet Laplacian except that an odd PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 11 number of half-lines meet at each pole (see Subsection 3.3), and that thenumber of lines meeting the interior boundary should be odd (resp. even) atthe boundary of an odd (resp. even) hole.4.2.
Equipartitions and nodal partitions of AB Hamiltonians.
Westart from constructions introduced in [26, 10]. Suppose Ω is a bounded,simply connected (thus, to simplify, we describe the case without holes),domain and that ∂ Ω is piecewise differentiable. Let D be a regular k -equipartition with energy Λ( D ) = l k (Ω) satisfying the PCC.We denote by X = X odd ( D ) = { X j } (cid:96)j =1 the critical points of the boundaryset N ( D ) of the partition for which an odd number of half-curves meet.For this family of points X , l k (Ω) is an eigenvalue of the AB Hamiltonianassociated with ˙Ω X , and we can explicitly construct the correspondingeigenfunction with k nodal domains described by { D i } .In [30] it was proven that there exists a family { u i } ki =1 of functions suchthat u i is a ground state of L D i and u i − u j is a second eigenfunction of L D ij when D i ∼ D j (here we have extended u i and u j by 0 outside of D i and D j ,respectively, and we recall that D ij = Int( D i ∪ D j )).The claim is that one can find a sequence ε i ( x ) of S -valued functions,where ε i is a suitable square root of exp( iϕ X ) in D i , such that (cid:80) i ε i ( x ) u i ( x )is an eigenfunction of the AB Hamiltonian T A X associated with the eigenvalue l k (Ω) = Λ( D ).4.3. The Berkolaiko–Cox–Marzuola construction in the Aharonov–Bohm approach.
We follow the approach of [7] but to be able to treatnon-admissible partitions we introduce the AB Hamiltonian attached to thepartition. Thus, let D be a k -partition in Ω. We denote by Γ = N ( D ) theboundary set of the partition in Ω, and by m k the multiplicity of l k (Ω) aseigenvalue of the magnetic AB Hamiltonian T A X , defined above.We consider the family { B σ } σ ∈ R of sesquilinear forms defined on themagnetic Sobolev space H , A (Ω) × H , A (Ω) (see L´ena [33] and also [23]) by( u, v ) (cid:55)→ B σ ( u, v ) = (cid:90) Ω ∇ A u · ∇ A v + σ (cid:90) Γ u v dS Γ , where A is the magnetic AB potential: A = A X and dS Γ is the inducedmeasure on (each arc of) Γ.We should explain how the integral over Γ is to be interpreted. For eacharc γ i in Γ ∩ ˙Ω X , we can define in its neighborhood V ( γ i ) in ˙Ω X a C ∞ square root exp(iΦ X /
2) of exp(iΦ X ) and we have exp(iΦ X / u ∈ H ( V ( γ i ))if u ∈ H A (Ω). We can then define (cid:90) γ i u v dS γ i := (cid:90) γ i (exp(iΦ X / u ) · (exp(iΦ X / v ) dS γ i , Note that by construction the D i ’s never contain any point of X . By Euler formula apath γ in D i can only contain an even number of points in X . Hence the square root iswell defined and the ground state energy of the Dirichlet Laplacian L D i is the same as theground state energy of H A X in D i . See [33]. where we use the standard trace for an element of H . Note that, with thisdefinition, the “magnetic trace space” on γ i is identified as H / A ( γ i ) := exp( − iΦ X / H / ( γ i ) . We further set H / A (Γ) := ⊕ i H / A ( γ i ), and writing (cid:90) Γ u v dS Γ = (cid:88) i (cid:90) γ i u v dS γ i , this permits to show that the sesquilinear form B σ is continuous.Associated with this sesquilinear form we have the corresponding magnetic-Robin AB Hamiltonian L σ defined as the Friedrichs extension.We also define L + ∞ as the corresponding AB magnetic Schr¨odinger oper-ator, with Dirichlet boundary conditions at ∂ Ω ∪ Γ.We collect some properties of the operators { L σ } . Proposition 4.1.
Assume that D is a weakly regular partition of Ω .The self-adjoint operators { L σ } , −∞ < σ ≤ + ∞ , have compact resolvents.Moreover, if σ ∈ R , then the domain of L σ consists of all elements u ∈ H , A (Ω) such that ( ∇ A ) u ∈ L (Ω) , and such that the following transmissionconditions are satisfied: If D i and D j are two neighbors in the partition D of Ω , and γ is a regular arc in ∂D i ∩ ∂D j , then, on γ , ν i · ∇ A u i = ν i · ∇ A u j = σ ( u j − u i ) , (4.1) where ν i is the exterior normal to D i (at a point of γ ) and u i denotes therestriction of u to D i .Proof. The proof follows in the same manner as for the Laplace operator,with small additions or modifications. We refer to [2, Proposition 2.2] forthe characterization of domain, and the compactness of the resolvent. Here,one should note that the magnetic Sobolev space H A (Ω) is continuouslyembedded in the ordinary Sobolev space H (Ω) if the fluxes around the polesare non-integers (see [33, Corollary 2.5]).For the transmission conditions along the boundary set, we refer to [24]. (cid:3) Given −∞ < σ ≤ + ∞ , we denote by { ˆ λ n ( σ ) } n ∈ N the analytic eigenvaluebranches of L σ , and we enumerate by { λ n ( σ ) } n ∈ N the increasing sequenceof eigenvalues of L σ , counted with multiplicity. As in [7, Lemma 2], aperturbative argument shows that σ (cid:55)→ ˆ λ n ( σ ) is either strictly increasing orequal to ˆ λ n (0), and the latter case only occurs when ˆ λ n (0) is an eigenvalueof L + ∞ . Proposition 4.2. As σ → + ∞ , λ n ( σ ) → λ n (+ ∞ ) . (4.2) Proof.
The resolvents of L σ converge to the resolvent of L + ∞ as σ → + ∞ ,see [2, Proposition 2.6] for the proof in the case of the Laplacian. It thenfollows (see [2, Proposition 2.8]) that the eigenvalues also converge. (cid:3) The operator L + ∞ can be identified as the direct sum of the AB magneticSchr¨odinger operators with vector potential A on each component D i of thepartition D , with Dirichlet boundary conditions on ∂D i . It remains to provethat we can gauge away the magnetic potential. For this we need PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 13
Lemma 4.3.
In each D j , the square root exp (cid:0) i Φ X (cid:1) can be defined as aunivalued function exp(i ϕ j ) in C ∞ ( D j ) . Moreover, ∇ A X exp(i ϕ j ) = exp(i ϕ j ) ∇ , in D j .Proof. It suffices to observe that for each X (cid:96) , exp (cid:0) i ϕ X(cid:96) (cid:1) has this property(distinguish the case when X (cid:96) ∈ ∂D j or not). (cid:3) We can now construct the magnetic Neumann–Poincar´e operator (calledΛ + ( ε ) + Λ − ( ε ) in [7] in the case without magnetic field). For this we proceedin the following way.For each D j , we consider ∂D j ∩ Ω. We introduce the magnetic Dirichlet–Neumann operator on ∂D j which associates, for ε >
0, to a function h ∈ H / A ( ∂D j ), vanishing on ∂ Ω ∩ ∂D j a solution u to (cid:40) T A u = ( l k + ε ) u in D j ,u = h on ∂D j . (4.3)Assuming first that A is regular,we define a pairing of elements in H − / A ( ∂D j )and H / A ( ∂D j ), inspired by how it is done in the non-magnetic case by theGreen–Riemann formula.If v ∈ H / A ( ∂D j ) there exists w ∈ H A ( D j ) such that (cid:40) − ( ∇ A ) w = 0 in D j w = v on ∂D j . The mapping v (cid:55)→ w is continuous from H / A ( ∂D j ) into H A ( D j ). Then,we set (cid:10) ν j · ∇ A u, v (cid:11) H − / A ( ∂D j ) ,H / A ( ∂D j ) := −(cid:104)∇ A u, ∇ A w (cid:105) + (cid:104) ( ∇ A ) u, w (cid:105) , (4.4)where ν j is the exterior normal derivative to ∂D j .Actually, to avoid possible problems with the singularities of A , we cancome back to the the case A = 0. We observe that exp(i ϕ j ) u belongs to H / ( ∂D j ), where ϕ j is defined in Lemma 4.3 (this will be our definition of H / A ( ∂D j ) in the case of singularities which coincides with the usual onein the regular case). So ν j · ∇ (exp(i ϕ j ) u ) belongs to H − / ( ∂D j ). Aftermultiplication with exp( − i ϕ j ) we end up in H − / A ( ∂D j ).We then define, for each D j , the reduced magnetic Dirichlet–Neumannoperator on H / A ( ∂D j ∩ Ω) by restricting the magnetic Dirichlet–Neumannoperator initially defined on H / A ( ∂D j ) and identifying H / A ( ∂D j ∩ Ω) toˆ H / A := { h ∈ H / A ( ∂D j ) , h = 0 on ∂ Ω ∩ ∂D j } :Λ j, A ( ε, l k ) h = ν j · ∇ A u | ∂D j ∩ Ω := exp( − i ϕ j ) (cid:0) ν j · ∇ (exp(i ϕ j ) u ) (cid:1) . We have to verify the compatibility of our definition of H / A in the commonboundaries of neighbors D i and D j , i.e. that the restriction of H / A ( D i ) to ∂D i ∩ ∂D j ∩ Ω coincides with the restriction of H / A ( D j ) to ∂D i ∩ ∂D j ∩ Ω.For this it suffices to observe that, for some constant c ij (cid:54) = 0,exp(i ˜ ϕ i ) = c ij · exp(i ˜ ϕ j ) where ˜ ϕ i (respectively ˜ ϕ j ) is the natural extension of ϕ i (respectively ϕ j ) in aneighborhood of ∂D i ∩ ∂D j ∩ ˙Ω X in ˙Ω X as a solution of d ˜ ϕ i = A (respectively d ˜ ϕ j = A ).At this point the Neumann–Poincar´e operator Λ NP A ( ε, D ) is defined as anoperator from H / A (Γ) into H − / A (Γ):Λ NP A ( ε, D ) = k (cid:88) j =1 ι j Λ j, A ( ε, l k ) r j , (4.5)where r j is the restriction of H / A (Γ) to H / A (Ω ∩ ∂D j ) and ι j is the extension(by 0) of the operator from H − / A (Ω ∩ ∂D j ) to H − / A (Γ). Proposition 4.4.
The operator Λ NP A ( ε, D ) is self-adjoint.Proof. The proof is similar to the non-magnetic case, and it is based onthe corresponding magnetic Green–Riemann formula. We consider onecomponent D i . Assume that u and v belong to H A ( D i ) and that ( ∇ A ) u and ( ∇ A ) v belong to L ( D i ). Then we claim that (cid:10) ν · ∇ A u, v | ∂D i (cid:11) H − / A ( ∂D i ) ,H / A ( ∂D i ) = −(cid:104)∇ A u, ∇ A v (cid:105) + (cid:104) ( ∇ A ) u, v (cid:105) . Indeed, according to (4.4), with v = v | ∂D i , (cid:10) ν · ∇ A u, v | ∂D i (cid:11) H − / A ( ∂D i ) ,H / A ( ∂D i ) = −(cid:104)∇ A u, ∇ A w (cid:105) + (cid:104) ( ∇ A ) u, w (cid:105) = −(cid:104)∇ A u, ∇ A v (cid:105) + (cid:104) ( ∇ A ) u, v (cid:105)− (cid:104)∇ A u, ∇ A ( w − v ) (cid:105) + (cid:104) ( ∇ A ) u, ( w − v ) (cid:105) = −(cid:104)∇ A u, ∇ A v (cid:105) + (cid:104) ( ∇ A ) u, v (cid:105) . In the last step we used the fact that w − v satisfies a Dirichlet condition at ∂D i , so the terms from the two preceeding lines cancel each other.With the additional condition that T A u = ( l k + ε ) u , we find that (cid:10) ν · ∇ A u, v | ∂D i (cid:11) H − / A ( ∂D i ) ,H / A ( ∂D i ) = −(cid:104)∇ A u, ∇ A v (cid:105) + ( l k + ε ) (cid:104) u, v (cid:105) . If we further assume that T A v = ( l k + ε ) v , then (cid:10) ν · ∇ A u, v | ∂D i (cid:11) H − / A ( ∂D i ) ,H / A ( ∂D i ) = (cid:10) u | ∂D i , ν · ∇ A v (cid:11) H / A ( ∂D i ) ,H − / A ( ∂D i ) . The self-adjointness follows. (cid:3)
Following [7], we denote by τ A ( ε, D ) the number of negative eigenvaluesof Λ NP A ( ε, D ). We introduce the defect Def( D ) of the partition D asDef( D ) := (cid:96) ( D ) − k ( D ) , where (cid:96) ( D ) denotes the minimal labelling of the eigenvalue l k of the ABHamiltonian T A , and k = k ( D ) is the number of components of the parti-tion D . We are ready to state our main result, and for simplicity we do itfor simply connected domains Ω (the only change for the general case wouldbe in the definition of the magnetic Aharonov–Bohm potential). PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 15
Theorem 4.5.
Let D be a regular k -equipartition of a simply connecteddomain Ω satisfying the PCC with energy l k = l k (Ω) . Let A = A X be theassociated Aharonov-Bohm potential. Then, for sufficiently small ε > , Def( D ) = 1 − dim ker( T A − l k ) + τ A ( ε, D ) . Lemma 4.6.
Assume that σ > . Then − σ is an eigenvalue of Λ NP A ( ε, D ) if, and only if, l k + ε is an eigenvalue of L σ . If this is the case, then themultiplicities agree.Proof. This is merely by construction, with the transmission conditions fromProposition 4.1. We refer to [2, Theorem 4.1] and to [7, Lemma 1]. (cid:3)
Proof of Theorem 4.5.
The proof is similar to the proof of equation (3) in [7].The first eigenvalue λ (+ ∞ ) of L + ∞ is also the first Dirichlet eigenvalueon each component of D , and hence it has multiplicity k ( D ). Moreover, itequals l k . Hence, lim σ → + ∞ λ n ( σ ) (cid:40) = l k , ≤ n ≤ k,> l k , n > k. The operator L = T A , on the other hand, has (cid:96) ( D ) + dim ker( T A − l k ) − l k , and exactly k ( D ) of them will convergeto l k as σ → + ∞ . This means that (cid:96) ( D ) + dim ker( T A − l k ) − − k ( D )eigenvalues of L σ will cross l k + ε for some finite σ >
0, if ε > NP A ( ε, D ), including counting multiplicity. (cid:3) Remark . It would be interesting to understand, like in the bipartitesituation, the link between the zero deficiency property1 − dim ker( T A − l k ) + τ A ( ε, D ) = 0 , and the minimal partition property.It is mentioned in [27, Remark 5.2] that if we have a minimal k -partitionthen we are in the Courant sharp situation for the corresponding AB Hamil-tonian T A , i.e. it has the zero deficiency property.The converse is true as recalled above for a bipartite partition but wrongin general. A counterexample is given for the square and k = 5 in [10,Fig. 19], which is kindly reproduced in Figure 4.1. We have on the left a5-partition with one critical odd point which is the nodal partition of the 5-theigenfunction of its associated AB operator, but is not minimal. We haveon the right a 5-partition with four critical odd points which is the nodalpartition of the 5-th eigenfunction of its associated AB operator, which isnot minimal. It is conjectured that a minimal 5-partition is indeed obtainedfor the middle configuration with also four odd critical points.4.4. The Berkolaiko–Cox–Marzola construction through double cov-ering lifting.
In many of the papers analyzing minimal partitions, the au-thors refer to a double covering argument. This point of view (which appearsfirst in [29] in the case of domains with holes) is essentially equivalent to theAharonov approach. We just mention the main lines of the argument. Onecan, in an abstract way, construct a double covering manifold (cid:101)
Ω := ˙Ω X R above Figure 4.1.
Three 5-equipartitions satisfying the PCC, with0 deficiency index. The middle one has minimal energy amongthese three.˙Ω X . One can then lift the initial spectral problem to one for the Laplaceoperator on this new (singular) manifold (cid:101) Ω. In this lifting, the K X -realeigenfunctions become eigenfunctions which are real and antisymmetric withrespect to the deck map (exchanging two points having the same projectionon ˙Ω X ).In the case of the disk, the construction is equivalent to considering theangular variable θ ∈ (0 , π ), and the deck map corresponds to the translationby 2 π . The nodal set of the 6-th eigenfunction gives by projection theMercedes star and the 11-th eigenvalue (which is the 5-th in the spaceof antiperiodic functions) gives by projection the candidate for a minimalthree-partition.Starting from an eigenfunction u of T A with zeroset Γ, the idea is now toapply the construction of Berkolaiko–Cox–Marzola to the Laplacian on (cid:101) Ω andto the lifted eigenfunction ˜ u , having in mind that this is an antisymmetriceigenfunction. The zero-set of ˜ u is Π − (Γ). One should then interpret thequantities for the covering in term of the basis.Hence we should define the Poincar´e-Neumann attached to the Laplacianon (cid:101) Ω and (cid:101)
Γ = Π − (Γ) and reinterpret it when restricted to antisymmetricfunctions on (cid:101) Γ. The spectrum of T A consists of the eigenvalues correspondingto the antisymmetric eigenfunctions of − ∆. Hence the labelling of the eigen-value of T A corresponds to the labelling of − ∆ restricted to the antisymmetricspace. 5. The cutting construction for general regularPCC-equipartitions
Example: the Mercedes star.
We first consider the case when wehave in Ω a 3-partition, with only one critical point (which has the topology ofthe Mercedes star). We can assume that the critical point is at 0 ∈ Ω and wedenote by Γ , Γ and Γ the three branches of the star. The partition consistsin three open sets denoted by D := (cid:98) D , D := (cid:98) D and D := (cid:98) D , wherewe have ∂ (cid:98) D ij ∩ Ω = Γ i ∪ Γ j . One such example is given in Figure 5.1. Startingfrom the “formal” Aharonov–Bohm point of view, we try to eliminate thereference to this operator by using a suitable square root exp(i ϕ X ( · ) / \ Γ . We note that exp(i ϕ X ( · ) /
2) can be well defined as
PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 17 𝐷 𝐷 𝐷 (a) Γ Γ Γ (b) Figure 5.1.
The unit disk with (a) a simple 3-equipartition(b) the boundary of this 3-equipartition.a univalued function on Ω \ Γ . If u ∈ H , A (Ω), ˆ u := exp( − i ϕ X / u belongsto H (Ω \ Γ ) with boundary condition ˆ u | ∂ Ω = 0 and ˆ u | Γ − = − ˆ u | Γ +1 . Wedenote this space as (cid:98) H (Ω \ Γ ).On this new Sobolev space (cid:98) H (Ω \ Γ ), we get the sesquilinear form(ˆ u, ˆ v ) (cid:55)→ ˆ B σ (ˆ u, ˆ v ) = (cid:88) j (cid:90) D j ∇ ˆ u · ∇ ˆ v dx + σ (cid:90) Γ ˆ u ˆ v dS Γ = (cid:90) Ω \ Γ ∇ ˆ u · ∇ ˆ v dx + σ (cid:90) Γ ˆ u ˆ v dS Γ . Here we note that on Γ the left trace of ˆ u ˆ v equals the right trace of ˆ u ˆ v . Thequestion is now to determine what is the transmission obtained on Γ, theoperator being the standard Laplacian. We write ˆ u | D j = ˆ u j ( j = 1 , ,
3) andwe can redescribe (cid:98) H (Ω \ Γ ). Then we can express the new transmissionrelations through Γ , Γ and Γ . The transmission conditions are unchangedon Γ and Γ . We find that − ∂ ν − ˆ u − ∂ ν ˆ u = σ ˆ u , ˆ u = ˆ u , on Γ , − ∂ ν − ˆ u − ∂ ν ˆ u = σ ˆ u , ˆ u = ˆ u , on Γ ,and ∂ ν − ˆ u − ∂ ν ˆ u = σ ˆ u , ˆ u = − ˆ u , on Γ .We next define the operator that replaces Λ − ( ε ), Λ + ( ε ) in [7]. Again, it isan operator from H / (Γ) to H − / (Γ).At this stage we have defined a realization of the Laplacian (cid:98) T Γ , (cid:98) Γ ( σ ) with (cid:98) Γ = Γ . For σ = 0, we get the operator ˆ L Ω \ (cid:98) Γ replacing the Dirichlet realization L Ω or the AB Hamiltonian in the magnetic Laplacian. For σ = + ∞ , we recoverthe Dirichlet Laplacian on the disjoint union of the D i ’s.We start from the triple (ˇ u , ˆ u , ˆ u ) and use the reduced Dirichlet-to-Neumann operators associated to each D i (i.e. the Dirichlet-to-Neumannoperator restricted to elements with trace 0 on ∂ Ω) to associate– in the case of D , with a pair (ˇ u ⊕ ˆ u ) in H / (Γ ) ⊕ H / (Γ ) theelement ˆ f ⊕ ˆ f in H − / (Γ ) ⊕ H − / (Γ ); – in the case of D , to a pair (ˆ u ⊕ ˆ u ) in H / (Γ ) ⊕ H / (Γ ) anelement ˆ g ⊕ ˆ g in H − / (Γ ) ⊕ H − / (Γ );– in the case of D , to a pair (ˆ u ⊕ ( − ˇ u )) in H / (Γ ) ⊕ H / (Γ ) anelement ˆ h ⊕ ˆ h in H − / (Γ ) ⊕ H − / (Γ ).Summing up, we get the map(ˇ u ⊕ ˆ u ⊕ ˆ u ) (cid:55)→ (( ˆ f + ˆ h ) ⊕ ( ˆ f + ˆ g ) ⊕ (ˆ g + ˆ h )) , which acts in H / (Γ) with the notation introduced in (4.5):ˆΛ NP ( ε, D , (cid:98) Γ) = (cid:88) i =1 ι i Λ i ( ε, l k )ˆ r i , where ˆ r = r , ˆ r (ˆ u , ˇ u ) = (ˆ u , − ˇ u ), and ˆ r = r .In this way, we avoid to discuss the artificial singularities introduced withAharonov–Bohm operators.In this formalism, we denote by (cid:98) τ ( ε, D , (cid:98) Γ) the number of negative eigen-values of ˆΛ NP ( ε, D , (cid:98) Γ). We introduce the defect (cid:100)
Def( D , (cid:98) Γ) of the partition D as (cid:100) Def( D ) := ˆ (cid:96) ( D , (cid:98) Γ) − k ( D ) , where ˆ (cid:96) ( D ) denotes the minimal labelling of the eigenvalue ˆ l k of the Hamil-tonian (cid:98) τ Γ , (cid:98) Γ . We can reformulate Theorem 4.5 in the following way: Theorem 5.1.
Let D a regular k -equipartition of a simply connected domain Ω satisfying the PCC with energy l k = l k (Ω) . Then, for sufficiently small ε > , (cid:100) Def( D ) = 1 − dim ker( ˆ L Ω \ (cid:98) Γ − l k ) + (cid:98) τ ( ε, D , (cid:98) Γ) . (5.1)What we have established above is the validity of the theorem if Ω is theMercedes star. Note that due to the symmetries one can have in this casea nicer more explicit expression for ˆΛ NP ( ε, D , (cid:98) Γ) (recall the analysis on thecircle from Section 2). The question is now how to choose (cid:98)
Γ in the generalsituation and to define ˆΛ NP ( ε, D , (cid:98) Γ).5.2.
The general case.
The question is now to extend what we have donefor the Mercedes star.5.2.1.
The choice of (cid:98) Γ . This is indeed quite analogous to what is done whenwe want to define the square root of z (cid:55)→ ( z − z )( z − z ) . . . ( z − z (cid:96) ) in amaximal domain of C . By defining branch cuts, we can then recover thedouble covering by gluing the two sheets along these branch cuts.In our case, we have in a addition a boundary set Γ containing the “odd”points X , . . . , X (cid:96) in Ω and what we have to prove is that Γ contains aclosed subset (cid:98) Γ corresponding to the branch cuts, which is minimal, in asense described below. These branch cuts are either connecting inside Γone odd point to (one point of) the boundary or connecting two odd points.We should have the property that we can then construct a square root ofexp(i θ ) denoted by exp(i θ/
2) which is univalued on Ω \ (cid:98) Γ and maximal inthe sense that it can not be extended to a larger open set. The set (cid:98)
Γ will ingeneral not be unique, but all we need is the existence. A natural notion was
PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 19 introduced in [29] called the slitting property and the only change is thatholes are replaced here by points (or) poles.For a given X = ( X , . . . , X (cid:96) ) in Ω (cid:96) with distinct X i , we say that a closedset N slits Ω with singularities at X if:– N is a weakly regular closed set in the sense of Section 3.2;– X odd ( N ) = ( X , . . . , X (cid:96) );– Ω \ N is connected.This definition was introduced to characterize the properties of the nodaldomain of the ground state of T A . (a) (b) (c)(d) (e) (f) Figure 5.2.
Slitting examples, (a) (b), (c), (d), (e), (f).Figure 5.2 is inspired by [29, Fig. 1], and shows some examples of regionswhich are slitting (but replace the holes in the picture by points in our case).Note that crossing points at even points are permitted (see Figure 5.2d).Note also that for the (cid:96) = 1 case (Figure 5.2a), a set which slits Ω consists ofone line which joins the outer boundary of Ω to the pole (Mercedes situation).We have explained above the no-hole situation. The case with holes is treatedin the same way, once we have selected some “odd” holes, and placed onepole X (cid:48) j in each of these odd holes. If a collection of paths slits a region thenno sub- or supercollection of these paths can also slit the region.In this formalism the main result is Proposition 5.2. If Γ is regular with corresponding X odd (Γ) = X and ∂ Ω odd (Γ) , then it contains a slitting set (cid:98) Γ with X odd ( (cid:98) Γ) = X odd (Γ) = X and ∂ Ω odd ( (cid:98) Γ) = ∂ Ω odd (Γ) . Then, once we have this slitting property, we have
Proposition 5.3.
Under the previous assumptions, there exists (cid:98) Γ such that { X , . . . , X d } ⊂ (cid:98) Γ ⊂ Γ and such that there exists in Ω \ (cid:98) Γ a univalued regularsquare root of exp(iΦ X ) which is maximal in the sense that it cannot beextended as a univalued regular function in an open set in Ω containingstrictly Ω \ (cid:98) Γ . The setting has a natural formulation in terms of graph theory. Thiscorresponds indeed in the nodal case to the notion of nodal graph (see forexample [32, Subsection 3.1]). We have a graph contained in Ω ⊂ R . Theboundary points and the singular points of Γ are the vertices and the regulararcs of Γ are the edges. The vertices of ∂ Ω and the odd vertices (an oddnumber of edges arrive at the vertex) in Ω play a special role. We can alsodefine the notion of odd hole by determining the parity of the number ofedges arriving at the boundary of the hole. Moreover, (cid:98)
Γ can be consideredas a subgraph of Γ. The graph translation is:
Lemma 5.4. If Γ is a graph in Ω with given “odd” set of vertices X odd (Γ) =( X , . . . , X (cid:96) ) and given “odd” holes, then there exists a subgraph (cid:98) Γ with thesame “odd” sets such that Ω \ (cid:98) Γ is connected.Proof (given by G. Berkolaiko). We first consider the case with no hole. Itis better to identify all the points of ∂ Ω and to look a the new graph as agraph (cid:101)
Γ on the sphere, ∂ Ω being the north pole P and Ω being the S \ { P } .To get the connexity property, it is enough to destroy all the cycles on thegraph. It is now enough to observe that if there is a cycle we can delete allthe elements of the cycle. But at each vertex of the cycle, only two edgesbelonging to the cycle arrive. Hence when destroying a cycle, this alwayspreserves the odd vertices and the even vertices. Finally, we observe that noodd vertex can disappear when deleting a cycle. This case is exemplified inFigure 5.3. (a) 𝑃 (b) (c) Figure 5.3. (a) A domain Ω. (b) The constructed graphon the sphere. The boundary of Ω is mapped to the point P .We remove one loop (dashed). (c) Back in Ω, the removedloop (cid:98) Γ is dashed.In the case with holes, we identify each component of ∂ Ω with a point andlook at a new graph (cid:101)
Γ on the sphere with Ω being S \ { P , . . . , P m } . Each P i corresponds to a component of ∂ Ω. Then it suffices to think in the previousproof that the points P (cid:96) corresponding to odd boundary components are oddvertices. This situation is exemplified in Figure 5.4. (cid:3) PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 21 (a) 𝑃 𝑃 (b) (c) Figure 5.4. (a) A non-simply connected domain Ω. (b) Thecorresponding graph on the sphere. The outer boundary isidentified at P while the inner boundary is identified at P .We remove two loops (dashed). (c) Back in Ω, this meansthat we removed the dashed curves.5.2.2. Proof of Theorem 5.1.
With the slitting lemma at hand, we have ageneral method to define (cid:98)
Γ, and then we can complete the proof in thegeneral setting, by following what was done in the case of the Mercedes star.Thus, we introduce a Sobolev space associated with the pair ( (cid:98) Γ , Γ \ (cid:98) Γ).We note that (cid:98)
Γ is a union of regular curves ˆ γ (cid:96) , ending at critical points, andwe can choose an orientation of ˆ γ (cid:96) so that, locally, in the neighborhood D ( x, r )of an interior point x ∈ ˆ γ (cid:96) , we can write D ( x, r ) \ ˆ γ (cid:96) = D + ( x, r ) ∪ D − ( x, r )permitting to define a trace on the left and on the right.Starting from H (Ω \ (cid:98) Γ), we introduce (cid:98) H (Ω \ (cid:98) Γ) := { u ∈ H (Ω \ (cid:98) Γ) , u | ˆ γ + (cid:96) = − u | ˆ γ − (cid:96) , u | ∂ Ω = 0 } . On this Sobolev space ˆ H (Ω \ ˆΓ), we get the sesquilinear form(ˆ u, ˆ v ) (cid:55)→ ˆ B σ (ˆ u, ˆ v ) = (cid:88) j (cid:90) D j ∇ ˆ u · ∇ ˆ v dx + σ (cid:90) Γ ˆ u ˆ v dS Γ = (cid:90) Ω \ (cid:98) Γ ∇ ˆ u · ∇ ˆ v dx + σ (cid:90) Γ ˆ u ˆ v dS Γ . We can then associate, via the Lax–Milgram theorem, to this sesquilinearform a realization (cid:98) T Γ , (cid:98) Γ ( σ ) of the Laplacian in Ω \ (cid:98) Γ with σ transmissionproperties on Γ \ (cid:98) Γ, Dirichlet condition on ∂ Ω and σ -Robin like condition on (cid:98) Γ. As in the Mercedes-star case, ˆ L Ω \ (cid:98) Γ corresponds to σ = 0.It remains to detail our definition of ˆΛ NP . The only point is to have aclear definition of ˆ r i (which is an immediate consequence of the choice of ourSobolev space. We have introduced an orientation on each regular componentof (cid:98) Γ. But H / ( (cid:98) Γ) can be identified with ⊕ (cid:96) H / (ˆ γ + (cid:96) ). So, when definingour Neumann–Poincar´e map attached to some D i , and when ˆ γ (cid:96) ⊂ ∂D i , weconsider the trace ε (cid:96) u (cid:96) with ε (cid:96) = +1 if D i is locally on the right side of ˆ γ (cid:96) and ε (cid:96) = − ∂D i ∩ Ω,we just proceed like in [7].
Comparison between two constructions. If D is an equipartition withboundary set Γ, we can observe that Ω \ (cid:98) Γ is a bipartite equipartition.Moreover, if it satisfies the PCC, it is a nodal partition. Finally, if D is aminimal partition then it is a Courant sharp nodal partition in Ω \ (cid:98) Γ (see [12]where this argument is used for the analysis of the Hexagonal conjecture). Itis then natural to compare our construction relative to D (seen as a partitionof Ω) with the Berkolaiko–Cox–Marzuola construction associated with D seenas a nodal partition in Ω \ (cid:98) Γ. The difference is that in the second case, werestrict the first construction to elements which vanish on (cid:98)
Γ and then projecton H − (Γ \ ˆΓ). Coming back to the definitions, it is then immediate to seethat an eigenvalue of the Neumann–Poincar´e second operator is actually aneigenvalue of the first Neumann–Poincar´e operator. It is then interesting tocompare the two formulas (5.1) and (1.1) for the pair (Ω \ ˆΓ , Γ \ ˆΓ). Figure 5.5.
The slitting example H of [12, Figure 24].To give a more explicit example, we continue the discussion of the circlefrom Section 2. Now (cid:98) Γ is just one point, say θ = 0. The construction in [7]leads to an ( N − × ( N −
1) matrix obtained by taking u = 0 and forgetting v . This leads to the matrix M λ := 12 α ( λ ) β ( λ ) 0 · · · β ( λ ) 2 α ( λ ) β ( λ ) · · · β ( λ ) 2 α ( λ ) · · · ...... ... . . . . . . β ( λ )0 0 · · · β ( λ ) 2 α ( λ ) . whose spectrum is given by (cid:110) α ( λ ) + β ( λ ) cos kπN (cid:111) N − k =1 . Hence M λ has the same eigenvalues as M λ except α − β . All its eigenvaluesare positive. Again we can verify for the energy ( N/ in this case that(1.1) holds with Ω = S \ { } and the same partition as in Section 2. PECTRAL FLOW FOR PAIR COMPATIBLE EQUIPARTITIONS 23
Acknowledgements
The first author would like to thank G. Berkolaiko for comments and hisprecious help in graph theory. We also thank T. Hoffmann-Ostenhof andG. Cox for useful comments.
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