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Diamagnetism versus Robin condition and concentration of ground states

We estimate the ground state energy for the magnetic Laplacian with a Robin condition. In a special asymptotic limit, we find that the magnetic field does not contribute to the two-term expansion of the ground state energy, thereby proving that the Robin condition weakens diamagnetism. We discuss a semi-classical version of the operator and prove that the ground states concentrate near the boundary points of maximal curvature.


Introduction
This paper is motivated by two different questions. The first question concerns the analysis of the ground state energy and the concentration of the ground states for the magnetic Laplacian with a Robin condition and a semi-classical parameter, and is a continuation of the work in [9,10]. The second question is around the influence of the Robin condition on diamagnetism. We will find that these two questions are intimately related and we will get satisfactory answers for both. Besides the concentration of the ground states near the points of maximal curvature, we will identify the optimal strength of the Robin condition/magnetic field such that diamagnetism occurs to leading order of the energy.
The results in this paper are valid in an open set Ω ⊂ R 2 . We will assume that the boundary Γ = ∂Ω of Ω is C 3 smooth, compact and consists of a finite number of connected components. Our assumptions allow for Ω to be an interior or exterior domain, and the smoothness of the boundary ensures the existence of a normal vector every where on the boundary. We will denote by ν the unit outward normal vector (field) of ∂Ω.
1.1. Concentration of ground states. Let us introduce the semi-classical magnetic Laplacian that we will study. Consider the magnetic potential: We are interested in the same magnetic Laplacian studied in [10] which involves four parameters, the strength of the magnetic field b > 0, the semi-classical parameter h > 0, two parameters γ ∈ R \ {0} and α ∈ R that will serve in defining the boundary condition. The operator is The parameters α and γ serve in controlling the 'strength' of the boundary condition in (1.1.5).
As we shall see, the sign of γ and the values of α have a strong influence on the spectrum of the magnetic Laplacian in (1.1.3). Notice that γ = 0 corresponds to the extensively studied magnetic Laplacian with Neumann condition (cf. [5,2]), while b = 0 corresponds to the Laplacian without a magnetic field. That justifies the assumption γ = 0 and b > 0. The ground state energy (lowest eigenvalue) of the operator in (1.1.3) is: . (1.1.7) We will study the asymptotic limit where the semi-classical parameter h tends to 0 + , while the parameters b, α and γ are assumed fixed. In this regime, we can reduce to the case b = 1 simply by observing that, for all b > 0, 1.8) and that as long as we assume b fixed, b −1 h → 0 when h → 0 + . The results in [10] distinguish between two situations. The first one corresponds to α ≥ 1/2 and is fairly understood: A two term asymptotic expansion of the ground state energy in (1.1.7) is established; the ground state energy is in the discrete spectrum (cf. [8]); and the ground states are localized near the boundary points where the curvature is maximal.
The second situation corresponds to α < 1 2 and is less understood. Here the sign of γ will play a dominant role. The contribution in this paper will clarify the situation when γ < 0. For γ > 0, the ground state energy satisfies For γ < 0, the behavior of the ground state energy is completely different and displayed as follows, Note that this asymptotic expansion does not involve the strength of the magnetic field b. Again, the ground state energy is an eigenvalue, as long as the semi-classical parameter h is sufficiently small (cf. [8]). In [10], it is proved that the ground states concentrate near the boundary (when α < 1 2 and γ < 0). The natural question is then weather one can refine the concentration near some special boundary points, e.g. points of maximal curvature. We will give an affirmative answer to this question in Theorem 1.1 below.
In the statement of Theorem 1.1, κ max is the maximum of the curvature along the boundary, n ∈ N is the smallest positive integer satisfying and e n (ζ) is the quantity that we will introduce in (2.2.5) below. As ζ → 0 + , e n (ζ) behaves like 1 4 ζ 2 (cf. Remark 2.6). Now we are ready to state: Theorem 1.1. Suppose that b > 0, α < 1 2 and γ < 0.
(1) As h → 0 + , the ground state energy in (1.1.7) satisfies where Notice that the asymptotic expansions for µ 1 (h; b, α, γ) are compatible in the cases α = 1 3 and α < 1 3 . Formally, we get the expansion for α < 1 3 by taking γ → −∞ in the case α = 1 3 . Theorem 1.1 adds two improvements to the results in [10] by • establishing a two-term expansion for the ground state energy; • refining the concentration of the ground states near the points of maximal curvature.
Remark 1.2. The magnetic field is assumed constant in Theorem 1.1, but the methods in the this paper should allow for dealing with non-constant C 1 magnetic fields.
1.2. Diamagnetism. Here we will discuss the question of diamagnetism. We will find that imposing a Robin condition may slow diamagnetism (and even neglect this effect). Let β ∈ R, H ≥ 0 and L β (H) be the self-adjoint operator in L 2 (Ω), (Ω) and ν · (∇ − iHA 0 )u + βu = 0 on ∂Ω} . Here the magnetic Sobolev space H 1 HA 0 (Ω) is introduced in (1.1.6). Note that, for H = 0, L β (0) is the Robin Laplacian, while for H > 0, L β (H) is the magnetic Laplacian with a (magnetic) Robin condition. Let σ L β (H) be the spectrum of the operator L β (H). We introduce the ground state energy,μ (1.2. 2) The diamagnetic inequality yields, for all β ∈ R and H > 0, In physical terms, this inequality refers to diamagnetism. It simply says that introducing a magnetic field increases the ground state energy. We will see that, when β → −∞, diamagnetism is weak in the sense that the differenceμ 1 (β; H) −μ 1 (β; 0) is small. This property is a unique feature for the Robin condition as it fails for the Neumann and Dirichelt conditions. On the contrary, in simply connected domains, a Neumann boundary condition induces strong diamagnetism (cf. [2]). The asymptotic analysis of the spectrum of the Robin Laplacian is studied in many papers, cf. [1,7,6,12,13,14]. In particular, as β → −∞, the ground state energy satisfies, µ 1 (β; 0) = −β 2 + βκ max + βo(1) . (1.
In the situation where H ≈ β σ and σ < 2, diamagnetism occurs as a correction term and will compete with the correction term coming from the curvature of the boundary. According to Theorem 1.3: • If 3 2 < σ < 2, then diamagnetism occurs in the second correction term while the influence of the curvature occurs in the third correction term ; • If σ = 3 2 , both diamagnetism and curvature corrections appear in the second correction term ; • If σ < 3 2 , dia-magnetism is weak and its contribution is negligible compared to the contribution of the curvature correction term (compare with (1.2.4)). Through this paper, the following notation will be used. C denotes a constant independent from the semi-classical parameter h. O(h ∞ ) is a quantity satisfying that, for all N ∈ N, there exist two constants h 0 ∈ (0, 1) and C N > 0 such that, for all h ∈ (0, h 0 ), The paper is organized as follows. In Section 2, we analyze three auxiliary differential operators useful to prove Theorem 1.1. The proof of Theorem 1.1 occupies all of Section 3. Finally, in Section 4, we explain how to get the result in Theorem 1.3 from the existing results on the semi-classical magnetic Laplacian with a Robin condition, in particular those in Theorem 1.1.

Analysis of auxiliary operators
2.1. 1D Laplacian on the half line. Here we introduce a simple 1D operator that will play a fundamental role in the next sections. This operator arises naturally in the analysis of the Robin Laplacian without magnetic field (cf. [12,7]). The operator is The spectrum of this operator is {−1} ∪ [0, ∞), and −1 is a simple eigenvalue with the L 2 normalized eigenfunction u 0 (τ ) = √ 2 exp(−τ ) .
Here ζ > 0 and ξ ∈ R are two parameters. Let us denote by (λ n (H harm (ζ, ξ))) the increasing sequence of the eigenvalues of H harm (ζ, ξ) counting multiplicities. We will study the aymptotic behavior of the eigenvalue as ζ → 0.
By comparison with the operator in (2.1.1), we get: For all ζ > 0 and ξ ∈ R, it holds, The lower bound in Lemma 2.1 can be improved as follows: There exists a universal constant A 0 > 0 such that, if 0 < ζ < 1 and |ξ| ≥ A 0 ζ, then Proof. Let u be an L 2 normalized ground state of the operator H harm (ζ, ξ). Let us write, We know that the lowest eigenvalue of the operator in (2.1.1) is −1. Let µ(A) be the eigenvalue of the operator , with the boundary condition u ′ (0) = −u(0). Now, it results from the min-max principle that When |ζ −1 ξ| is sufficiently large, we get µ(ζ −1 ξ) ≥ 1 2 , which is a result of the following two facts proved in [10], lim We will prove that: , and a collection of vector functions, Furthermore,
Let us define For simplicity of the notation, we will write The remainder R 1 is We choose the coefficients and the functions in (2.2.3) so that all the terms on the left side vanish. This is possible since the operator − d 2 dτ 2 + 1 can be inverted in the orthogonal complement of the eigenfunction u 0 . That way we choose, Step 2: The iteration process. Suppose that we have constructed (µ j ) n j=1 and (u j ) n j=1 such that for a collection (v n,p ) of Schwartz functions that do not depend on ζ and ξ, and the function f n,ζ,ξ satisfies, where C is a constant independent of ζ and ξ.
We expand (H − λ n+1 )w n+1 and rearrange the terms in the form, where the functions v n+1,p and g n+1,p are expressed in terms of the functions u j,q and the real numbers µ j,q .
All what we have to do now is to select the functions u n+1,p+1 and the real numbers µ n+1,p+1 such that To that end, we select µ n+1,p+1 such that, Finally, we define the function u n+1,p+1 as follows, As a consequence of Theorem 2.3, Lemma 2.1 and the spectral theorem, we get: Definition 2.5. Let n ∈ N and A 0 be the universal constant in Lemma 2.2. For all ζ ∈ (0, 1), we define the following quantity

2.3.2.
Lower bound for the principal eigenvalue of the operator H ζ,β,ξ,h . In the next two propositions, we determine refined lower bounds of the eigenvalue λ 1 (H ζ,β,ξ,h ). The bound is valid as h → 0 + and is uniform with respect to the parameters ξ, ζ and β.

Analysis of the semi-classical Laplacian with a weak magnetic field
3.1. Semi-classical Laplacian with weak magnetic field. We will introduce a new semiclassical magnetic Laplacian but with a Robin condition not involving the parameters α and γ.
These two parameters will be absorbed by a new (small) parameter ζ.
For h > 0 and ζ > 0, we introduce the operator Let σ(P h,ζ ) be the spectrum of the operator P h,ζ . We introduce the ground state energy, There is a relationship between the ground state energies in (1.1.7) and (3.1.4) displayed as follows: Now Theorem 1.1 follows from: It holds the following.
The proof of the item (4) in Theorem 3.1 follows from Theorems 3.3 and 3.9.
3.2. Boundary coordinates. We will perform various computations of trial functions supported in a tubular neighborhood of the boundary. To single out the influence of the boundary curvature, we need a special coordinate system displaying the arc-length along the boundary and the normal distance to the boundary. We will refer to such coordinates as boundary coordinates. These are the same coordinates used in the semi-classical analysis of the magnetic Laplcian (cf. [5,2]). The boundary coordinates are valid in every connected component of the boundary. For simplicity, we will suppose that ∂Ω has one connected component; if more than one connected component exists, then we use the coordinates in each connected component independently. Let The smoothness of the boundary yields the existence of a constant t 0 > 0 such that, upon defining In the new coordinates, the components of the vector field A 0 are given as follows, The new magnetic potential A 0 = ( A 1 , A 2 ) satisfies, For all u ∈ L 2 (V δ ), we assign the function u defined in the new coordinates as follows u(s, t) := u(Φ(s, t)). Consequently, for all u ∈ H 1 (V t 0 ), we have, with u = u • Φ,

2.5) and
Vt 0 ∩∂Ω Finally, we recall a useful gauge transformation that we borrow from [5,2]. Let x 0 ∈ ∂Ω and V x 0 be a neighborhood of x 0 in Ω. There exists a smooth function φ x 0 in Φ −1 (V x 0 ) such that, in the boundary coordinates, 3.3. Upper bound for the principal eigenvalue. In the rest of this paper, we will use the following notation. For all ǫ > 0 and For all ζ ∈ (0, 1), define where n ∈ N is the smallest positive integer satisfying (2n + 2)ǫ > 1 2 . Proposition 3.2. Under the assumptions in Theorem 3.1, there exist two constants C > 0 and h 0 ∈ (0, 1) such that, for all h ∈ (0, h 0 ), the ground state energy in (3.1.4) satisfies, Here b ǫ (ζ) is as in (3.3.1).
Proof. The proof consists of constructing a trial function v h (x) and computing its energy. This trial function will be defined via the boundary coordinates (s, t) in (3.2.1). Select x 0 ∈ ∂Ω such that κ(s(x 0 )) = κ max is equal to the maximal curvature. We may choose the coordinates (s, t) in (3.2.1) such that s(x 0 ) = 0. Let V x 0 be a neighborhood of the point x 0 in Ω, and ϕ 0 = ϕ x 0 be the function defined in V x 0 and satisfying (3.2.7). The construction of the trial function v h and the computation of its energy will be done for the cases ǫ < 1 4 , ǫ = 1 4 and ǫ > 1 4 independently. The case ǫ < 1 4 . Let n ∈ N be the largest positive integer such that (2n + 2)ǫ > 1 2 . Recall the definition of f n (ζ, ξ) in (2.2.6). Select ξ n = ξ n (ζ) such that f n (ζ, ξ n ) = e n (ζ) = min{f n (ζ, ξ) : |ξ| ≤ ζ max(A 0 , 1)} and |ξ n | ≤ ζ max(A 0 , 1) .
The trial function v h is defined using the (s, t)-coordinates and the relation in (3.2 The upper bound in Proposition 3.2 follows from the min-max principle and the following two estimates:

3.5)
where r * = 1 + min (2n + 2)ǫ, 1 2 + ǫ 2 is given in (3.3.2). The estimate in (3.3.4) is easy to obtain in light of the expression of v h and the formula in (3.2.5). For the estimate in (3.3.5), notice that, after expressing the operator P ζ,h in the boundary coordinates (s, t), we get (compare with (3.2.4)) Note that,
Returning back to the t variable then integrating with respect to the s variable, we get, Now, we insert this and (3.3.9) into (3.3.6) to get, To finish the proof, we notice that κ max = κ(0) and in the support of the function u h , we have |κ(s) − κ(0)| ≤ Ch 1/8 .
The case ǫ = 1 4 . Now the trial function v h is defined using the (s, t)-coordinates and the relation in where the constant c and the function χ 1 are as in (3.3.3), and w 1 is the function defined in Theorem 2.3. Performing a calculation similar to the one done for the case ǫ < 1 4 (in particular, using (2.3.7) for δ = 1 16 and β = κ(s)) we get, The min-max principle now yields the desired upper bound for λ 1 (h, ζ). The case ǫ > 1 4 . Here we simply take the same trial state for the case without a magnetic field but times a phase (cf. [7,12]). Precisely, we define v h as follows, where the constant c and the function χ 1 are as in (3.3.3) and u 0 (τ ) = √ 2 exp(−τ ). Easy calculations similar to those done for ǫ < 1 4 give us (cf. [7,12]) The min-max principle now yields the upper bound for λ 1 (h, ζ).
Proof of Theorem 3.3. The proof is similar to that of Theorem 5.1 in [7]. Let t(x) = dist(x, ∂Ω) and Φ(x) = exp( α t(x) h 1/2 ). We perform an integration by parts to write the following identity, Consider a partition of unity of R Define . Associated with this partition of unity, we have the simple standard decomposition where (by Lemma 3.4)
Step 1. Localization near the boundary. Consider a partition of unity of R, We have the decomposition and by (3.5.5), Thus, Step 2. Analysis near the boundary. Let us cover the boundary ∂Ω by a family of open disks (B(x j , h 1/8 )). Let (f j ) ⊂ C ∞ (∂Ω) be a partition of unity in ∂Ω such that, for all j, We extend f j in the tubular neighborhood {dist(x, ∂Ω) < 2h ρ } of the boundary via the formula We decompose the boundary term in (3.5.7) as follows, (3.5.8) We will write a lower bound for each term q h,ζ (f j χ 1,h w) as follows. First, let us denote by κ j = κ(s(x j )) .
By smoothness of the scalar curvature and boundedness of the boundary, we know that That way we get the following pointwise lower bound in every B(x j , h σ ), Let φ j = φ x j be the function satisfying (3.2.7) in B(x j , 2h σ + 2h ρ ). Define the function v j =f jχ1,h w e −iϕ j . We express the quadratic form q h,ζ (f j χ 1,h w) in boundary coordinates and then we use the aforementioned inequalities to write, Note that, for t ∈ (0, h 1 2 −ρ ) and τ = h − 1 2 t, |∆ β,τ | ≤ C(|β| + 1)τ . Thus, we can apply the results in Sec. 2.3.
An important ingredient in the proof of Theorem 3.9 is: Lemma 3.10. Let ǫ > 0 and ρ be as in (3.5.2). Let ρ ′ ∈ (ρ, 1 2 ). There exist two constantsC > 0 and h 0 ∈ (0, 1) such that, for all h ∈ (0, h 0 ) and u in the form domain of the operator P h,ζ , and q h,ζ is the quadratic form in (3.1.3).
Proof. Letμ be the ground state energy of the operator P h,ζ − V h,ζ . We will prove thatμ > 0.