Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions

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1 Numerical analsis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions V. Bonnaillie-Noël and B. Helffer June 27, 21 Abstract This paper is devoted to present numerical simulations and interpret theoreticall the results for determining the minimal k-partitions of a domain Ω as considered in [14]. More precisel using the double covering approach introduced b B. Helffer, M. and T. Hoffmann-Ostenhof, M. Owen and further developed for questions of isospectralit b the authors in collaboration with T. Hoffmann-Ostenhof and S. Terracini in [14, 4], we analze the variation of the eigenvalues of the one pole Aharonov-Bohm Hamiltonian on the square and the nodal picture of the associated eigenfunctions as a function of the pole. This leads us to discover new candidates for minimal k-partitions of the square with a specific topological tpe and without an smmetric assumption, contrar to our previous works [5, 4]. This illustrates also recent results of B. Noris and S. Terracini, see [2, 21]. This finall supports or disproves conjectures for the minimal 3 and 5-partitions on the square. 1 Introduction 1.1 Minimal partitions For a given partition D of an open set Ω b k disjoint open subsets D i, we consider Λ(D) = ma i=1,...,k λ(d i), (1.1) where λ(d i ) is the groundstate energ of the Dirichlet Laplacian on D i. We denote the infimum of Λ over all k-partitions of Ω b L k (Ω) = inf Λ(D). (1.2) D O k We look for minimal k-partitions, i. e. partitions D = (D 1,..., D k ), such that L k (Ω) = Λ(D). We recall that these minimal k-partitions, whose eistence was proven in [8, 1, 9], share with nodal domains man properties of regularit, ecept that the number of half-lines IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, F-3517 Bruz, France, Virginie.Bonnaillie@bretagne.ens-cachan.fr Laboratoire de Mathématiques, Bat. 425, Univ Paris-Sud and CNRS, F-9145 Orsa Cede, France, Bernard.Helffer@math.u-psud.fr 1

2 meeting (with equal angle) at critical points of their boundar set can be odd [14]. Here b critical points we mean points which are at the intersection of at least three distincts D i s. Moreover, it was shown in [14] that, if all these numbers are even, then the k- minimal partition consists indeed of the k-nodal domains of some eigenfunction of the Dirichlet Laplacian in Ω. In [5], we have combined results of [14] and [15] with efficient numerical computations to ehibit some candidates to be minimal 3-partitions for the square, the disk,.... This approach was based on the assumption that the minimal 3-partition should inherit from one of the smmetries of the domain. This permits a reduction to a more standard spectral analsis and consequentl can onl give smmetric candidates. Using two different smmetries of the square, we get the surprise of finding two candidates D 1 and D 2 with Λ(D 1 ) = Λ(D 2 ) ( ) and give numerical evidence that the unique critical point for these partitions is at the center of the square. These candidates are represented in Figure 1(a). This leads naturall to questions of isospectralit which were solved using the (a) Candidates with different smmetries. (b) Asmmetric candidate. Figure 1: Candidates for the minimal 3-partition of the square. Aharonov-Bohm Hamiltonian with a singularit at the center of the square, see [4]. This kind of arguments also appears in a similar contet in [18] and [17]. Using this operator could provide new asmmetric candidates for the 3-minimal partition (see Figure 1(b)) and it is one of the aims of this paper to ehibit them. 1.2 Aharonov-Bohm Hamiltonian Let us recall some definitions and results about the Aharonov-Bohm Hamiltonian (for short X-Hamiltonian) with a singularit at X introduced in [4, 12] and motivated b the work of Berger-Rubinstein [2]. We denote b X = (, ) the coordinates of the pole and consider the magnetic potential with renormalized flu Φ 2π = 1 2 A X (, ) = (A X 1 (, ), A X 2 (, )) = 1 2 ( r 2, r 2 at X: ). (1.3) We know that the magnetic field vanishes identicall in Ω X. The X-Hamiltonian is defined b considering the Friedrichs etension starting from C ( Ω X ) and the associated differential operator is A X := (D A X 1 ) 2 + (D A X 2 ) 2 with D = i and D = i. (1.4) Let K X be the antilinear operator K X = e iθ X Γ, 2

3 with ( ) + i( ) = e iθ X, and Γ the comple conjugation operator Γu = ū. We sa that a function u is K X -real, if it satisfies K X u = u. Then the operator A X is preserving the K X -real functions and we can consider a basis of K X -real eigenfunctions. Hence we onl analze the restriction of the X-Hamiltonian to the K X -real space L 2 K X where L 2 K X ( Ω X ) = {u L 2 ( Ω X ), K X u = u }. It was shown that the nodal set of such a K X -real eigenfunction has the same structure as the nodal set of an eigenfunction of the Laplacian ecept that an odd number of half-lines should meet at X. When no ambiguit eists, we omit sometimes the reference to X and write more simpl θ, K, L 2 K, A. 1.3 Main goals Although we will come back to man of these points in the net sections let us comment on some of the difficulties we met in this analsis. As mentioned previousl, we have proposed in [5] some smmetric candidates for the minimal 3-partition. If we do not assume a priori smmetries for a minimal 3-partition, a first method, inspired b [7], is to test the following iterative method (see [6]) : Initialization. Let D = (D1, D 2, D 3 ) be a 3-partition of Ω. Iteration. For n 1, we define the partition D n = (D1 n, Dn 2, Dn 3 ) b D n 1 = Dn 1 3, (D2 n, Dn 3 ) is the nodal partition associated to the second eigenvector of the Dirichlet Laplacian on Int(Ω \ D1 n). If the algorithm converges to the partition D = (D 1, D 2, D 3 ), then Λ(D) = λ 1 (D 1 ) = λ 1 (D 2 ) = λ 1 (D 3 ). The results obtained in [6] are at the moment puzzling. Depending on the initial data, on the accurac and on the form of the domain, all possible situations occur: convergence to the candidate, no convergence, convergence to a non-minimal 3- partition. The case of the equilateral triangle is ver strange, the authors get indeed for one of the model a convergence to a three-partition whose energ is clearl above the epected energ, whose singular point is at the center of the smmetr and which is NOT an eigenvalue of the Aharonov-Bohm Hamiltonian. Another method followed b [11] looks also interesting but the paper does not give enough details for permitting an analsis of its efficienc. In an case, admitting that there eists a perfect good iterative numerical method, it remains interesting on the mathematical level to see that the obtained candidate for a minimal partition is (or is not) a nodal partition for some Hamiltonian. We will come back to this question in the conclusion. When working on this problem, we realize that we get as a b-product a nice illustration of the general question of analzing the deformations of the nodal sets and the transition between different nodal structures when varing a parameter. There are actuall ver few theoretical papers on this question and we also eplain the role of the smmetries for solving some questions of avoided crossings or effective crossings, see Section 5.3. This question is ver difficult to solve numericall. We will push a numerical analsis associated with the X-Hamiltonians with several goals: 3

4 Illustrate the fact that the two smmetric candidates (see Figure 1(a)) for minimal 3-partitions on the square belong actuall to a continuous famil of non necessaril smmetric candidates (see Figure 1(b) and Figure 8). Check, b moving the pole X of the X-Hamiltonian, the conjecture that the singular point of the minimal 3-partition of the square is at the center. Understand and illustrate the mechanism of deformation of the nodal set, and hence etend or guess, in connection with recent papers of B. Noris and S. Terracini [2, 21], some of the properties described in Berger-Rubinstein [2] and Helffer M.-and-T. Hoffmann-Ostenhof Owen [12] for the ground state energ (see also [1]). Finall, let us mention that an etended version of this paper, with more computations, is available in [3]. We have chosen here to focus on the most interesting phenomena. 1.4 Organization of the paper In Section 2, we eplain how we implement the computations on the double covering of the punctured square. In Section 3, we appl Courant s theorem for comparing the eigenvalues of the Dirichlet Laplacian on the square to the Aharonov-Bohm eigenvalues associated to this puncturing. Section 4 analzes the dependence of the eigenfunctions of the Dirichlet Laplacian on the double covering with respect to the puncturing point. Section 5 is more specificall devoted to the analsis of the behavior of the nodal sets and eigenvalues when the poles belong respectivel to the perpendicular bisector = 1 2, the diagonal =, which correspond to cases when some smmetr of the square is respected in the puncturing. We treat also the case of the ais = as an eample of a generic situation. Section 6 describes the possible applications of our analsis of nodal sets to the research of minimal partitions with a given topological tpe. We conclude b the presentation of a conjecture motivated b our computations. Acknowledgements. The authors would like to thank Thomas Hoffmann-Ostenhof, Luc Hillairet and Susanna Terracini for useful discussions around this topics. We have been in particular stimulated b successive versions of [2]. Discussions about numerics with François Alouges and Grégor Vial were also ver fruitful. G. Vial helps us moreover b realizing numerous meshes for the computations and also for the detection of the nodal lines. This paper has been partiall written during the sta of the authors at the Erwin Schrödinger Institute from Ma to Jul 29 and the authors are grateful for the ver good working conditions and also for the fruitful discussions with other participants of the workshop on Topics in Spectral Theor. 2 Numerical implementation The X-Hamiltonian has a singularit at the pole X and the eigenfunctions are complevalued. For these reasons, we prefer to deal with the Dirichlet Laplacian on the doublecovering Ω R X whose eigenfunctions are real-valued. Some of these eigenfunctions, which will be described below, are directl related with the K X -real eigenfunctions of the X- Hamiltonian, as mentioned in [4, Section 6.3]. For the construction of the double covering, we choose a simple line γ X joining the pole to the boundar such that Ω \ γ X is simpl connected. This path permits to go from one sheet to the other one. 4

5 The numerical results were realized b using the Finite Element Librar Mélina (see [19]). The method is completel standard but the new idea is to work on the double covering of a pointed domain. The computations consist onl of the determination of the eigenfunctions of a Dirichlet Laplacian on a double covering domain. Nevertheless, since we are interested in the nodal lines of these eigenfunctions, computations have to be quite accurate and we choose the package Mélina permitting the implementation of high order elements. The main point of the numerical part consists in meshing the double covering Ω R X of the punctured domain Ω \ {X}. To do this, we use the two-dimensional mesh generator Triangle (see [22]). Let us eplain in more details how we proceed. Let Ω be the square [, 1] [, 1] and X be a point in [, 1 2 ] [, 1 2 ]. We start with meshing the domain Ω so that (see Fig. 2): the segment joining (, ) to the pole X, in red in Figure 2, does not go through an element of the mesh, the segment [(, ); X] is the union of edges of an even number of triangles, the pole X is the verte of some triangles. Figure 2: Mesh of the double covering Ω R X. This first mesh is essentiall done for the first sheet and we repeat this mesh for the second sheet. To obtain a mesh of the double covering Ω R X, we choose as a cutting line the segment γ X = [(, ); X] and we have to echange the verte along the segment [(, ); X] between the first and second sheet. Then we remove the point X of the second sheet b equaling it to the verte X of the first sheet. For the numerics, X will be chosen on the lattice P = {( i 1, j 1 ), 1 i, j 5}. Theoreticall, the eigenvalues and eigenfunctions depend onl on the pole and are independent of the cut chosen for our construction. The introduction of the segment [(, ); X] is onl a technical point and we have verified that the numerical computations of the eigenfunctions and eigenvalues are (with a rather good accurac 1 3 ) independent of the choice of the line joining the pole to the boundar, that is the line between the first and second sheet. Man computations corresponding to two different choices of cutting lines are available on the web page (see [6]): The computed eigenvalues are given all along this paper with an approimation at In the following, we use a P 6 approimation with at least 6 elements. To detect the nodal lines, we use a program realized b G. Vial. The idea is that it is ver eas to compute the zero set of linear functions. In our case, we deal with a function which is piecewise P k and given b a finite element method. We know the values of this function at some points. As soon as we have these values, we can replace this function b a 5

6 new function which is piecewise linear. For this, we introduce some new points b an interpolation method. Then we detect the zero set of this new function. 3 A few theoretical comparison theorems 3.1 Notation We denote b Ω the square [, 1] [, 1] and b C = ( 1 2, 1 2 ) the center of the square. We compute the eigenfunctions of the Laplacian on the double covering Ω R X of Ω X = Ω \ {X}. B a smmetr argument, it is enough to consider X = (, ) in the quarter square [, 1 2 ] [, 1 2 ]. There are two was of labelling the eigenvalues. We can label it in the standard wa and then denote b λ k ( Ω R X ) the k-th eigenvalue of the Dirichlet Laplacian on Ω R X. We can also take account of the smmetr relative to the deck map DX R associating with a given point in the covering the distinct point with same projection b the covering map πx R of Ω R X onto Ω X. This splits the spectrum into two independent spectra relative to two orthogonal spaces in L 2 ( Ω R X ). The eigenvalues correspond either to eigenfunctions lifted from the eigenfunctions (of the Dirichlet Laplacian) 1 on the square b the covering map (sometimes called DX R -smmetric because the are smmetric with respect to the deck map), or to eigenfunctions which are DX R -antismmetric with respect to the deck map. We also call them X-eigenvalues because the can be seen as eigenvalues of an - Hamiltonian with a pole X creating a renormalized flu equal to 1 2. We shortl write X-Hamiltonian if we want to make eplicit the reference to the pole. We denote b λ X j = λ j ( Ω X ) the j-th eigenvalue of the -Hamiltonian with pole at X. In consequence, for an pole X and an integer k, the eigenvalue λ k ( Ω R X ) of the Dirichlet Laplacian on the double covering is either an eigenvalue λ j (Ω) of the square, either an eigenvalue λ X j of the -Hamiltonian on Ω X with pole at X. 3.2 Eigenvalues of the square The eigenvalues of the square are well known and given b the double sequence π 2 (m 2 +n 2 ) with m N \ {}, n N \ {}, with corresponding basis of eigenfunctions given b Ω (, ) φ mn (, ) := sin(mπ) sin(nπ). Labelling the eigenvalues in increasing order leads to the sequence denoted b λ k (Ω), k N. Table 1 gives the first eight eigenvalues and the nodal set of the associated eigenfunctions belonging to the above basis. The second, fifth and seventh eigenvalues are double and consequentl, it is also natural to look at the nodal sets of linear combinations in order to determine all the possible nodal configurations associated with this eigenvalue. We notice that the DX R -smmetric spectrum of the Dirichlet Laplacian on the double covering Ω R X is the spectrum of the square and is independent of the pole. This is a consequence of the fact that the spectra of the Dirichlet Laplacian in Ω and Ω X are the 1 We sometimes speak more shortl of spectrum of the square. 6

7 Eigenvalues of the square (m, n)-labelling Nodal sets for φ mn λ 1 (Ω) = 2π (1, 1) λ 2 (Ω) = λ 3 (Ω) = 5π (2, 1), (1, 2) λ 4 (Ω) = 8π (2, 2) λ 5 (Ω) = λ 6 (Ω) = 1π (3, 1), (1, 3) λ 7 (Ω) = λ 8 (Ω) = 13π (3, 2), (2, 3) Table 1: First eight eigenvalues of the Dirichlet Laplacian on Ω and nodal sets for the associated basis φ mn. same, the puncturing point being of capacit. So it is more the X-spectrum which is of interest because depending on the position of the pole. Nevertheless, the standard labelling of all the eigenvalues on Ω R X can pla a role when appling Courant s theorem. Of course we have λ 1 ( Ω R X ) = λ 1(Ω). 3.3 Theoretical estimates of the eigenvalues This subsection is concerned with the comparison between the spectrum on the square, the spectrum on the double covering Ω R X and the X-spectrum. We propose some equalities and upper bounds between the eigenvalues essentiall based on the minima principle and on the Courant s nodal theorem which is recalled now. Theorem 3.1 Let k 1, λ k (D) be the k-th eigenvalue for the Dirichlet Laplacian on D. associated eigenfunction has at most k nodal domains. Then an We would also appl this theorem for the K X -real eigenfunctions of the X-Hamiltonian on Ω X. Equivalentl, this corresponds to a Courant nodal theorem for the DX R-antismme- tric eigenfunctions on Ω R X, alread discussed in [4]. Combination of the Courant nodal theorem and Ma-Min principle for the X-Hamiltonian leads to the following proposition. Proposition 3.2 Let X [, 1 2 ] [, 1 2 ], then λ 1 (Ω) = λ 1 ( Ω R X), and for k = 2, 4, 5, 7, 9, 11, there eists an integer l k such that λ X 1 = λ 2 ( Ω R X), λ X 2 = λ 3 ( Ω R X), (3.1) λ X 2 < λ 2 (Ω) = λ l2 ( Ω R X) with l 2 4 (with multiplicit at least 2), (3.2) λ X 3 λ 4 (Ω) = λ l4 ( Ω R X) with l 4 7, (3.3) λ 5 (Ω) = λ l5 ( Ω R X) with l 5 8, (3.4) λ X 5 λ 7 (Ω) = λ l7 ( Ω R X) with l 7 12, (3.5) λ 9 (Ω) = λ l9 ( Ω R X) with l 9 14, (3.6) λ X 8 λ 11 (Ω) = λ l11 ( Ω R X) with l (3.7) If X belongs to the perpendicular bisectors of the square, we have more accuratel: l 4 = l 4 (X) 8, (3.8) 7

8 λ X 6 λ 7 (Ω). (3.9) Remark 3.3 The multiplicit of λ l (Ω) as eigenvalue of the Dirichlet Laplacian on Ω R X is of course larger or equal to its multiplicit on Ω. This could permit to improve some inequalities above when we can find for a given pole an eigenfunction u of the Dirichlet Laplacian on Ω vanishing at the pole. The number of nodal domains of the lifted smmetric function on the covering is then 2µ(u) instead of 2µ(u) 1 where µ(u) is the number of nodal domains of u. To find this eigenfunction could be easier when the eigenspace is of higher dimension. This appears for eample for λ 2 (Ω). Proof : Let us first prove (3.2). We first observe that, for an X [, 1 2 ] [, 1 2 ], there eists an eigenfunction u X of the Dirichlet Laplacian associated with λ 2 (Ω) and u X (X) =. We have just to look for u X of the form u X = αφ 1,2 +βφ 2,1 with (α, β) (, ) satisfing αφ 1,2 (X) + βφ 2,1 (X) =. B lifting on Ω R X, this gives a DR X-smmetric eigenfunction u X πx R for the Dirichlet Laplacian on Ω R X with four nodal domains and associated with λ 2 (Ω). Hence b Courant s theorem, λ 2 (Ω) = λ l2 ( Ω R X ) with l 2 4. To establish the first inequalit in (3.2), we consider the functions ma(u X, ) and ma( u X, ), which span a two-dimensional space in the form domain of the X-Hamiltonian for which the energ is less than λ 2 (Ω). We can conclude b the minima principle. It is eas to see that the inequalit is strict. Hence at this stage we also get (3.2). Let us now prove (3.3). Using the three functions obtained b restriction to one nodal domain of the function φ 2,2 (then etended b ) which does not contain X, we obtain a 3- dimensional space of functions in the form domain of the X-Hamiltonian for which the energ is less than λ 4 (Ω) (or a 4-dimension space if X is on the perpendicular bisector to the side of the square because we can in this case get a 4-dimensional space, see Remark 3.3). We then conclude b the minima principle. The relation with λ l4 ( Ω R X ) is an application of the Courant s nodal theorem using the function φ 2,2 πx R. Relation (3.4) is a consequence of (3.3). For (3.5), we can this time use the function φ 3,2 which has at least 5 nodal domains not containing X. For X on the perpendicular bisector, we get (3.9). The function φ 4,1 has at least 3 nodal domains not containing X. Using (3.5) and the multiplicit of λ 9 (Ω), we obtain (3.6). Using the function φ 3,3 which has at least 8 nodal domains not containing X, we deduce (3.7). The lower bound for l 7, l 9 and l 11 results immediatel of the upper bounds of λ X 5 b λ 7 (Ω) (hence b λ 9 (Ω)) and of λ X 8 b λ 11 (Ω) established in (3.5) and (3.7). Lemma 3.4 The nodal set of the second K X -real eigenfunction u X 2 consists of one line joining the pole X to the boundar. Proof : We know from [12] that a piecewise regular line in the nodal set should join the pole X to the boundar. Another piece in the nodal set should necessar create an additional nodal domain which will lead to λ 2 λ X 2 in contradiction to (3.2). The inequalit λ X 1 λ 1 (Ω) is of course a particular case of the diamagnetic inequalit. We will observe on the pictures that the situation is much more complicate for the ecited states. Ecept in the case of additional smmetries where some monotonicit will be proven, we have no theoretical results. 8

9 Remark 3.5 We will see in Figures 11, 12 and 16, that the upper bounds (3.2), (3.3) and (3.5) in Proposition 3.2 for the X-eigenvalues are optimal in the sense that we can find a pole such that the upper bound is false with a smaller eigenvalue of the square. 4 Behavior of the eigenvalues on the double covering of the punctured square, when moving the pole In this section, we start to discuss the influence of the location of the puncturing point X (or pole) on the topological structure of the nodal set of the first eigenfunctions. 4.1 Behavior when the pole tends to the boundar It has been announced b B. Noris and S. Terracini [2, 21] that the k-th X-eigenvalue of the punctured square tends to the k-th eigenvalue of the Dirichlet Laplacian on the square as the pole tends to the boundar (see also [16] for connected results). The also establish in [2] the continuit with respect to a pole and prove that X λ X k is of class C 1 if λ X k is simple. Because after a translation b X, we get a fied operator with moving regular boundar and fied pole at (, ), the regularit is actuall eas. These results are illustrated in Figures 3 7 which represent the eigenvalues λ k ( Ω R X ), k = 2, 3, 6, 7, according to the location of the pole X P and Table 2 which gives the first 12 eigenvalues of the Dirichlet Laplacian on Ω R X for three points X: one near the boundar denoted b A, one at the center denoted b C and one other denoted b B. n λ n ( Ω R A ) λ n( Ω R B ) λ n( Ω R C ) A B C Table 2: First 12 eigenvalues of the Dirichlet Laplacian on Ω R A, ΩR B and Ω R C, with A = ( 1 1, 1 1 ), B = ( 1 1, 2 5 ), C = ( 1 2, 1 2 ). 4.2 Eigenvalues 2 to 5 We observe numericall, see Figure 3, that for X P, the function X λ 2 ( Ω R X ) has a global maimum, denoted b λ ma 2 for X = C and is minimal when X belongs to the boundar = or =. This minimum equals λ 2 (Ω). Moreover we do not observe other critical points in P. Looking at Figure 4, we observe numericall that the function X λ 3 ( Ω R X ) behaves conversel: it has a global minimum, denoted b λmin 3, for X = C 9

10 and the maimum is reached at the boundar = or = and equals λ 3 (Ω). We have monotonicit along lines joining a point of the boundar to the center C. Furthermore, we notice that λ ma 2 = λ min Figure 3: X λ 2 ( Ω R X ) for X P Figure 4: X λ 3 ( Ω R X ) for X P. Figure 5 gives the eigenvalues and the nodal lines of the eigenfunctions associated with the second and third eigenvalues of the Dirichlet Laplacian on Ω R X on the first and second lines respectivel. The j-th column corresponds to the domain Ω R X j with X j = ( 1 5, j 1 ), j = 1,..., 5. These figures are illustration of the theor of Berger Rubinstein [2] and Helffer Hoffmann-Ostenhof Hoffmann-Ostenhof Owen [12] (see also [1]). For the ground state energ, we recover the theorem of these authors that the nodal set is composed of a line joining the pole to the boundar. We observe that the nodal line in the first case is choosing a kind of minimal distance between the pole and the boundar whereas the nodal line in the second case seems to choose a kind of maimal distance. We do not have a rigorous eplanation for this propert ecept that it should be related to the theorem proved in [2, 12] that λ X 1 is the infimum over the Dirichlet eigenvalue of the Laplacian in Ω \ γ where γ is a regular path joining the pole X to the boundar. We also recover the two last equations in (3.1). Using (3.2), we have proved that λ 2 (Ω) λ 5 ( Ω R X ). We observe numericall (see also ahead Figures 11 and 12 for poles along a smmetr ais and Figure 16) that, for an X P, we have λ 4 ( Ω R X) = λ 5 ( Ω R X) = λ 2 (Ω). (4.1) 1

11 Figure 5: Nodal set for the eigenfunctions associated with λ k ( Ω R X ), k = 2, 3, for poles X = ( 1 5, j 1 ), 1 j Eigenvalues 6 and Figure 6: λ 6 ( Ω R X ), in function of the pole X P Figure 7: λ 7 ( Ω R X ), in function of the pole X P. Numerics shows that the 6-th eigenvalue λ 6 ( Ω R X ) is minimal at the boundar and has a unique maimum λ ma reached for the pole at the center. We do not observe other critical points. The 7-th eigenvalue λ 7 ( Ω R X ) is minimal when the pole is at the center and its minimum λ min 7 is equal to λ ma 6. When the pole is at the center, the zero set of the 6-th eigenfunction provides, b projection, a candidate for a 3-partition and λ ma 6 is the conjectured value for L 3 (Ω). We observe that the 7-th eigenvalue becomes constant equal 11

12 to λ ma 7 = λ 4 (Ω) = 8π 2 as a function of the pole when the pole is close to the boundar. This corresponds to a crossing when X approaches the boundar between the spectrum of the square (i.e. the DX R -smmetric spectrum on the covering) and the X-dependent X-spectrum (i.e. the DX R-antismmetric spectrum on the covering Ω R X ). Appling relation (3.3), we have proved theoreticall that λ 4 (Ω) λ 7 ( Ω R X ). We observe numericall that this relation is optimal in the sense that we have equalit for X close to the boundar. Considering the linear combination of the eigenfunctions u 6 and u 7 associated respectivel with λ 6 ( Ω R C ) and λ 7( Ω R C ), with C = ( 1 2, 1 2 ), we can construct a famil of 3-partitions with the same energ. Figure 8 gives the projection b πc R of the nodal set for the functions tu 6 + (1 t)u 7 with t = k/8, k =,..., 8. Figure 8: Continuous famil of 3-partitions with the same energ. It is interesting to discuss if we can prove the numericall observed inequalit λ X 3 λ 2 (Ω). (4.2) This is directl related to the conjecture proposed b S. Terracini 2 : Conjecture 4.1 Ecept at the center X = C = ( 1 2, 1 2 ), λx 3 is simple and the corresponding nodal set of the K X -real eigenfunction is the union of a line joining the pole to the boundar and of another line joining two points of the boundar. We note indeed that if the conjecture is true, we will get (4.2) b the minima principle. 4.4 Eigenvalues 8 to 1 Figures 9 1 represent the numerical computations of λ 8 ( Ω R X ) and λ 9( Ω R X ) for X P. We observe numericall that the function X λ 8 ( Ω R X ) has a unique maimum denoted b λ ma 8 at a point C 1 on the diagonal and X λ 9 ( Ω R X ) reaches its unique minimum, λmin 9, at this point. We can recover this behavior on Figure 11 where are drawn the eigenvalues for poles on the diagonal. Numericall, λ ma 8 = λ min 9 and we come back to this equalit in Subsection 5 where we look at the nodal lines of the eigenfunctions associated with λ 8 ( Ω R X ) and λ 9( Ω R X ) and predict the eistence of the point C 1, see Figure 14(a). According to (3.8), we have proved that λ 8 ( Ω R X ) λ 7(Ω). This theoretical upper bound is rough and the numerics suggests that we have in fact the better bound λ 8 ( Ω R X ) λ 5(Ω). Moreover C 1 is singular for the maps X λ 8 ( Ω R X ) and λ 9( Ω R X ). We observe numericall that, for an X P, we have What we have proven in (3.4) is weaker. 2 Personal communication λ 1 ( Ω R X) = λ 5 (Ω). (4.3) 12

13 Figure 9: X λ 8 ( Ω R X ) for X P Figure 1: X λ 9 ( Ω R X ) for X P. 5 Moving the pole along smmetr ais 5.1 Analsis of the smmetries Let us begin with some considerations about the X-Hamiltonian on the X-punctured square, using the smmetr along the perpendicular bisector = 1 2 or the diagonal =. We refer to [4] for more details. The square is invariant under the smmetries We consider the antilinear operators σ 1 (, ) = (, 1 ) and σ 2 (, ) = (, ). Σ c j = ΓΣ j, j = 1, 2, where Γ is the comple conjugation (Γu = u) and Σ j is associated with σ j b the relation Σ 1 u(, ) = u(, 1 ) and Σ 2 u(, ) = u(, ). We use the smmetr of the X-punctured square to give an orthogonal decomposition of L 2 K = L2 K X : L 2 K = L 2 K,Σ j L 2 K,a Σ j, (5.1) where L 2 K,Σ j = {u L 2 K, Σ c ju = u }, and L 2 K,a Σ j = {u L 2 K, Σ c ju = u }. (5.2) As established in [4, Lemma 5.6], we can prove that: 13

14 if X = (, 1 2 ): if u C ( Ω X ) L 2 K,Σ 1, then its nodal set contains [, ] { 1 2 }, if u C ( Ω X ) L 2 K,aΣ 1, then its nodal set contains [, 1] { 1 2 }, if X = (, ): if u C ( Ω X ) L 2 K,Σ 2, then the nodal set of u contains {(, ), < < }, if u C ( Ω X ) L 2 K,aΣ 2, then the nodal set of u contains {(, ), < < 1}. Dealing with a mied Dirichlet-Neumann condition on the half-domain, we deduce that the eigenvalues for which the eigenfunctions are smmetric are increasing with respect to, whereas the eigenvalues for which the eigenfunctions are antismmetric are decreasing with respect to. 5.2 Spectral variation Figures 11 and 12 give the eigenvalues for poles along the ais = 1 2 and = respectivel and < 1 2. Poles are denoted b X() = (, 1 2 ) and ˇX() = (, ). The below mentioned smmetr (resp. antismmetr) is in this section with respect to Σ c j (see (5.2)) and denoted b Σ j (resp. aσ j ) on the figures where j = 1 when we consider poles X() and j = 2 for poles ˇX(). Let us first mention some numerical observations available for these two configurations: (a) lim λ X() k ˇX() = λ k (Ω) and lim λk = λ k (Ω). (b) For an integer k 3, λ C 2k+1 = λc 2k+2. (c) λ 1 ( Ω R X() ) and λ 1( Ω RˇX() ) equal λ 1 (Ω), in adequation with the theoretical result (3.1). ˇX() (d) λ 2 ( Ω R X() ) = λx() 1 and λ 2 ( Ω RˇX() ) = λ1 are strictl increasing from [, 1 2 ] onto [λ 1(Ω), λ C 1 ] and the eigenfunctions are smmetric. The equalit between λ 2 ( Ω R X ) and λx 1 was proved in (3.1). ˇX() (e) λ 3 ( Ω R X() ) = λx() 2 and λ 3 ( Ω RˇX() ) = λ2 are strictl decreasing from [, 1 2 ] onto [λc 2, λ 2 (Ω)] and the eigenfunctions are antismmetric. The equalit between λ 3 ( Ω R X ) and λx 2 was proved in (3.1). (f) λ 4 ( Ω R X ) = λ 5( Ω R X ) = λ 2(Ω), for X = X() or X = ˇX(). This numerical observation is more accurate than the theoretical result deduced from (3.2): λ 4 ( Ω R X ) λ 2(Ω). This relation seems to be an equalit for an X, see Subsection 4.2. ˇX() (g) λ 6 ( Ω R X() ) = λx() 3 and λ 6 ( Ω RˇX() ) = λ3 are strictl increasing from [, 1 2 ] onto [λ 3(Ω), λ C 3 ] and the eigenfunctions are smmetric. Once the smmetr is admitted, the monotonicit results directl of a domain monotonicit. Let us now discuss properties specific to each smmetr. 14

15 Spectral variation for poles along the ais = 1 2. We observe that when X() = (, 1 2 ), λx() k is monotonicall increasing for k = 1, 3, 5, whereas it is decreasing for k = 2, ! 9 (")=! 1 (")=17# 2 a$ 1 $ 1! 9 B 2! 1! (" R 2 X )! (" R 3 X )! 4 14! 7 (")=! 8 (")=13# 2 $ 1 A 2 a$ 1! 8 B 1! 5! (" R 6 X )! ! 5 (")=! 6 (")=1# 2 a$ 1 $ 1 $ 1! 7! 6 A 1! 5 $ 1! 8! (" R 9 X )! (" R 1 X )! (" R 11 X ) 8 6 4! 4 (")=8# 2! 2 (")=! 3 (")=5# 2 a$ 1 $ 1 a$ 1! 4! 3! 2! 12! (" R 13 X )! (" R 14 X )! (" R 15 X )! 16! 17! 18 2! 1 (")=2# 2 $ 1! =1/2, Figure 11: Moving the pole along the ais = 1/2. We introduce A j = X(a j ), B j = X(b j ) specific points which can be seen on Figure 11. We observe numericall: (h 1 ) λ 7 ( Ω R X() ) = λx() 4 is strictl decreasing from [, 1 2 ] onto [λc 4, λ 4 (Ω)] and the eigenfunctions are antismmetric. (i 1 ) λ 8 ( Ω R X() ) = λ 4(Ω). We have proved in (3.8) that λ 8 ( Ω R X() ) λ 4(Ω) and we observe that this upper bound is actuall an equalit. We notice that there is a gap between λ 4 (Ω) and λ 5 (Ω) where there is no eigenvalue λ k ( Ω R X ) for X on the perpendicular bisector. This observation is no more true for poles on the diagonal (see Figure 12). (j 1 ) λ 9 ( Ω R X() ) = λ 1( Ω R X() ) = λ 5(Ω). (k 1 ) λ 11 ( Ω R X() ) = λx() 5 is strictl increasing from [, 1 2 ] onto [λ 5(Ω), λ C 5 ] and the eigenfunctions are smmetric. This observation shows that the theoretical upper bound λ 11 ( Ω R X ) λ 7(Ω) deduced from (3.5) can not be improved. (l 1 ) λ 12 ( Ω R X() ) = λx() 6 is strictl increasing from [, a 1 ] onto [λ 6 (Ω), λ A 1 6 ] and the eigenfunctions are smmetric. It is strictl decreasing from [a 1, 1 2 ] onto [λ C 6, λ A 1 6 ] and the eigenfunctions are antismmetric. This illustrates theoretical result deduced from (3.9): λ 12 ( Ω R X() ) λ 7(Ω) and shows that this result is optimal. 15

16 (m 1 ) λ 13 ( Ω R X() ) = λx() 7 is strictl decreasing from [, a 1 ] onto [λ 7 (Ω), λ A 1 7 ] and the eigenfunctions are antismmetric. It is strictl increasing from [a 1, 1 2 ] onto [λ C 7, λ A 1 7 ] and the eigenfunctions are smmetric. We observe then that λ 13 ( Ω R X ) can be bounded from above b λ 7(Ω) whereas we have proved in (3.6) the upper bound b λ 9 (Ω). (n 1 ) λ 14 ( Ω R X() ) equals λ 8(Ω) on [, b 1 ]. It equals λ X() 8 on [b 1, 1 2 ] and is strictl decreasing from [b 1, 1 2 ] onto [λc 8, λ 8 (Ω)] with antismmetric eigenfunctions. (o 1 ) λ 15 ( Ω R X() ) = λ 8(Ω). Spectral variation for poles along the ais =. Figure 12 gives the eigenvalues for poles along the diagonal line of the square = with < 1/ ! 9 (")=! 1 (")=17# 2 a$ 2 $ 2! 9! (" R 1 X )! 2! 3! 4 14! 7 (")=! 8 (")=13# 2 $ 2 C 3 a$ 2! 8 D 3! 5! 6! ! 5 (")=! 6 (")=1# 2 a$ 2 $! 2 7 $ 2! C 6 2 $! 2 5! 8! (" R 9 X )! 1! (" R 11 X ) 8 6 4! 4 (")=8# 2! 2 (")=! 3 (")=5# 2 $ 2 a$ 2 $ C a$ D 1 $ 2 a$ 2 D 2 a$ 2! 4! 3! 2! 12! 13! 14! (" R 15 X )! 16! 17! 18 2! 1 (")=2# 2 $ 2! = Figure 12: Moving the poles on the diagonal. We introduce C j = ˇX(c j ), D j = ˇX(d j ) specific crossing points appearing on the figure. Then, we observe: (h 2 ) λ 7 ( Ω RˇX() ˇX() ) equals λ 4 (Ω) on [, d 1 ] and λ4 on [d 1, 1 2 ] where it is strictl decreasing onto [λ C 4, λ 4 (Ω)] and the eigenfunctions are antismmetric. This numerical computations show that the theoretical estimate λ 7 ( Ω R X ) λ 4(Ω), deduced from (3.3) is optimal. (i 2 ) λ 8 ( Ω RˇX() ˇX() ) equals λ4 on [, d 1 ] and λ 4 (Ω) on [d 1, 1 2 ]. It is strictl increasing from [, c 1 ] onto [λ 4 (Ω), λ 4 ( Ω R C 1 )] with smmetric eigenfunctions and strictl decreasing from [c 1, d 1 ] onto [λ 4 (Ω), λ 4 ( Ω R C 1 )] with antismmetric eigenfunctions. This illustrates that (3.4) is optimal. 16

17 (j 2 ) λ 9 ( Ω RˇX() ˇX() ) equals λ5 on [, d 2 ] and λ 5 (Ω) on [d 2, 1 2 ]. It is strictl decreasing from [, c 1 ] onto [λ 4 ( Ω R C 1 ), λ 5 (Ω)] with antismmetric eigenfunctions and strictl increasing from [c 1, d 1 ] onto [λ 4 ( Ω R C 1 ), λ 5 (Ω)] with smmetric eigenfunctions. (k 2 ) λ 1 ( Ω RˇX() ) = λ 5 (Ω). (l 2 ) λ 11 ( Ω RˇX() ˇX() ) equals λ 5 (Ω) on [, d 2 ] and λ5 on [d 2, 1 2 ] where it is strictl increasing onto [λ 5 (Ω), λ5 C ] and the eigenfunctions are smmetric. This illustrates the fact that relation (3.5) is optimal. (m 2 ) λ 12 ( Ω RˇX() ˇX() ) = λ6 is strictl increasing from [, c 2 ] onto [λ 6 (Ω), λ C 2 6 ] and the eigenfunctions are smmetric. It is strictl decreasing from [c 2, 1 2 ] onto [λ C 6, λ C 2 6 ] and the eigenfunctions are antismmetric. (n 2 ) λ 13 ( Ω RˇX() ˇX() ) = λ7 is strictl decreasing from [, c 2 ] onto [λ 7 (Ω), λ C 2 7 ] and the eigenfunctions are antismmetric. It is strictl increasing from [c 2, 1 2 ] onto [λ C 7, λ C 2 7 ] and the eigenfunctions are smmetric. We then observe λ 13 ( Ω RˇX() ) λ 7 (Ω) whereas we have proved the weaker upper bound b λ 9 (Ω) in (3.6). (o 2 ) λ 14 ( Ω RˇX() ˇX() ) equals λ 7 (Ω) on [, d 3 ]. It equals λ8 on [d 3, 1 2 ] and is strictl decreasing from [d 3, 1 2 ] onto [λc 8, λ 8 (Ω)] with antismmetric eigenfunctions. (p 2 ) λ 15 ( Ω RˇX() ) = λ 8 (Ω). 5.3 Echange of smmetr and crossing points When moving the pole on one bisector or one diagonal, and for each eigenvalue of multiplicit 1, the corresponding K X -real eigenfunction should be either smmetric or antismmetric with respect to Σ c j. Figure 11 suggests that there eist two poles A 1 = (a 1, 1 2 ) and A 2 = (a 2, 1 2 ) on the perpendicular bisector such that λ 12( Ω R A 1 ) and λ 16 ( Ω R A 2 ) are eigenvalues of multiplicit 2. Taking the Aharonov-Bohm point of view, this corresponds to a crossing between λ X() 6 and λ X() 7 for = a 1, with a 1 ] 42 1, 43 1 [ and to a crossing between λ X() 8 and λ X() 9 at = a 2, with a 2 ] 28 [. The nodal sets of the 1, 29 1 corresponding eigenfunctions are given in Figure 13. The first line gives the eigenvalues λ X 6, λ X 8 and the associated nodal sets, and the second line λ X 7, λ X 9 and the corresponding nodal set for X along the perpendicular bisector and close to A 1, A 2. Figure 12 suggests that there are 3 points, C 1, C 2 and C 3 on the diagonal such that λ 8 ( Ω R C 1 ), λ 12 ( Ω R C 2 ) and λ 16 ( Ω R C 3 ) are eigenvalues of multiplicit 2. This corresponds to a ˇX() ˇX() crossing between λ4 and λ5 at = c 1, with c 1 ] 28 1, 29 1 [. Similarl, there is ˇX() ˇX() a crossing between λ6 and λ7 at = c 2, with c 2 ] 36 1, 37 1 [, and also between ˇX() ˇX() λ8 and λ9 at = c 3, with c 3 ] 23 1, 24 1 [. The nodal set of the corresponding eigenfunctions are given in Figures 14. Looking at Figures 13 and 14, we can verif that there eists an echange of smmetr 3 predicting the eistence of the points A j and C j. 3 Look at the horizontal or diagonal nodal line joining the pole to the boundar! 17

18 i (a) k = 6, 7 for poles X = ( 1, 12 ), i = 42, 43. i (b) k = 8, 9 for poles X = ( 1, 12 ), i = 28, 29. X() Figure 13: Change of smmetr on the nodal sets associated with λk (a) k = 4, 5 for poles X = i i ( 1, 1 ), i = 28, 29. (b) k = 6, 7 for poles X = i i ( 1, 1 ), i = 36, 37. (c) k = 8, 9 for poles X = i i ( 1, 1 ), i = 23, 24. Figure 14: Nodal set for the eigenfunctions associated with λx k 5.4 Nodal deformation: an eample Figure 15 gives the deformation mechanism for the nodal set associated with the fifth i eigenvalue of the X-Hamiltonian for poles X = ( 1, 12 ), 1 i 5, on the perpendicular bisector of one side of the square. Between the fourth and fifth subfigures, we have a nodal structure where there are two double points at the boundar Figure 15: Nodal set for the 5-th eigenfunction of the -Hamiltonian with poles X = i ( 1, 12 ), i = 1, 7, 3, 42, 43, 44, 45,

19 5.5 Moving the pole without respecting the smmetries of the square Figure 16 gives the eigenvalues λ k ( Ω R X ) for 1 k 12 when the pole X belongs to the line = We choose this ais to ehibit a case without smmetr and we notice that the -eigenvalues λ X k are no longer monotone with respect to when X = (, ). The result should be the same for an arbitrar line (ecept the perpendicular bisector and the diagonal). We choose to present the simulations for this line because this line contains enough points in P to use the previous numerical computations. 18! 9 (")=! 1 (")=17# ! 7 (")=! 8 (")=13# 2! 9! 8! 1! (" R 2 X )! (" R 3 X )! (" R 4 X )! 5! 6! (" R 7 X )! (" R 8 X )! (" R 9 X )! 1 12! 7! 11! 12! 5 (")=! 6 (")=1# 2 1! 6! 5! 13! (" R 14 X )! (" R 15 X )! 16! 4 (")=8# 2! 17 8! 18! 4 6! (")=! (")=5# 2 2 3! 3 4! 2 2! 1 (")=2# 2! , =5+/2 Figure 16: Moving the pole along the ais = , 1 2. It would be interesting to make computations for a finer grid of X = (, ) for around 4 to detect possible crossings between λ X 5, λ X 6 and λ X 7. 6 Nodal sets and minimal partitions The analsis of λ X 5 and λ X 6 leads us to guess numericall the eistence of a double eigenvalue when X is the center. As for the pairs λ X 3 and λ X 4 which lead us to the famil of candidates for the minimal 3-partitions of the square (see Subsection 4.3), we are led to produce a candidate for a minimal 5-partition for the square, with the propert that it is minimal inside the class of 5-partitions which can be lifted to Ω R C. Although, we observe that it is Courant-sharp for the C-Hamiltonian. This time, neither numerics nor theor is giving the eistence of a continuous famil of 5-partitions. Actuall, one knows from elementar results on the perturbation of harmonic polnomials of order 2 that the perpendicular crossing of two lines will genericall disappear b perturbation. λ C 5 = λ 11 ( Ω R C ) is not Courant-sharp4 for the Dirichlet Laplacian on Ω R C 4 The inde is 11 and not 1. 19

20 In the unit ball of the 2-dimensional eigenspace of λ 11 ( Ω R C ) we onl find four eigenfunctions leading to four distinct configurations whose projection on one sheet has five domains. These eigenfunctions are smmetric or antismmetric with respect to one of the four bisectors of the square (see Figure 17). The other configurations seem to have (see below) four smmetric (for the deck map) pairs of domains. Looking at the linear combination tu 11 + (1 t)u 12 of the eigenfunction associated with λ 11 ( Ω R C ) and λ 12( Ω R C ), we observe in Figure 17 that the triple point is ver unstable and onl appears for t and t 2 when we consider t 1. Figure 17: Nodal sets of the linear combination of u 11 and u 12 tu 11 + (1 t)u 12 with t = 1 2 (, 25, 55, 96, 125, 138, 175, 194, 2). Of course it is interesting to compare with what can be obtained b looking at other topological tpes for the minimal 5-partitions. We recall that these tpes can be classified b using Euler formula (see [13] for the case of 3-partitions). Inspired b [11], we look for a partition which has the smmetries of the square and four critical points. We get two tpes of models and using the smmetries, we can reduce to a Dirichlet-Neumann problem on a triangle corresponding to the eighth of the square (see Figure 18 where we impose Neumann conditions on dashed lines). Moving the Neumann boundar on one side like in Figure 18: Dirichlet-Neumann problem on the eighth of the square. [5] leads to two candidates. Numerical computations demonstrate a lower energ in one case which coincides with one of the pictures in [11] (see Figure 19). λ 11 ( Ω R X ) = λdn 2 = λ DN 2 = Figure 19: Three candidates for the 5-partition of the square. Remark 6.1 Note that in the case of the disk a similar analsis leads to a different answer. The 2

21 partition of the disk b five halfras with equal angle has a lower energ than the minimal 5-partition with four singular points (see Figure 2). We note that, on the basis of standard computations (see for eample (A1) and (A5) in [14], Appendi A) this energ corresponds to the eleventh eigenvalue of the Dirichlet problem on the double covering on the punctured disk (hence is not Courant-sharp) but corresponds to the fifth eigenvalue of the Aharonov- Bohm spectrum on the punctured disk at the center. Hence it is Courant-sharp in the sense developed in [15] (for the sphere) and it shows the minimalit of this 5-partition inside the class of the 5-partitions of the disk having a unique critical point which is, in addition, at the center Figure 2: Two candidates for the 5-partition of the disk. 7 Conclusion We have eplored rather sstematicall how minimal partitions could be obtained b looking at nodal domains of a problem on the double covering of a punctured square. We have analzed the behavior of the nodal set when moving the pole in the square. This has permitted to confirm the status of main candidate for some 3-partitions in the case of the square. This has also permitted to ehibit a natural candidate for a minimal 5-partition which finall appears to be less favorable than another partition with four critical points. This is a starting point for a program which can be developed in at least two directions: analze other domains, do the same work b considering the double covering of a multi-punctured domain and moving the poles. This program is related to the following conjecture Conjecture 7.1 Let Ω be a simpl connected open set of R 2, L k (Ω) = inf inf L l N X 1,...,X l k ( Ω X1,...,X l ). Here, for l distinct points X = (X 1,..., X l ) in Ω, L k (Ω X1,...,X l ) is defined as follows. First we can etend our construction of an Aharonov-Bohm Hamiltonian in the case of a configuration with l points (putting a (renormalized) flu 1 2 at each of these points). We can also construct (see [12]) the antilinear operator K X and consider the K X -real eigenfunctions. L k ( Ω X1,...,X l ) denotes the smallest eigenvalue of the (X 1,..., X l ) Hamiltonian for which there is an eigenfunction with k-nodal domains. Let us present a few eamples to illustrate the conjecture. When k = 2, there is no need to consider punctured Ω s. The infimum is obtained for l =. When k = 3, it is possible 21

22 to show that it is enough, to minimize over l =, l = 1 and l = 2. In the case of the disk and the square, it is proven that the infimum cannot be for l = and we conjecture that the infimum is for l = 1 and attained for the punctured domain at the center. For k = 5, in the case of the square, it seems that the infimum is for l = 4 and for l = 1 in the case of the disk. Let us eplain ver briefl wh this conjecture is natural. Considering a minimal k- partition, we denote b X 1,..., X l the critical points of the partition corresponding to an odd number of meeting half-lines. Then we suspect that L k (Ω) = λ k ( Ω X1,...,X l ) (Courant-sharp situation). Conversel, an famil of nodal domains of an Aharonov- Bohm operator on Ω X1,...,X l corresponding to L k gives a k-partition. Using the Euler formula, see [13], we obtain easil that the maimal number of critical points with an odd number of meeting half-lines l is bounded from above b 2k 3. In other words, when the minimal partition is not nodal, we conjecture that it is actuall the projection of a nodal partition of suitable eigenfunction on the double covering for a suitable puncturing X. References [1] B. Alziar, J. Fleckinger-Pellé, P. Takáč. Eigenfunctions and Hard inequalities for a magnetic Schrödinger operator in R 2. Math. Methods Appl. Sci. 26(13) (23) [2] J. Berger, J. Rubinstein. On the zero set of the wave function in superconductivit. Comm. Math. Phs. 22(3) (1999) [3] V. Bonnaillie-Noël, B. Helffer. Numerical analsis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions. Preprint ESI 2193 (29). [4] V. Bonnaillie-Noël, B. Helffer, T. Hoffmann-Ostenhof. Aharonov-bohm Hamiltonians, isospectralit and minimal partitions. J. Phs. A: Math. Theor. 42(18) (29) 18523, 2pp. [5] V. Bonnaillie-Noël, B. Helffer, G. Vial. Numerical simulations for nodal domains and spectral minimal partitions. ESAIM Control Optim. Calc. Var. 16(1) (21) [6] V. Bonnaillie-Noël, G. Vial. Computations for nodal domains and spectral minimal partitions. (27). [7] F. Bozorgnia. Numerical algorithm for spatial segregation of competitive sstems. SIAM J. Scientific Computing 31(5) (29) [8] M. Conti, S. Terracini, G. Verzini. An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198(1) (23) [9] M. Conti, S. Terracini, G. Verzini. On a class of optimal partition problems related to the Fučík spectrum and to the monotonicit formulae. Calc. Var. Partial Differential Equations 22(1) (25)

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