Flux and symmetry effects on quantum tunneling

Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for a recently established tunneling approximation, thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

1. Introduction 1.1.Tunneling and flux effects.The tunneling induced by symmetries is an interesting phenomenon in spectral theory featuring an exponentially small splitting between the ground state and the next excited state energies.The magnetic flux has an effect on the eigenvalue multiplicity which can lead to oscillatory patterns in the spectrum: as the magnetic flux varies, the eigenvalues may cross and split infinitely many times.Hence it is interesting to look at the interaction between symmetry and flux effects.We explore this question by investigating examples of operators involving the magnetic Laplacian (−ih∇ − A) 2 perturbed in various ways by an electric potential, a boundary condition or a magnetic field discontinuity.We observe interesting flux effects, manifested in endless eigenvalue crossings, when adding symmetry assumptions on the electric potential, the boundary of the domain or the magnetic field discontinuity set.
A braid structure in the distribution of the low lying eigenvalues was predicted heuristically [8,Sec. 15.2.4] and confirmed numerically [4] for the magnetic Laplacian on an equilateral triangle with Neumann boundary condition and constant magnetic field.We confirm this prediction by giving a proof for smoothed equilateral triangles and for other examples on the full plane (electro-magnetic Laplacian and magnetic steps).
Let us introduce a mathematical definition of (semi-classical) braid structure of lowest eigenvalues.Consider a family of unbounded self-adjoint operators (T h ) h∈(0,1] on a Hilbert space H. Let us assume that, for every h ∈ (0, 1], T h is semi-bounded from below and denote by λ 1 (T h ), λ 2 (T h ), • • • the discrete eigenvalues below the essential spectrum of T h , counted with multiplicity.In practical examples, the parameter h will be the semi-classical parameter, which tends to 0 and can result as the inverse of the magnetic field's intensity in problems involving strong magnetic fields.
Definition 1.1.The lowest eigenvalues of T h , λ 1 (T h ) and λ 2 (T h ), are said to have a braid structure, if According to Definition 1.1, the eigenvalues λ 1 (T h ) and λ 2 (T h ) exhibit infinitely many crossings and splittings as the parameter h varies in a right neighborhood of 0 (see Fig. 1).This phenomenon has also been observed in non-simply connected domains when considering a magnetic Laplacian and assuming an Aharonov-Bohm magnetic potential.Furthermore, it has been proven in [18] that these crossings and splittings occur in response to variations in the magnetic flux.
For the semi-classical magnetic Laplacian on a simply connected domain with Neumann boundary conditions, the spectrum is related with the spectral properties of an operator which is defined on the boundary.Hence we actually work on another nonsimply connected domain (i.e. the boundary) and therefore flux effects are expected to exhibit crossings and splittings of eigenvalues.However, this is not the case when the boundary curvature has for example a unique non-degenerate maximum.In this case the eigenvalues split in the semi-classical limit [9].It is when non-degenerate minima are exchanged in the presence of symmetries (like in the case of an ellipse or a smoothed equilateral triangle), that eigenvalue crossings are expected to occur along with tunneling effects [5,8].We will also prove such a behavior for the electro-magnetic Laplacian with 'potential' wells located on the vertices of an equilateral triangle, which to our knowledge is quite novel.1.2.Electro-magnetic tunneling.The analysis in this paper yields new results on the electro-magnetic Laplacian on R 2 , where b, h are positive parameters, A = 1 2 (−x 2 , x 1 ) is the vector field generating the unit uniform magnetic field, curl A = 1, and V is a smooth function.

1/ℎ
Figure 1.A schematic figure of eigenvalues with a braid structure, occurring in the presence of trilateral symmetry.The ground state energy has multiplicity 2 infinitely many times.Observe also that the energy of the second state may have multiplicity 2 while the ground state energy is a simple eigenvalue.
What we call the wells are the points where V attains its minimum.The pure electric case where b = 0 was settled for any number of wells n in [16].We would like to address the case where b > 0 and n ≥ 2. For n = 2, this problem was considered in [17] and revisited recently in [7,14].The article [17] follows a perturbative approach (i.e.considers the case b relatively small) and assumes the analyticity of the electric potential V , while the results in [7,14] hold for any b > 0 but under the assumption that V is non-positive and defined as a superposition of radially symmetric compactly supported functions.1.2.1.Double wells.Suppose that the electric potential V is as follows where v 0 ∈ C ∞ (R + ) vanishes on [a, +∞), negative-valued on [0, a) and has a unique and non-degenerate minimum at 0. The wells of V are then z 1 and z 2 .We prove the following theorem that, in particular, rules out eigenvalue crossings for double wells.
i) The asymptotics in (1.2) was obtained earlier in [14] for b = 1 but under the assumption that ii) The use of (1.3) in [14] was technical.In fact, assuming (1.3), it is proved in [7] that where c h (v 0 , L) is the hopping coefficient that will be introduced in (3.15) later on.The accurate approximation of ln c h (v 0 , L) was then carried out in [14].iii) For b ̸ = 1, by a change of semi-classical parameter, the condition in (1.3) reads as follows Clearly, this is a very strong condition on L which in particular prevents us from considering the limit b ↘ 0. The novelty in Theorem 1.2 is in improving the previous condition which could appear as artificial.However, it is still an open question whether (1.2) holds for 2a < L < 4a.iv) The dependence on b in the expression of E b,L (v 0 ) can actually be made more explicit.We have indeed where S(•, L) will be introduced in (3.18) later on.The leading term of E b,L (v 0 ) in the limit b ↘ 0 was calculated in [14, Prop.6.7], and it is consistent with the existing results [15,21] without a magnetic field, b = 0, thereby showing a sort of continuity of the tunneling estimate with respect to the magnetic field's strength.1.2.2.Three wells and eigenvalue crossings.Suppose now that the electric potential V is as follows where v 0 is the same function as in (1.5) and that the wells z 1 , z 2 , z 3 are located on the vertices of an equilateral triangle with side length L. We prove then the existence of a braid structure in the sense of Definition 1.1.
Theorem 1.4.Assuming b > 0 is fixed and V is given as in (1.5) with then the lowest egigenvalues of L h,b has a braid structure.Moreover, with E b,L (v 0 ) the same negative quantity as in Theorem 1.2.
Not only Theorem 1.4 establishes the existence of infinitely many eigenvalue crossings and splittings, but it also establishes an accurate estimate for the magnetic tunneling induced by three symmetric potential wells, thereby extending the recent results of [7,14] on double wells.1.3.Geometrically induced braid structure.We present here results on the pure magnetic Laplacian where the eigenvalue crossings are induced by a combination of the geometry and the flux in the semi-classical limit.We shall describe the results when Ω is a smoothed triangle (see Fig 2).That is, Ω is a simply connected domain, invariant under rotation by 2π/3, with three points of maximum curvature that are equidistant with respect to the arc-length distance on the boundary.
Such magnetic fields have been called 'magnetic steps' in the literature, and the semi-classical limit for the operator L B h has been studied recently in [3,10] (and in [11] for ϑ = −1).The bound states of the system become increasingly concentrated along the discontinuity of B in the semi-classical limit.Consequently, we expect the emergence of flux effects, and they are indeed realized when Ω is a smoothed triangle with 3-fold symmetry.
Theorem 1.6.Assume that Ω is a smoothed triangle, invariant under the rotation by 2π/3 (see Fig. 2).The lowest eigenvalues of the operator L B h have in the semi-classical limit a braid structure.We therefore have an example where the eigenvalues of the Landau Hamiltonian on the full plane cross and split infinitely many times.This is the consequence of having a sign changing magnetic field with a discontinuity along a simple smooth curve.
Recently, an estimate for the tunneling induced by a smooth magnetic field that vanishes non-degenerately along a smooth curve has been established in [1].It seems natural to expect the existence of a braid structure in that setting too when adding symmetry assumptions.1.4.Organization.The paper is organized as follows.Since we work with various symmetry configurations, Section 2 is devoted to an abstract spectral reduction to a hermitian matrix (so often called the interaction matrix in the literature on tunneling effects [15,13]).This provides us with a robust methodology when analyzing tunneling effects in various settings.Loosely speaking, all we need is the construction of adequate quasi-modes.
In Section 3 we discuss the electro-magnetic Laplacian.Our investigation has two ingredients, the first is to appply the abstract methodology in Section 2, and the second is to control the errors produced by the interaction terms; the later task is achieved by using the analysis in the recent work [14].Section 3 concludes with the proofs of Theorems 1.2 and 1.4.
In Section 4, we prove Theorem 1.5 by applying the results of Section 2. The control of the error terms and the computation of the interaction term were done in [5].
Finally, in Section 5, we prove Theorem 1.6 by applying the constructions in Section 2. We will be more succinct here since the analysis is similar to Section 4. The control of the error terms and the asymptotics of the interaction terms were done in [10] and are actually rather close to those in [5].

Abstract framework for a spectral reduction to a hermitian matrix
In this section we consider a family of operators dependent on a positive semi-classical parameter h ≪ 1.Assuming the existence of certain quasi-modes (see Assumption 2.1 below), we can approximate the eigenvalues of the operator with those of the matrix of its restriction on a specific basis (Proposition 2.4 below).Assuming an additional symmetry hypothesis (Assumption 2.6 below) we can relabel the eigenvalues and spot their possible crossings and splittings as the semi-classical parameter approaches 0 (Eqs.(2.20), (2.21) and Paragraphs 2.4.2, 2.4.3).
2.1.Preliminaries.Consider a Hilbert space H endowed with an inner product ⟨•, •⟩ and a family of self-adjoint unbounded operators T h : D h → H, h ∈ (0, 1].Assume furthermore that, for every h ∈ (0, 1], T h is semi-bounded from below and has a sequence of discrete eigenvalues counted with multiplicity.By the min-max principle the eigenvalues can be represented as We will work under additional assumptions on the operators (T h ) h∈(0,1] .Assumption 2.1 (n wells).Let n ≥ 2 be an integer.There exist positive constants S 1 , S 2 , S 3 , c, p, q and h 0 ∈ (0, 1] such that p < q and for all h ∈ (0, h 0 ], there exists a subspace E h = span(u h,1 , . . ., u h,n ) ⊂ D h such that: (1) max 1≤i≤n ∥T h u h,i ∥ = O(e −S 1 /h ). ( It results from (2) above that dim(E h ) = n for h small enough.As a consequence of Assumption 2.1, we now prove that the operator T h has precisely n eigenvalues that are exponentially small in h, and there is a gap to λ n+1 (h), since it is at least of polynomial size in h.Proposition 2.2.Under Assumption 2.1, there exist positive constants C, h 1 such that, for h ∈ (0, h 1 ], λ n (h) ≤ Ce −S 1 /h .(2.1)In particular λ n (h) < λ n+1 (h) for h sufficiently small.
Proof.Since dim(E h ) = n, we can use the min-max principle.Let ϕ = j α j u h,j be in E h .Then, the triangle inequality and Cauchy-Schwarz inequality, together with (1) and (2) from Assumption 2.1, provide the existence of constants A 1 and A 2 , such that if h is small enough, On the other hand, we can use (2) of Assumption 2.1 to bound the norm of ϕ from below.We get indeed constants A 3 , A 4 and A 5 such that, if h is small enough, This gives the bound in (2.1).Combining this bound with (4) in Assumption 2.1 we conclude that that λ n+1 (h) > λ n (h) if h is sufficiently small.□ We want to link the quasi mode constructions {u h,j } to the low-lying eigenvalues of T h .To do this, we want to show that the symmetric matrix U h = (u j,k ), does not differ much (component-wise) from the matrix W h that will be constructed as the restriction of T h to the eigenspace written in an orthonormal basis.We will do the approximation in two steps.We first consider the projected functions and show that the norms ∥v h,j − u h,j ∥ are small.Since the span of {u h,j } is ndimensional by Assumption 2.1 (2), it will follow that the {v h,j } are linearly independent, and thus constitute a basis for F h .
Proposition 2.3.If Assumption 2.1 holds for E h , then for h > 0 sufficiently small, we have dim(F h ) = n, and the vectors form a basis of F h .Moreover they satisfy Proof.Since we count multiplicities, we know in general that dim(F h ) ≥ n.However, by (4) in Assumption 2.1, we get from Proposition 2.2 that dim( On the other hand, According to Assumption 2.1 (1) and (2), and Proposition 2.2, Combining these inequalities we get (2.4).From this and Assumption 2.1 (2), we find that {v h,1 , . . ., v h,n } are linearly independent, and hence a basis for F h .□ 2.2.Reduction to a matrix through a suitable orthonormal basis.The aim in this subsection is to find an orthonormal basis for F h such that the matrix of the restriction of T h in this basis can be well approximated.Later in the applications to multiple wells problems this matrix will be according to the previous literature called the interaction matrix.The basis {v h,j } of F h that we just constructed will, in general, not be orthogonal.We construct, by a symmetry-preserving Gram-Schmidt procedure an orthonormal basis {w h,j }.The matrix W h will be the matrix of T h restricted to F h , written in this new basis {w h,j }.
Let us denote by G h = (g ij (h)) 1≤i,j≤n the Gram matrix of the basis {v h,1 , . . ., v h,n } of F h , where Since the {v h,j } are linearly independent, the Gram matrix becomes positive definite, so is well defined and positive definite.We obtain an orthonormal basis . . .
We consider the restriction of T h to the space F h and denote by W h = (w ij ) 1≤i,j≤n its matrix in the basis V h , so w ij = ⟨T h w h,i , w h,j ⟩.The matrix W h is hermitian, with eigenvalues {λ 1 (h), . . ., λ n (h)}.The next proposition controls how W h is approximated by the matrix U h defined by (2.2).
Step 1. Proposition 2.3 says that With I denoting the n × n identity matrix, we get from (2) in Assumption 2.1, where in the second estimate in (2.10), we simply combine the first estimate and the definition of Λ h .
Step 2. We may write Using this identity, (2.10) and Proposition 2.3, we get Then, we use By Proposition 2.3 and (1) in Assumption 2.1, we have

So we get
which together with (2.11) implies (2.9).□ An immediate consequence of Proposition 2.4 is an improved lower bound on the lowest eigenvalue λ 1 (h).

2). By Proposition 2.4, we have
As we will see in the next subsection, we can actually say much more about the spectrum of these matrices, when we impose some symmetry condition involving T h and the choice of the u h,j .
2.3.Implementing invariance assumptions.Our task in this subsection is to analyze the case when the matrix W h of T h | F h enjoys certain invariance properties.We shall see that this corresponds to what occurs in the case of symmetric wells in the applications, starting from the double well case as mathematically considered by E. Harrell [12] and later extended to the multiple wells case in [15,16,20,21].Here we mainly follow in a more abstract way [16] and the heuristic presentation given in [8].We denote by Z n the cyclic group of order n and by g → ρ(g) a faithful unitary representation of Z n in H.We denote by a n its generator, so a n n = e where e is the identity element of the group.
In addition to the properties in Assumption 2.1, we assume Assumption 2.6.
(1) The operator T h commutes with ρ(g) for all g ∈ Z n . ( Remark 2.7.In the applications considered in this article, the Hilbert space will be where Ω is a domain in R 2 .We first consider the unitary representation ρ 0 of Z n as the group G n of the n-fold rotations, i.e. the representation such that We let the rotation g n act on functions as This gives by extension to any element of is a domain invariant by G n and we then define ρ by Equivalently to Assumption (2.6), we can then write in this case The operator T h commutes with M (g n ). ( 6 permits to treat more general situations which for example occur in the case of manifolds or in the case of higher dimension.
Remark 2.9.If n = 2 we can define another group of symmetry G defined by the reflection g2 ( x 1 x 2 ) = ( −x 1 x 2 ).We can then consider a variant of Assumption 2.6 by instead assuming that the domain Ω is invariant by the reflection g2 and that u h,2 = M (g 2 )u h,1 .This symmetry invariance was assumed by the papers considering the magnetic tunneling induced by the geometry of the domain [1,5,10,19].Notice that M (g 2 ) does not commute with T h and that we have consequently to compose M (g 2 ) with the complex conjugation Γ in order to get Proof.Recall that {w h,1 , . . ., w h,n } is defined in (2.6) by the Gram matrix starting from the basis consisting of the vectors v h,i = Π F h u h,i , 1 ≤ i ≤ n.It suffices to observe that the projector Π F h on the eigenspace F h commutes with ρ(g).Actually, if {w h,1 , . . ., w h,n } is an orthonormal basis of F h consisting of eigenvectors T h , then, since T h commutes with ρ(a n ), we get that ρ(a n )w h,1 , . . ., ρ(a n )w h,n are eigenvectors of T h and form an orthonormal basis of F h .Consequently The matrix of ρ(a n ) in the basis V h is the same as the matrix of the shift operator τ on ℓ 2 (Z/nZ), whose matrix is given by where δ i,k denotes the Kronecker symbol, with i computed in Z/nZ.When n = 2 and n = 3, the matrix τ is respectively given by 0 1 1 0 and We observe that τ n−1 = τ −1 = τ * .The property that the operator T h commutes with ρ(a) implies that the matrix W h (of for some coefficients I 0 (h), . . ., I n−1 (h) ∈ C.Here τ 0 denotes the identity matrix.The Hermitian property of W h gives, in addition, (2.17) Notice that the matrix U h introduced in Proposition 2.4 satisfies the same properties as W h .Hence we can also write for some coefficients J 0 (h), . . ., J n−1 (h) ∈ C and the Hermitian property of U h also implies (2.19)All these invariant matrices (W h or U h ) share the property to be diagonalizable in the same orthonormal basis of eigenfunctions e k (k = 1, . . ., n) whose coordinates in our selected basis are given by It is then easy to compute the corresponding eigenvalues.
In particular, we get an explicit representation of the eigenvalues of W h which illustrates when n = 3, 4, the possibility of eigenvalue crossings (i.e.change of multiplicity).
• When n = 2, W h assumes the form ).This matrix has three eigenvalues • When n = 4, we meet the matrix with I 2 real and I 1 = ρe iθ , ρ ≥ 0, θ ∈ [0, 2π).We refer to [8] for a further discussion of this case.Figure 3 illustrates the braid startucture of the eigenvalues of the matrix W h .

Electro-magnetic tunneling
3.1.Introduction.In this section, the Hilbert space is H = L 2 (R 2 ), and for given n ∈ N we consider g = g n to be rotation around the origin by 2π/n, and M (g) to be as in (2.14).We are interested in the spectrum of the electro-magnetic Schrödinger operator where b, h > 0, and Notice that A generates the constant magnetic field curl A = 1.For the potential V we assume that V is invariant by the rotation g n .
Moreover, we assume that The minimum of V is attained at n non-degenerate minima. (3.2d) Then it results from the invariance property of V that these minima are n equidistant points of R 2 \ {0}.We will refer to these points as the wells.
Notice that, when dealing with a fixed b > 0 we can reduce the analysis to the case where b = 1 by introducing an effective semi-classical parameter ℏ = b −1 h so that So we will assume henceforth that b = 1.To relate with the discussion in Section 2, our operator T h is the electro-magnetic Laplacian shifted by a certain scalar λ(h).
Note that the assumption in (3.2c) implies that T h commutes with M (g).Hence the condition of invariance of the preceding section holds.The shift constant λ(h) in (3.4) will be chosen as the ground state energy of a reference single well operator.

3.2.
Single well ground states.Let us first discuss the one well operator.

Preliminary discussion and assumptions.
There are various possible approaches to create one well problems in the presence of multiple wells.The approach considered in [15,16] starts from a general electric potential V and introduce suitable Dirichlet conditions to create an infinite barrier leading to a one well problem.The so-called LCAO3 approach, frequently used in Physics and Atomic Chemistry, and considered in [7,14], particularly applies when V is a superposition of single well potentials.
As in [7,14], we consider a radial single well potential.More precisely, we assume in this section that We choose D(0, a) as the smallest closed disc containing supp v 0 , i.e.
3.2.2.Preliminary inequalities.As in [14], we will encounter various errors of exponential order which are defined in terms of the function v 0 and the constant With (v 0 , L) as above, a = a(v 0 ) and with we introduce the four constants (we will often skip the depends on v 0 and L) Notice also that, since v 0 vanishes on [a, +∞), we can express the constant S a as follows It is important when comparing the errors to compare the three constants introduced in (3.9).
(3.12) Since v 0 ≤ 0, L sw h is smaller (in the sense of comparison of self-adjoint operators) than the Landau Hamiltonian (−ih∇ − A) 2 .Hence we have by the min-max principle λ(h) ≤ h . (3.13) In light of the conditions on v 0 in (3.5), we know from [14, Thm.1.1] that λ(h) is a simple eigenvalue and that L sw h has a positive radial ground state satisfying, for any relatively compact domain (3.14)An important term in the context of tunneling is the hopping coefficient defined in [7,14] (in the case n = 2) by where a = a(v 0 ) (see (3.6)).
As observed in [7], the hopping coefficient is a negative real number and an accurate estimate of it was established recently in [14, Sec.6 & Eq.(6.25)], which we recall below. 4roposition 3.2.Assume that v 0 and L satisfy (3.5) and (3.7).Then there exists a positive constant S(v 0 , L) such that Moreover, where S a (v 0 , L) and Ŝ(v 0 , L) are introduced in (3.9).
We shall use Proposition 3.2 later on in the proof of Proposition 3.5 when dealing with n potential wells (n ≥ 2).Let us recall the definition of S(v 0 , L) given in [14, Eq. (6.18) and (6.20)].We have where a = a(v 0 ) is introduced in (3.6) and, with d introduced in (3.6), The following proposition deals with a term similar to the hopping coefficient and plays a key role in the approximation of various error terms that we will encounter later, e.g. when we verify Assumption 2.1 for an electro-magnetic Schrödinger operator with multiple wells, as in Proposition 3.4.
Proof.We can estimate w in the same way as the hopping coefficient c h (v 0 , L) was estimated in [14, Prop.5.1].
The only difference is that in the expression of c h (v 0 , L) (see (3.15)) we encounter the potential energy term v 0 in the integrand and the integral is consequently over the smaller disc D(0, a).Hence the new term to estimate corresponds to w 2 below, w 3 being of the same type as w 1 after a symmetry.
We decompose the integral defining w into three terms Step 1: Contribution of the integral in D(0, a).We express the integral defining w 1 in polar coordinates where S a is introduced in (3.9b).Notice that the penultimate identity follows from the fact that Therefore, we have that Step 2: Contribution of the integral in D(0, L) \ D(0, L − a).We argue as in Step 1. Expressing the integral defining w 3 in polar coordinates we get where Therefore, we have that Step 3: Contribution of the integral in D(0, L − a) \ D(0, a).We express the integral defining w 2 as follows where .
Of importance to us is that which follows by the same argument as used in the proof of [14, Prop.6.5]; for convenience, we provide details in Appendix A. We get eventually w 2 = O(e (−Sa+o(1))/h ).
(3.27)With (3.21) and (3.23), this achieves the proof of the proposition.□ 3.3.Verifying Assumption 2.1-superposition of single well potentials.We study the specific case where the potential V is given by where v 0 is the non-positive radial function satisfying (3.5) and (we identify C and R 2 ) The wells in this setting are the points {z k } 1≤k≤n which are selected so that dist(z k , z k+1 ) = L. (3.30) Let us verify that V satisfies (3.2c).Our construction of the points z k is such that z k+1 = gz k , for k ∈ Z/nZ.Since v 0 is radial, we have Consequently, (3.2c) holds and T h commutes with M (g).We still have to check that Assumption 2.1 holds with the choice in Assumption 2.8 (3).We introduce the functions where • (z k ) 1≤k≤n are the points introduced in (3.29); • A is the vector field introduced in (3.1); ) is a radial cut-off function satisfying χ = 1 on D(0, L) and supp χ ⊂ D(0, L + η) with η ∈ (0, 1) fixed arbitrarily.The phase term in (3.31) is due to the effect of magnetic translation, which ensures that The above constructions ensures that Assumption 2.8 (3) holds (since z • A(g −1 n x) = (g n z) • A(x)).Moreover, the next proposition shows that Assumption 2.1 holds too.Proposition 3.4.Let T h and λ(h) be as in (3.4) and (3.12) respectively.The conditions in Assumption 2.1 hold with the following choices: (a) {u h,1 , . . ., u h,n } are as in (3.31);(b) any constants S 1 , S 2 , S 3 , p, q satisfying S 1 ∈ (0, Ŝa ), S 2 ∈ (0, 2 Ŝa ), S 3 ∈ (0, Ŝa ), p ∈ (0, 1], q ∈ (1, 2), (3.33)where Ŝa is introduced in (3.9c). Proof.
Step 1.We have by (3.32), where , we get by using (3.14) and the decay of the ground state u h that ∥r h,k ∥ = O(h −1/2 e − Ŝa/h ). (3.35) Step 2. We have where S 0 is introduced in (3.9a).
Let us now consider i ̸ = j.We first inspect the case where |z i − z j | = L, which occurs only if j = i ± 1.By a change of variable, we check that (thanks to the invariance by rotation) We perform the following decomposition where, In D(z 1 , L), the cut-off functions in the definitions of u h,1 and u h,2 are equal to 1.By a change of variable, we get 2h dx , and by Lemma 3.3, we get where in the last step we used the inequalities Ŝa < S a < S 0 from Proposition 3.1.
) by a similar argument.
Step 3. Let us now estimate λ 1 (h) from below.Consider a partition of unity on Pick a normalized ground state f h of L h .Then we have, We have the decomposition formula By the min-max principle we have and (with (3.13) in mind) Hence there exists h 0 > 0 such that for h ∈ (0, h 0 ] The next proposition allows us to verify Assumption 2.11. where c h (v 0 , L) is the hopping coefficient introduced in (3.15), Ŝa (v 0 , L) is introduced in (3.9c) and Moreover, if 2 Ŝa (v 0 , L) > S(v 0 , L) then we have where S(v 0 , L) is introduced in (3.18).
Proof.From (3.34), we have We now move to estimate J 1 (h).First let us recall that the symmetry relations in Assumption 2.8 (3) imply that u h,1 = M (g)u h,n and5 Similarly to the previous estimate of J 0 (h), by (3.32) and (3.34) we have ), where . By a translation, we get v 0 (y)u h (y + z n − z 1 )u h (y)e −i(zn−z 1 )•A(y)/h dy. With , we have z n = (ℓ n , 0) and Observing that |z n − z 1 | = L, we write z n − z 1 = Le iβn where β n ∈ (0, π].We denote by R βn the rotation by β n around the origin.Noticing that The change of variable y → x = R −1 βn y yields where we used that the functions v 0 , u h are radial and that Now we have that e −iϕn/2h J app 1 (h) = c h (v 0 , L) and by Proposition 3.2 with 0 < S(v 0 , L) < Ŝ.The same asymptotics hold for |J 1 (h)| if we assume that 2 Ŝa > S(v 0 , L) .□ From now on, we work under the following assumption on (v 0 , L).
We fix the choice of S 1 , S 2 , S 3 as in (3.33) but with the additional condition that This is possible under Assumption 3.6 since S(v 0 , L) < 2 Ŝa (v 0 , L).Therefore, (2.22) holds with S = S(v 0 , L) and Proposition 3.5 ensures that the other conditions in Assumption 2.11 hold too6 .
3.4.The case n = 2: Double wells.We choose here V to be the double well potential defined by (3.28) for n = 2.We therefore have two wells
Theorem 3.8.Assuming the conditions in (3.5) and in Assumption 3.6 are fulfilled, then the following asymptotics holds, where S(v 0 , L) is the constant introduced in (3.18).
3.4.1.Discussion.Let us introduce the following classes7 of admissible (v 0 , L), where v 0 satisfies the conditions in (3.5), and ) and (3.37) holds}, By Proposition 3.1, we have A HKS ⊂ A HKS and by Theorem 3.8 A HKS ⊂ A .
The earlier results in [7,14] yield that However, our results are much stronger since A FSW is a proper subset of A HKS .

3.5.
The case n = 3: Three wells and flux effects.Let us assume that n = 3 so that V defined by (3.28) is a potential with three wells which are located on the vertices of an equilateral triangle with side length L and let D ⊂ R 2 be its interior.The area of D is then By Proposition 3.5, we have Recall that by (2.27), If for a given constant c 0 > 0 we introduce the set then it results from (3.39) that (since θ(h) ∈ [0, 2π) in the definition of I 1 (h)) Now by paragraph 2.4.3 we get crossing of eigenvalues quantified via the following functions Theorem 3.9.Assume that the conditions in (3.5) are fulfilled and that V is defined by (3.28) with n = 3.If Assumption 3.6 holds, then there is a relabeling µ 1 (h), µ 2 (h), µ 3 (h) of the eigenvalues λ 1 (h), λ 2 (h), λ 3 (h) of the electro-magnetic Schrödinger operator L h such that the asymptotics hold for all L > 2a(v 0 ).Moreover, there exists a sequence h 1 (k), h 2 (k), h 3 (k) k≥k 0 which converges to 0 such that, for all k ≥ k 0 we have Consider constants δ i , i = 0, 1, 2, 3, such that Notice that h * 3 (k) > h * 0 (k + 1) and by (3.41), there exists k 0 ≥ 1 such that, for all k ≥ k 0 , we have By continuity, we select (Ω) and we will assume (see below for the invariance assumption) that we are in the situation considered in Remark 2.7.We assume that Γ is a simple curve and denote its length by |Γ| = 2L.Let R/2LZ ∋ s → γ(s) be the arc-length parameterization of Γ such that the unit tangent vector τ (s) := γ(s) turns counterclockwise.Let us denote by k the curvature along Γ defined as follows γ(s) = k(s)ν(s) where ν(s) is the unit normal to Γ at γ(s) pointing inward Ω.
Definition 4.1.We introduce the maximal curvature along Γ as follows and call a point z 0 ∈ Γ a curvature well if z 0 = γ(s 0 ) and k(s 0 ) = k max .The point z 0 is said to be a non-degenerate curvature well if furthermore k ′′ (s 0 ) < 0. We denote by Γ 0 the set of curvature wells.
Consider a positive integer n and the rotation by 2π/n denoted by g n .We assume that Ω is invariant by the rotation g n (4.2a) and that we have n non-degenerate curvature wells The symmetry assumption yields that and The magnetic Neumann Laplacian.We are interested in the magnetic Neumann Laplacian L N h = (−ih∇ − A) 2 , in the Hilbert space L 2 (Ω), where the magnetic field B = curl A = 1 is uniform 9 and generated by the magnetic potential A introduced in (3.1).The operator L N h acts on functions u ∈ H 2 (Ω) satisfying the (magnetic) Neumann condition ν • (−ih∇ − A)u| Γ = 0.The case of double curvature wells corresponding to n = 2 was treated in [5], where the symmetry was generated by the reflection g2 (see Remark 2.9).Here we focus on the case with n ≥ 3 curvature wells and where the symmetry is generated by a rotation.When n = 3, a typical example is the smoothed triangle (Fig. 2).When Ω is an equilateral triangle the heuristic discussion is given in [8] but no rigorous result can be given since the authors were unable to have a sufficiently accurate control of the tunneling.Numerically, this has been computed in [4] which in particular gives the enlightening picture predicting eigenvalue crossings (Fig. 4).
The investigation of L N h can be connected with Section 2 after shifting by a constant ℓ(h) and taking the operator T h as follows The shift constant ℓ(h) will be defined by the ground state energy of an operator with a single curvature well.Two constants are important in our analysis.First we meet a magnetic flux like term that controls the eigenvalue crossings and is defined by Secondly the following constant controls the strength of the tunneling and is defined by The definition of Φ 0 and S n involves universal constants related to the de Gennes model.We recall that, for ξ ∈ R, µ N 1 (ξ) denotes the lowest eigenvalue of the Neumann realization of the Harmonic oscillator on the semi-axis R + : Minimizing over ξ ∈ R we get the de Gennes constant Denoting by u 0 the positive L 2 -normalized ground state of h N [ξ 0 ], the constant C 1 appearing in (4.4b) is defined by There exists a geometric constant ε 0 > 0 such that the above transformation is invertible when 0 < t < ε 0 ; the image of Φ is the tubular neighborhood of Γ Γ(ε 0 ) = {x ∈ Ω : dist(x, ∂Ω) < ε 0 } (4.7) and for x ∈ Γ(ε 0 ), (s, t) = T −1 (x) means that s is the arc-length coordinate of the projection of x on Γ and t is the normal distance from x to Γ.We will refer to s as the tangential variable and to t as the normal variable.
Moreover, it transforms the Hilbert space L 2 (R/2LZ) × (0, ε 0 ); a ds dt to the Hilbert space L 2 Γh ; ãh dσ dτ , and the operator Lh to h Ñh where Let us consider Ñh , in L 2 Γh ; ãh dσdτ , with domain10 The study of the eigenvalues of L N h , can then be compared with those of Ñh (see [5,Prop. 2.7]).Proposition 4.2.Let N ∈ N and S n be the constant introduced in (4.4b).There exist K > S n , C, h 0 > 0 such that, for all h ∈ (0, h 0 ] and 1 ≤ k ≤ N , we have, where λ k (L N h ) and λ k ( Ñh ) are the k-th eigenvalues, counting multiplicity, of the operators L N h and Ñh , respectively.Recall that we are interested in applying the results in Section 2 to the operator T h obtained by shifting the operator L N h (see (4.3)).Effectively, that is related to the operator obtained by doing the corresponding shift to the operator Ñh , Our next task is to verify Assumptions 2.1 and 2.8 (they will both hold for Th and T h ) and this will require the construction of certain quasi-modes (u h,i ) 1≤i≤n .Let us make the following two observations: The assumption in (4.2a) implies that Th commutes with M (g n ), hence the condition of invariance by rotation holds.(ii) For every fixed labeling n, the eigenfunctions of Th corresponding to the n'th eigenvalue decay exponentially in the (rescaled) tangential variable; more precisely, given an eigenfunction f n of Th with corresponding eigenvalue λ n ( Th , there exist positive constants C n , h n , α n such that The relevance of (i) above is that we can construct quasi-modes of Th obeying the symmetry invariance properties as in Assumption 2.8, and in turn we can use these quasi-modes to produce quasi-modes for the initial operator T h in (4.3).The observation in (ii) asserts that the eigenfunctions of Th , once rescaled to the initial tangential variable t = h 1/2 τ , can be ignored in the interior of the domain Ω.
the same as the space with the flat measure L 2 Γh ; dσdτ with equivalent norm.
The expression of the operator Ñh (and hence Th ) involves the effective semi-classical parameter ℏ := h 1/2 .This leads us to adjust our setting by working with the operators and where Now we consider the operator with domain Dom(N We denote by λ(ℏ) the ground state energy of the operator N (1) ℏ ; it is simple and can be expanded as follows [9,6] where (δ 1,j ) j≥1 are real constants and k 2 = k ′′ 2 (0) < 0. Moreover there exist real constants (δ 2,j ) j≥1 such that the second eigenvalue satisfies and there is a spectral gap Functions in the domain of N (1) ℏ can be extended to the full half-plane R 2 + = R × R + after multiplication by a suitable cutoff function.We could have considered the single well problem in an alternative manner by truncating the curvature and extending it by 0 outside Γ(1) ℏ,η and also the weight function a ℏ to get an operator in R 2 + .Due to the exponential decay of bound states, the spectra agree up to exponentially small errors that are negligible compared with the estimate of the tunneling [5, Prop.
Note that the action of L B h on Γ(ε 0 ) is transformed to (after, possibly, a gauge transformation) where the constant γ 0 and the function b ϑ are introduced in (4.9) and (5.4) respectively.
That is essentially done in [10].We introduce the following 'distance' where C * (ϑ) ∈ C \ {0} is a constant independent of ℏ.

1. 3 . 1 .Theorem 1 . 5 .
The magnetic Neumann Laplacian under constant magnetic field.In the Hilbert space L 2 (Ω), we consider the magnetic Neumann Laplacian L N h = (−ih∇ − A) 2 on Ω, with uniform magnetic field curl A = 1 and (magnetic) Neumann condition ν • (−ih∇ − A)u| Γ = 0. Let the domain Ω be a smoothed triangle, invariant under the rotation by 2π/3 (see Fig.2).The lowest eigenvalues of the operator L N h have a braid structure.The presence of the Neumann boundary condition plays a vital role in the preceding theorem.This condition is responsible for the semi-classical localization of the bound states near the boundary of Ω and this is this localization that gives rise to the observed flux effects.

Figure 2 .
Figure 2. Illustration of the domain Ω: a smoothed triangle invariant under the rotation by 2π/3.

Proof.
To get the asymptotics in (3.43), we use the computations in Paragraph 2.4.3 and (3.39).Then we approximate |J 1 (h)| by Proposition 3.5.

4 . 4 . 1 . 1 .
and therefore we get the eigenvalue crossings as indicated in Paragraph 2.4.3.□ Smoothed triangles and Neumann boundary condition 4.1.Introduction.Geometric setting.In this section, Ω is a bounded open set of R 2 with C ∞ boundary Γ.The Hilbert space is H = L 2
dx is the magnetic flux in Ω .