On the semiclassical spectrum of the Dirichlet-Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.


Introduction
In this article we consider the magnetic Pauli operator defined on a bounded and simply-connected domain Ω ⊂ R 2 subject to Dirichlet boundary conditions. This operator is the model Hamiltonian of a non-relativistic spin- 1 2 particle, constraint to move in Ω, interacting with a magnetic field that is perpendicular to the plane.
Formally the Pauli operator acts on two-dimensional spinors and it is given by where h > 0 is a semiclassical parameter and σ is a two-dimensional vector whose components are the Pauli matrices σ 1 and σ 2 . The magnetic field B enters in the operator through an associated magnetic vector potential A = (A 1 , A 2 ) that satisfies Assuming that the magnetic field is positive and few other mild conditions we provide precise asymptotic estimates for the low energy eigenvalues of P h in the semiclassical limit (i.e., as h → 0). Let us roughly explain our results. Let λ k (h) be the k-th eigenvalue of P h counting multiplicity. Assuming that the boundary of Ω is C 2 , we show that there exist α > 0, θ 0 ∈ (0, 1] such that the following holds: For all k ∈ N * , there exists C k > 0 such that, as h → 0, (1)) .
In particular, this result establishes the simplicity of the eigenvalues in this regime. The constants α > 0 and C k are directly related to the magnetic field and the geometry of Ω and C k is expressed in terms of SegalBargmann and Hardy norms that are naturally associated to the magnetic field. In the case when Ω is a disk and B is radially symmetric we compute C k explicitly and find that θ 0 = 1. This improves by large the known results about the Dirichlet-Pauli operator [6,11] (for details see Section 1.3.2).
These results may be reformulated in terms of the large magnetic field limit by a simple scaling argument. Indeed, µ k (b) = b 2 λ k (1/b), where µ k (b) is the k-th eigenvalue of [σ · (−i∇ − bA)] 2 .
Our results can also be used to describe the spectrum of the magnetic Laplacian with constant magnetic field B 0 . For instance, when Ω is bounded, strictly convex with a boundary of class C 1,γ (γ > 0), the k-th eigenvalue of (−ih∇ − A) 2 with Dirichlet boundary conditions, denoted by µ k (h), satisfies, for some c, C > 0 and h small enough, In particular, the first eigenvalues of the magnetic Laplacian are simple in the semiclassical limit. This asymptotic simplicity was not known before and (1.1) is the most accurate known estimate of the magnetic eigenvalues in the case of the constant magnetic field and Dirichlet boundary conditions (See [10, Section 4] and Section 1.3.2). Our study presents a new approach that establishes several connections with various aspects of analysis as Cauchy-Riemann operators, uniformisation, and, to some extent, Toeplitz operators. We may hope that this work will cast a new light on the magnetic Schrödinger operators.

Setting and main results.
Let Ω ⊂ R 2 be an open set. All along the paper Ω will satisfy the following assumption. Assumption 1. 1. Ω is bounded and simply connected.
Consider a magnetic field B ∈ C ∞ (Ω, R). An associated vector potential A : Ω −→ R 2 is a function such that B = ∂ 1 A 2 − ∂ 2 A 1 . We will use the following special choice of vector potential. The vector field A = (−∂ 2 φ, ∂ 1 φ) T := ∇φ ⊥ is a vector potential associated with B.
In this paper, B will be positive (and thus φ subharmonic) so that In particular, the minimum of φ will be negative and attained in Ω. Note also that the exterior normal derivative of φ, denoted by ∂ n φ, is positive on ∂Ω if Ω is C 2 [7, Hopf's Lemma, Section 6.4.2].
Notation 1. We denote ·, · the C n (n 1) scalar product (antilinear w.r.t. the left argument), ·, · L 2 (U ) the L 2 scalar product on the set U, · L 2 (U ) the L 2 norm on U and · L ∞ (U ) the L ∞ norm on U. We use o and O for the standard Laudau symbols.
1.2. The Dirichlet-Pauli operator. This paper is devoted to the Dirichlet-Pauli operator (P h , Dom(P h )) defined for all h > 0 on Dom(P h ) := H 2 (Ω; C 2 ) ∩ H 1 0 (Ω; C 2 ) , and whose action is given by the second order differential operator (1.3) Here p = −ih∇, and and σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices: and σ·x = σ 1 x 1 +σ 2 x 2 +σ 3 x 3 for x = (x 1 , x 2 , x 3 ) and σ·x = σ 1 x 1 +σ 2 x 2 for x = (x 1 , x 2 ). In terms of quadratic form, we have by partial integration, for all ψ ∈ Dom(P h ), Note that we have the following relation, for all x, y ∈ R 3 , where 1 2 is the identity matrix of C 2 . The operator P h is selfadjoint and has compact resolvent. This paper is mainly devoted to the investigation of the lower eigenvalues of P h .
Notation 2. Let (λ k (h)) k∈N * (h > 0) denote the increasing sequence of eigenvalues of the operator P h , each one being repeated according to its multiplicity. By the min-max theorem, . (1.6) Under the assumption that B > 0 on Ω, the lowest eigenvalues of P h are the eigenvalues of L − h . More precisely, our main result states that for any fixed k ∈ N * and h > 0 small enough, λ k (h) is the k-th eigenvalue of the Schrödinger operator L − h . 1.3. Results and relations with the existing literature.

Main theorem.
Notation 3. Let us denote by H (Ω) and H (C) the sets of holomorphic functions on Ω and C. We consider the following (anisotropic) Segal-Bargmann space We also introduce a weighted Hardy space Here, x min ∈ Ω and Hess x min φ ∈ R 2×2 are defined in Theorem 1.3 below, n(s) is the outward pointing unit normal to Ω, and ∂ n φ(s) is the normal derivative of φ on ∂Ω at s ∈ ∂Ω. We also define for P ∈ H 2 (Ω), A ⊂ H 2 (Ω), The main results of this paper are gathered in the following theorem.
Assume that Ω is C 2 , satisfies Assumption 1.1, and (a) B 0 := inf{B(x), x ∈ Ω} > 0, (b) the minimum of φ is attained at a unique point x min , (c) the minimum is non-degenerate, i.e., the Hessian matrix Hess x min φ at x min (or z min if seen as a complex number) is positive definite.
Then, there exists θ 0 ∈ (0, 1] such that for all fixed k ∈ N * , A precise definition of θ 0 is given in Remark 1.10.
Assuming that Ω is the disk of radius 1 centered at 0, and that B is radial, we have Remark 1.4. Assume that B = B 0 > 0 and that Ω is strictly convex, then φ has a unique and non-degenerate minimum (see [13,14] and also [11, for n ∈ N and u ∈ H 2 (Ω) (see also the proof of Lemma 3.5). This ensures that the space H 2 k (Ω) defined in (1.7) is a closed vector subspace of H 2 (Ω) and that dist H (z − z min ) k−1 , H 2 k (Ω) > 0 (see [3,Corollary 5.4]) since (z − z min ) k−1 / ∈ H 2 k (Ω).
Remark 1.7. In the case when B is radial on the unit disk Ω = D(0, 1), we get, using Fourier series, that (z n ) n 0 is an orthogonal basis for N B and N H which are up to normalization factors, the Szegö polynomials [4,Theorem 10.8]. In particular, In addition, P k−2 is N B -orthogonal to z k−1 so that , the lowest eigenvalue of the operator L + h . For fixed h > 0, there exists k(h) ∈ N * such that ν 1 (h) = λ k(h) (h). By (1.3), we have ν 1 (h) 2B 0 h and thus ν 1 (h) does not converge to 0 with exponential speed. Actually, Theorem 1.3 ensures that This accumulation of eigenvalues near 0 in the semiclassical is related to the fact that the corresponding eigenfunctions are close to be functions in the SegalBargmann space B 2 (C) which is of infinite dimension.
Actually, we can even see from our analysis that there is a class of magnetic fields for which θ 0 = 1. We introduce (1.8) Here, ∂ s denotes the tangential derivative. Then, for anyB ∈ B and G ∈ M Ω , we get θ 0 = 1 and lim inf for the magnetic field B = |G ′ (z)| 2B • G(z). This follows from the fact that the function Using the Riemann mapping theorem, we can deduce the following lower bound for Ω with Dini-continuous boundary. Its proof can be found in Section 5.4. Corollary 1.11. Assume that Ω is bounded, simply connected and that ∂Ω is Dinicontinuous. Assume also (a)-(c) of Theorem 1.3. Let k ∈ N * . Then, there exist c k , C k > 0 and h 0 > 0 such that, for all h ∈ (0, h 0 ), Remark 1.12. Note also that our proof ensures that the constants C k , c k can be chosen so that C k /c k does not depend on k ∈ N * .
Our results can be used to describe the spectrum of the magnetic Laplacian with constant magnetic field (see Remark 1.4). Corollary 1.13. Assume that Ω is bounded, strictly convex and that ∂Ω is Dinicontinuous. Assume also that (a)-(c) of Theorem 1.3 hold and that B is constant.
Then, the k-th eigenvalue of (−ih∇−A) 2 with Dirichlet boundary conditions, denoted by µ k (h), satisfies, for some c, C > 0 and h small enough, In particular, the first eigenvalues of the magnetic Laplacian are simple in the semiclassical limit. ii. In [11] (simply connected case) and [12] (general case), Helffer and Sundqvist have proved, under assumption (a), that lim h→0 h ln λ 1 (h) = 2φ min .
Moreover, under the assumptions (a), (b) and (c) of Theorem 1.3, their theorem [11,Theorem 4.2], implies the following upper bound for the first eigenvalue Note that Theorem 1.3 (i) provides a better upper bound even for k = 1. They also establish the following lower bound by means of rough considerations: is the first eigenvalue of the corresponding magnetic Dirichlet Laplacian. This estimate is itself an improvement of [5, Theorem 2.1]. Corollary 1.11 is an optimal improvement in terms of the order of magnitude of the pre-factor of the exponential. It also improves the existing results by considering the excited eigenvalues. Describing the behavior of the prefactor is not a purely technical question. Indeed, it is directly related to the simplicity of the eigenvalues and even governs the asymptotic behavior of the spectral gaps. This simplicity was not known before, except in the case of constant magnetic field on a disk. iii. The problem of estimating the spectrum of the Dirichlet-Pauli operator is closely connected to the spectral analysis of the Witten Laplacian (see for instance [11,Remark 1.6] and the references therein). For example, in this context, the ground state energy is whereas, as known in the literature in the present paper, we will focus on (see also Lemma 2.4). Considering real-valued functions v in (1.11) reduces to (1.10). In this sense, (1.11) gives rise to a "less elliptic" minimization problem.
1.4. About the intuition and strategy of the proof. In this paragraph we discuss the main lines of our strategy. It is intended to reveal the intuition behind some of our proofs. We will focus mostly on the ground-state energy, which is given by (1.6) as . (1.12) It is easy to guess from (1.3) that the ground state energy has to have the form ψ = (u, 0) T . This is consistent with the physical intuition that, for low energies, the spin of the particle should be parallel to the magnetic field.
The variational problem above can be re-written by means of a suitable transformation as where, ∂ z = (∂ 1 + i∂ 2 )/2 and φ is the unique solution to ∆φ = B in Ω with Dirichlet boundary conditions (see Definition 1.2). This connection between the spectral analysis of the Dirichlet-Pauli and Cauchy-Riemann operators is known in the literature (see e.g. [6,11,2] and [17]), and we describe it in Section 2.
In order to study the problem in (1.13) it is helpful to consider the following heuristics concerning F h (v, φ). Observation 1.14. A minimizer v h wants to be an analytic function in the interior of Ω but, due to the boundary conditions, has to have a different behaviour close to the boundary. So, if we set Ω δ := {x ∈ Ω, dist(x, ∂Ω) δ} for δ > 0, we expect that v h behaves almost as an analytic function on U with Ω δ ⊂ U ⊂ Ω. Moreover, this tendency is enhanced in the semiclassical limit when the presence of the magnetic field becomes stronger. Hence, we also expect that δ → 0 as h → 0 in some way.
We comment below how we make Observation 1.14 more precise, for the moment let us just mention that throughout this discussion we work with δ such that (1.14) As a consequence of Observation 1.14 we expect that where T δ := Ω \ Ω δ . An essential ingredient in our method is the analysis of the minimization problem associated with the RHS of (1.15). The main ideas go as follows: Assume first that Ω is the disk D(0, 1). By writing the integrand |∂ z v h | 2 e −2φ/h in tubular coordinates (see Item i. from the proof of Lemma 3.7 ) and Taylor expanding φ around any point at the boundary ∂Ω we get, for δ satisfying (1.14), (see also the proof of Lemma 5.5), where ∂ n φ ≡ ∂ n φ(s) is the normal derivative at the boundary (see Notation 3).
Observe that if ∂ n φ is a constant along the boundary, then it equals the flux Φ. In this case, as explained in Item iv. of the proof of Lemma 5.5, the problem of finding a non-trivial solution of can be reduced to a sum (labeled in the Fourier index) of one-dimensional problems that we solve explicitly in Lemma A.1.
For the particular case of v having only the non-negative Fourier modes on ∂Ω δ (i.e., v δ = m 0v δ,m e ims ) we find that (see Lemma 5.5) where the last equality is a trivial consequence of (1.14). Moreover, by Lemma A.1, the latter inequality is saturated when v δ =v δ,0 . Concerning, the assumption on v, recall that analytic functions on the disk have only Fourier modes for m 0. Notice that if B is rotationally symmetric ∂ n φ is constant. If ∂ n φ is not a constant we can give a suitable estimate using that min ∂Ω ∂ n φ > 0. We extend the previous analysis to more general geometries by using the Riemann mapping theorem.
There is another important point to take into account, this time concerning G h (v, φ).
Observation 1.15. Recall that φ 0 has an absolute, non-degenerate, minimum at x min . Hence, the weighted norm of v h , G h (v, φ), should have a tendency to concentrate around x min . This is made precise in Lemma 5.3 below. Moreover, observe that using Laplace's method, one formally gets that, as h → 0, Observations 1.14 and 1.15 reveal the importance of the behaviour of a minimizer around the boundary and close to x min , respectively. In addition, this behaviour is naturally captured through the norms N H and N B given in Definition 3, which, in turn, provide a natural Hilbert space structure to select linear independent test functions which are used to estimate the excited energies.
In order to show our result we give upper and lower bound to the variational problem (1.13). This is done in Sections 3 and 5, respectively. Concerning the upper bound: In view of the previous discussion it is natural to choose a trial function (at least for the disk, see Remark 3.2) v = ωχ where ω is an analytic function in Ω and χ is such that χ ↾ Ω δ = 1 and decays smoothly to zero towards ∂Ω. We pick χ ↾ T δ as an optimizer of the problem (1.18). For λ k (h), we choose ω to be a polynomial of degree (k − 1). In particular, for the ground-state energy, ω is constant and in view of (1.22) and (1.20) we readily see how the claimed upper bound (at least for the disk with radial magnetic field) is obtained.
As for the lower bound, as a preliminary step, we discuss in Section 4 some ellipticity properties related to the magnetic Cauchy-Riemann operators. Our main result there is Theorem 4.6. It provides elliptic estimates for the magnetic Cauchy-Riemann operators on the orthogonal of the kernel which consists, up to an exponential weight, in holomorphic functions. The findings of Section 4 are crucial to prove Proposition 5.4, which gives estimates on the behaviour described in Observation 1.14. Indeed, Proposition 5.4, together with the upper bound, roughly states that the non-analytic part of v h on any open set contained in Ω is, in the semiclassical limit, exponentially small in a sufficiently strong norm. At least for the disk with radial magnetic field, we can argue on how to get the lower bound if we assume v h to be analytic on an open set U with D(0, 1 − δ) ⊂ U ⊂ D(0, 1). Notice that (1.22) holds. Moreover, by Cauchy's Theorem Let us finally remark that actually, since the function v in (1.20) depends on h, Laplace's method cannot be performed so easily. Instead, after the change of scale y = x−x min h 1/2 , one has formally the Bargmann norm appearing: Ultimately, in the case of the disk with radial magnetic field, Problem 1.13 reduces formally to ( 1.22) which can be computed easily due to the orthogonality of the polynomials (z n ) n 0 in the Hilbert spaces H 2 (Ω) and B 2 (C) (see Remark 1.7). Of course, special attention has to be paid on the domains of integration and the sets where the holomorphic tests functions live. In the non-radial case however, we strongly use the multi-scale structure of (1.22) to get the result of Theorem 1.3 (see Section 5.3). Note that the constant θ 0 of Theorem 1.3 which appears in the computation of (1.22) somehow measures a symmetry breaking rate (see Remark 1.10 and Lemma 5.6).

Change of gauge
The following result allows to remove the magnetic field up to sandwiching the Dirac operator with a suitable matrix.
The proof follows from the next two lemmas and Definition 1.2 (see also [17,Theorem 7.3]).
Lemma 2.2. Let f : C → C be an entire function and A, B be two square matrices such that AB = −BA. Then, Proof. By Lemma 2.2, we have for k = 1, 2 that Thus, by the Leibniz rule, It remains to notice that −iσσ 3 = σ ⊥ := (−σ 2 , σ 1 ) so that We obtain then the following result.

Upper bounds
This section is devoted to the proof of the following upper bounds.
Using the min-max principle, this would give (3.1). Formula (2.2) suggests to take functions of the form v(x) = χ(x)w(x) , where i. w is holomorphic on a neighborhood on Ω, ii. the function χ : Ω → [0, 1] is a Lipschitzian function satisfying the Dirichlet boundary condition and being 1 away from a fixed neighborhood of the boundary. In particular, there exists ℓ 0 ∈ (0, d(x min , ∂Ω)) such that where d is the usual Euclidean distance.
Remark 3.2. The most naive test functions set could be the following Note however that in the radial case C sup (k) = C sup (k). We will rather use functions compatible with the Hardy space structure to get the bound of Proposition 3.1, as explained below.
Notation 4. Let us denote by (P n ) n∈N the N B -orthogonal family such that P n (Z) = Z n + n−1 j=0 b n,j Z j obtained after a Gram-Schmidt process on (1, Z, . . . , Z n , . . . ). Since P n is N B -orthogonal to P n−1 , we have Let us now define the k-dimensional vector space V h,k,sup by At the end of the proof, m will be sent to +∞. Note that we will not need the uniformity of the semiclassical estimates with respect to m. That is why the parameter m does not appear in our notations. Note that w n,h , being a non trivial holomorphic function, does not vanish identically at the boundary. To fulfill the Dirichlet condition, we have to add a cutoff function (see below).

Remark 3.4.
Consider Since Q n belongs to H 2 (Ω) ⊂ H 1 (Ω; C), the functions w n,h : x → ω n,h (x 1 + ix 2 ) and χ w n,h do not belong necessarily to H 1 (Ω; C) and H 1 0 (Ω; C) respectively. That is why we introduced Q n,m . Note that to get H 1 0 (Ω; C) test functions, it suffices to impose that χ is compactly supported in Ω. With this strategy, our proof can be adapted to the case where Ω is non necessarily simply connected.

3.2.
Estimate of the L 2 -norm. The aim of this section is to prove the following estimate.
Proof. Let α ∈ 1 3 , 1 2 , n, n ′ ∈ {0, . . . , k − 1}. In the proof, three types of terms will appear after a change of scale around x min : P n , P n ′ B , P n , Q n ′ ,m B and Q n,m , Q n ′ ,m B where ·, · B is the scalar product associated with N B . Since the polynomials (P n ) n∈N are N B -orthogonal, we have P n , P n ′ B = 0 if n = n ′ and we will prove that Q n,m , Q n ′ ,m B = O(h) and by Cauchy-Schwarz inequality P n , Q n ′ ,m B = O(h 1/2 ). More precisely, we have: i. Let us estimate the weighted scalar products related to P n for the weighted L 2 -norm.
Using the Taylor expansion of φ at x min , we get, for all x ∈ D(x min , h α ), 2h By using the change of coordinates we find where the last equality follows from Assumption (c) in Theorem 1.3. We recall Assumptions (b) and (c) of Theorem 1.3. Then, by the Taylor expansion of φ at x min , we deduce that where λ min > 0 is the lowest eigenvalue of Hess x min φ. Since P n is of degree n, there exists C > 0 such that Using this with (3.9), we get From (3.8) and (3.10), we find ii. Let us now deal with the weighted scalar products related to the Q n,m . Let u ∈ H 2 (Ω) and z 0 ∈ D(z min , h α ). By the Cauchy formula (see [4,Theorem 10.4]) and the Cauchy-Schwarz inequality, With the Taylor formula for u = Q n,m at z min , this gives Using (3.6), this implies (3.13) Using (3.9) and Q n,m ∈ W 1,∞ (Ω) ⊂ L 2 (Ω), we get (3.14) With (3.13) and (3.14), we deduce With the Cauchy-Schwarz inequality, and (3.15), iii. Let us now consider the scalar products involving the P n and the Q n ′ ,m . Using (3.11), (3.16), and the Cauchy-Schwarz inequality, we get (3.17) The conclusion follows by expanding the square in the left-hand-side of (3.5) and by using (3.11), (3.16), (3.17) .
Remark 3.6. From Lemma 3.5, we deduce that the vectors {χw j,h , 0 j k − 1} are linearly independent for h small enough.

3.3.
Estimate of the energy. The aim of this section is to bound from above the energy on an appropriate subspace.
Proof. Let χ be any function satisfying (3.2). We have where we have used that |∇χ| 2 = 4|∂ z χ| 2 since χ is real and ∂ z w h = 0. The proof is now divided into three steps. First, we introduce tubular coordinates near the boundary, then we make an explicit choice of χ, and finally, we control the remainders.
i. We only need to define χ in a neighborhood of Γ = ∂Ω. To do this, we use the tubular coordinates given by the map for t 0 small enough, γ being a parametrization of Γ with |γ ′ (s)| = 1 for all s, and n(s) the unit outward pointing normal at point γ(s) (see e.g. [8, §F]). We let η −1 (x) = (s(x), t(x)) , for all x ∈ η R/ |Γ|Z × (0, t 0 ) , the inverse map to η. We let, for all x ∈ Ω, The parameter ε > 0 and the function ρ are to be determined. We assume that ρ(s, 0) = 0 and ρ(s, t) = 1 when t ε. We will choose ε = o(h 1 2 ). Since the metric induced by the change of variable is the Euclidean metric modulo O(ε), we get Since Q n,m ∈ W 1,∞ (Ω), we have ∂ tQn,m • η ∈ L ∞ (Γ × (0, ε)) and by using the Taylor expansion ofw near t = 0, we get 18) and · ℓ 2 is the canonical Euclidian norm on C k . Then, ii. For the right hand side of (3.19) to be small, we choose ρ to minimize ∂ t ρ far from the boundary. The optimization of gives us the weight ∂ n φ. More precisely, Lemma A.1 with α = 2∂ n φ/h > 0 suggests to consider the trial state defined, for t ε, by ρ(s, t) = 1 − e −2t∂nφ(s)/h 1 − e −2ε∂nφ(s)/h , and by 1 otherwise. By Lemma A.1, we get We can choose ε = h| log h| so that .
is well defined and continuous (since the degree of P j is j). Note, in particular, that where c h is defined in (3.18). Since N H is a norm, and recalling Remark 3.6, we see that the application N h is a norm when h ∈ (0, h 0 ]. N 0 is also a norm (as we can see by using the Hardy norm and Q j,m ∈ H 2 k (Ω)). Let us define so that, for all h ∈ [0, h 0 ], and all c ∈ C k , Let us now estimate N H (w h ). From the triangle inequality, we get Then, from degree considerations and the triangle inequality, we get, for 1 j k − 2, Then, This ends the proof. (3.4) and χ h in Lemma 3.7. By Lemmas 3.5 and 3.7, we get

Proof of Proposition 3.1. Let us define
Taking the limit m → +∞, it follows lim sup h→0 h k−1 e −2φ min /h λ k (h) C sup (k) .

3.5.
Computation of C sup (k) in the radial case. Let k ∈ N * . Let us assume that Ω is the disk of radius R centered at 0, and that B is radial. In this case x min = 0, ∂ n φ is constant and Hess x min φ = B(0)Id/2. Thus, , and we notice that P n (z) = z n (see Notation 4) so that Note that this formula extends the upper bound obtained in [11] for constant magnetic fields on the disc.

On the magnetic Cauchy-Riemann operators
In this section, U will denote an open bounded subset of R 2 . It will be either Ω itself, or a smaller open set.
As we already observed (see (1.3)), the Dirichlet-Pauli operator, considered only as a differential operator, is the square of the magnetic Dirac operator σ · (p −A). It can be written as where d h,A and d × h,A are the magnetic Cauchy-Riemann operators:

4.1.
Properties of d 1,0 and d * 1,0 . In this part, we study the operators d h,A and d * h,A in the non-magnetic case B = 0 with h = 1 in order to get describe their properties in this simplified setting in which −∆ = d * 1,0 d 1,0 . Various aspects of this section can be related to the spectral analysis of the "zig-zag" operator (see [16]). The next section will be related to the magnetic case that is needed in our study.
and d * 1,0 acts as d × 1,0 . In particular, and there exists C > 0 such that, for all v ∈ ker(d * . Proof. Let u ∈ Dom(d 1,0 ) = H 1 0 (U; C). One easily checks that d 1,0 u 2 L 2 (U ) = ∇u 2 L 2 (U ) . Hence, the Poincaré inequality ensures that (d 1,0 , Dom(d 1,0 )) is a closed operator with closed range. Then, by definition of the domain of the adjoint, By density, this equality can be extended to w ∈ H 1 0 (U; C). This shows, by definition, , where the last equality follows from the elliptic regularity of the Laplacian. In particular, we get, for all w ∈ H 1 .

4.2.
Properties of d h,A and d * h,A . Let us introduce some notations related with the Riemann mapping theorem.
In the following, we gather some standards properties related with d h,A and d * h,A . We will use the following lemma.
These formulas can be extended to u ∈ H 1 0 (U; C).

Proof. It follows from integrations by parts and the fact that
The extension to u ∈ H 1 0 (U; C) follows by density. Remark 4.3. From Lemma 4.2, we deduce 4 , that for all u ∈ H 1 0 (U; C), and . Definition 4.5. We define the self-adjoint operators (L ± h , Dom(L ± h )) as the operators acting as on the respective domains

4.3.
Semiclassical elliptic estimates for the magnetic Cauchy-Riemann operator.
Note that Ω δ is actually an analytic manifold.
The following theorem is a crucial ingredient in the proof of the lower bound of λ k (h). It is intimately related to the spectral supersymmetry of Dirac operators [17, Theorem 5.5 and Corollary 5.6].
, where we used Notation 6. Theorem 4.6 follows from the following two lemmas.
Proof. For notational simplicity, we let U = Ω δ and we write d h, Remark 4.3). Thus, we get L + h w L 2 (U ) 2hB 0 w L 2 (U ) . By integration by parts and the Cauchy-Schwarz inequality, we have This ensures that and the conclusion follows.
For notational simplicity, we let U = Ω δ and we write d h,A for d h,A,U .
With the same notations as in the proof of Lemma 4.7 (u = d h,A w), we have Using Assumption (a), we get 4) so that using (4.3) and (4.4), there exists C > 0 such that .
Since A is bounded, . Thus, . (4.5) ii. Let us now deal with the derivatives of order two. From the explicit expression of Taking the L 2 -norm and using (4.5), we get .
Using a standard ellipticity result for the Dirichlet Laplacian, we find . (4.6) The uniformity of the constant with respect to δ ∈ (0, δ 0 ) can be checked in the classical proof of elliptic regularity. Alternatively, using Riemann mapping theorem, we send Ω on the unit disk. Then, we perform a change of scale for each δ to send D(0, 1 − δ) onto D(0, 1) and use a standard ellipticity result on D(0, 1). Here, δ appears as a regular parameter in the coefficients of the elliptic operator. Note that , and, since u = d h,A w, . (4.7) iii. A classical trace result combined with (4.7) and Lemma 4.7 gives , where it can again be checked using the same techniques that C does not depend on δ ∈ (0, δ 0 ).

Lower bounds
The aim of this section is to establish the following proposition.
Proposition 5.1. Assume that Ω is C 2 and satisfies Assumption 1.1. There exists a constant θ 0 ∈ (0, 1] such that for all k ∈ N * , lim inf If Ω = D(0, 1) and B is radial, we have

5.1.
Inside approximation by the zero-modes. Let k ∈ N * . Let us consider an orthonormal family (v j,h ) 1 j k (for the scalar product of L 2 (e −2φ/h dx)) associated with the eigenvalues (λ j (h)) 1 j k . We define In this section, we will see that the general upper bound proved in the last section implies that all v h ∈ E h wants to be holomorphic inside Ω.
This result will be used in the proof of Lemma 5.3 to compute the weighted L 2 norm of v h on Ω in term of its weighted L 2 norm on a shrinking neighborhood of x min .
Proof. We have λ k (h) = h −k+1 O(e 2φ min /h ) (see Proposition 3.1). By using the orthogonality of the v j,h , one gets Since v h satisfies the Dirichlet boundary condition and by integration by parts, we find It remains to use the Poincaré inequality.
We can now prove a concentration lemma.
where δ 0 is defined in Proposition 4.4.
Proof. Let us remark that the second limit is a consequence of the first one. We have By (3.9) and Lemma 5.2, we deduce that and the conclusion follows.

5.1.2.
Interior approximation. Now that we know that v h is localized inside Ω, let us explain why it is close to be a holomorphic function.
Notation 7. Let us denote by Π h,δ the orthogonal projection on the kernel of −i∂ z (i.e. the SegalBargmann functions on Ω δ which is defined in Theorem 4.6) for the L 2 -scalar product ·, e −2φ/h · L 2 (Ω δ ) .
We notice that, if where Π h,A,Ω δ was defined in Notation 5.
Proof. For all v h ∈ E h , we have where we used Lemma 5.3 to get the last inequality.
Let v h ∈ E h be such that Π h,δ v h = 0. Recalling Proposition 3.1, we have and v h = 0 on Ω δ so that Π h,δ is injective on E h and (c) follows.
Note that the Szegö projection preserves the L 2 holomorphic functions.
Notation 9. We let E := min where F , c 1 are defined in Notation 6.

By (5.4) and Lemma 5.3,
Thus, coming back to Ω δ (without forgetting the Jacobian of F ), Then, by using the (weighted) Hardy norm, we have iii. Using Proposition 5.4 and Lemma 5.3, we get Combing this with (5.7) and Proposition 3.1, we find iv. Using the Taylor expansion of φ at x min , we get, for all for x ∈ D(x min , h α ) , and the conclusion follows.
Remark 5.8. Lemma 5.6 shows in particular that .
In the next section, we will essentially provide a lower bound ofλ k (h). Note that if we could replace H 2 (Ω δ ) by the set of polynomials, then, we would get the bound presented in Remark 3.2. Nevertheless, there is no hope to do so since in general, , (This inequality is an equality in the radial case). We still have to work to get the lower bound of Theorem 1.3.

5.3.
Reduction to a polynomial subspace: Proof of Proposition 5.1. We can now prove Proposition 5.1.
Since Ω is regular enough, the Riemann mapping theorem ensures that The conclusion follows.

Appendix A. A unidimensional optimization problem
The goal of this section is to minimize, for each fixed s, the quantity, This leads to the following lemma. Proof. i. Since α > 0, we have that F α,ε (ρ) ε 0 |ρ ′ (ℓ)| 2 dℓ for all ρ ∈ V . There exists C > 0 such that, for all ρ ∈ V , This ensures that any minimizing sequence (ρ n ) n∈N ⊂ V is bounded in H 1 (I) and any H 1 -weak limit is a minimizer of inf{F α,ε (ρ) , ρ ∈ V }. ii. F 1/2 α,ε is an euclidian norm on V so that F α,ε is strictly convex and the minimizer is unique. iii. At a minimum ρ, the Euler-Lagrange equation is (e αℓ ρ ′ ) ′ = 0 .