Diamagnetism versus Robin condition and concentration of ground states

Abstract

We estimate the ground state energy for the magnetic Laplacian with a Robin boundary 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 boundary 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.

Keywords

magnetic Laplacian Robin boundary condition semi-classical analysis

1. Introduction

This paper is motivated by two different questions. The first question is around the influence of the Robin boundary condition on diamagnetism. The second 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]. 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 $Ω \subset R^{2}$ . We will assume that the boundary $Γ = \partial Ω$ 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 $\partial Ω$ .

Since the boundary is assumed smooth, there exists a geometric constant $t_{0} \in (0, 1)$ such that, if $x \in R^{2}$ satisfies $dist (x, \partial Ω) < t_{0}$ , then we may assign a unique point $p (x) \in \partial Ω$ such that $dist (x, p (x)) = dist (x, \partial Ω)$ . The function $κ (\cdot)$ denotes the curvature along the boundary and $κ_{max} = {max}_{x \in \partial Ω} κ (x)$ .

1.1. Diamagnetism

In this section, we will discuss the question of diamagnetism. We will find that imposing a Robin boundary condition may slow diamagnetism (and even neglect this effect).

Let us introduce the magnetic Laplacian that we will study. First, consider the magnetic potential: $\begin{matrix} (1.1) & R^{2} ∋ (x_{1}, x_{2}) \mapsto A_{0} (x_{1}, x_{2}) = (- x_{2}, 0) . \end{matrix}$ This magnetic potential generates the constant magnetic field: $\begin{matrix} (1.2) & B : = curl A_{0} = 1 . \end{matrix}$ Let $β \in R$ , $H ⩾ 0$ and $L^{β} (H)$ be the self-adjoint operator in $L^{2} (Ω)$ , $\begin{matrix} (1.3) & L^{β} (H) = - {(\nabla - i H A_{0})}^{2} \end{matrix}$ with domain $\begin{matrix} D (L^{β} (H)) = {u \in H_{H A_{0}}^{1} (Ω) : {(\nabla - i H A_{0})}^{2} u \in L^{2} (Ω) and ν \cdot (\nabla - i H A_{0}) u + β u = 0 on \partial Ω} . \end{matrix}$ Here, for $t ⩾ 0$ , the magnetic Sobolev space $H_{t A_{0}}^{1} (Ω)$ is introduced as follows $\begin{matrix} (1.4) & H_{t A_{0}}^{1} (Ω) = {u \in L^{2} (Ω) : (\nabla - i t A_{0}) u \in L^{2} (Ω)} . \end{matrix}$ For $t = 0$ , $H_{t A_{0}}^{1} (Ω)$ is simply the Sobolev space $H^{1} (Ω)$ . The same is true when the domain Ω is bounded.

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 boundary condition. Let $σ (L^{β} (H))$ be the spectrum of the operator $L^{β} (H)$ . We introduce the ground state energy, $\begin{matrix} (1.5) & {\tilde{μ}}_{1} (β; H) = inf σ (L^{β} (H)) . \end{matrix}$ The diamagnetic inequality yields, for all $β \in R$ and $H > 0$ , $\begin{matrix} (1.6) & {\tilde{μ}}_{1} (β; H) - {\tilde{μ}}_{1} (β; 0) ⩾ 0 . \end{matrix}$ 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 $β \to - \infty$ , diamagnetism is weak in the sense that the difference ${\tilde{μ}}_{1} (β; H) - {\tilde{μ}}_{1} (β; 0)$ is small. This property is a unique feature for the Robin condition as it fails for the Neumann and Dirichelt boundary conditions. On the contrary, in simply connected domains, a Neumann boundary condition induces strong diamagnetism (cf. [4]).

The asymptotic analysis of the spectrum of the Robin Laplacian is studied in many papers, cf. [2,5,6,8,15–17]. In particular, as $β \to - \infty$ , the ground state energy satisfies, $\begin{matrix} (1.7) & {\tilde{μ}}_{1} (β; 0) = - β^{2} + β κ_{max} + β o (1) . \end{matrix}$ When the strength of the magnetic field is a fixed constant (i.e. the parameter H is independent of β), the diamagnetic inequality and the min–max principle yield that the asymptotic expansion in (1.7) remains true for ${\tilde{μ}}_{1} (β; H)$ . However, in the limit $β \to - \infty$ , it is natural to allow for H to depend on β.

The strong field limit $H \to \infty$ and β fixed is analyzed in [14]. However, this regime does not deviate from the known results for the Neumann operator where $β = 0$ (cf. [3,10]).

We will write an asymptotic expansion for the magnetic ground state energy ${\tilde{μ}}_{1} (β; H)$ valid when $β \to - \infty$ and $H \to \infty$ simultaneously. This is the content of Theorem 1.1 below. In particular, we will get a fair knowledge about the difference in (1.6) which measures the strength of diamagnetism.

In the statement of Theorem 1.1, two spectral quantities will appear. These are the two real-valued functions $Θ (\cdot)$ and $e_{n} (\cdot)$ that we will define below.

The function $Θ (\cdot)$ is introduced in [10] and defined as follows. For $γ \in R$ , define the ground state energy $\begin{matrix} (1.8) & Θ (γ) = inf_{ξ \in R} (inf_{u \in B^{1} (R_{+}) ∖ {0}} \frac{\int_{0}^{\infty} (| u^{'} (t) |^{2} + | (t - ξ) u |^{2}) d t + γ | u (0) |^{2}}{‖ u ‖_{L^{2} (R_{+})}^{2}}), \end{matrix}$ where $\begin{matrix} B^{1} (R_{+}) = {u \in L^{2} (R_{+}) : u^{'}, t u \in L^{2} (R_{+})} . \end{matrix}$ For every natural number $n ⩾ 1$ , the function $e_{n} (\cdot)$ is defined in (2.9) below. We will not introduce it explicitly at the moment, but we mention one important property of it (cf. Remark 2.6) $\begin{matrix} (1.9) & e_{n} (ζ) = \frac{1}{4} ζ^{2} + O (ζ^{4}) as ζ \to 0 . \end{matrix}$ Let us collect some useful properties of the function $Θ (\cdot)$ :

The function $Θ (\cdot)$ is smooth, increasing and $Θ (γ) > - γ^{2}$ for all $γ ⩽ 0$ (cf. [10]);

$0 < Θ (0) < 1$ (cf. [1,3]).

Finally,

κ_{max}

denotes the maximum of the (signed) curvature on

\partial Ω

Now we are ready to state:

Theorem 1.1.
Let $σ > 0$ , $0 < c_{1} < c_{2}$ and $β_{0} < 0$ . Suppose that $\begin{matrix} β < β_{0} and c_{1} | β |^{σ} ⩽ H ⩽ c_{2} | β |^{σ} . \end{matrix}$ As $β \to - \infty$ , the ground state energy ${\tilde{μ}}_{1} (β; H)$ satisfies:
If $σ > 2$ , $\begin{matrix} {\tilde{μ}}_{1} (β; H) = Θ (0) H + H o (1) . \end{matrix}$

If $σ = 2$ , $\begin{matrix} {\tilde{μ}}_{1} (β; H) = H Θ (β H^{- 1 / 2}) + H o (1) . \end{matrix}$

If $\frac{3}{2} < σ < 2$ , $\begin{matrix} {\tilde{μ}}_{1} (β; H) = - β^{2} + e_{n} (H β^{- 2}) β^{2} + β κ_{max} + β o (1), \end{matrix}$ where n is the smallest positive integer satisfying $\begin{matrix} (n + 1) (2 - σ) > \frac{1}{2} \end{matrix}$ and $e_{n} (\cdot)$ satisfies ( 1.9 ).

If $σ = \frac{3}{2}$ , $\begin{matrix} {\tilde{μ}}_{1} (β; H) = - β^{2} + (\frac{H^{2} β^{- 3}}{4} + κ_{max}) β + β o (1) . \end{matrix}$

If $σ < \frac{3}{2}$ , $\begin{matrix} {\tilde{μ}}_{1} (β; H) = - β^{2} + κ_{max} β + β o (1) . \end{matrix}$

Theorem 1.1 suggests that, in the limit $β \to - \infty$ , diamagnetism occurs to leading order when the strength of the magnetic field satisfies $H \approx β^{σ}$ and $σ ⩾ 2$ .

In the situation where $H \approx β^{σ}$ 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 1.1:
If $\frac{3}{2} < σ < 2$ , then diamagnetism occurs in the second correction term while the influence of the curvature occurs in the third correction term;

If $σ = \frac{3}{2}$ , both diamagnetism and curvature corrections appear in the second correction term;

If $σ < \frac{3}{2}$ , dia-magnetism is weak and its contribution is negligible compared to the contribution of the curvature correction term (compare with (1.7)).

1.2. Concentration of ground states

Here, 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 $γ \in R ∖ {0}$ and $α \in R$ that will serve in defining the boundary condition. The operator is $\begin{matrix} (1.10) & L_{h, b, Ω}^{α, γ} = - {(h \nabla - i b A_{0})}^{2} in L^{2} (Ω), \end{matrix}$ with a boundary condition of the third type (Robin condition) $\begin{matrix} (1.11) & ν \cdot (h \nabla - i b A_{0}) u + h^{α} γ u = 0 on \partial Ω . \end{matrix}$ The magnetic potential $A_{0}$ is introduced in (1.1). This operator can be defined via Friedrich’s Theorem and the closed semi-bounded quadratic form, $\begin{matrix} (1.12) & Q_{h, b, Ω}^{α, γ} (u) : = {‖ (h \nabla - i b A_{0}) u ‖}_{L^{2} (Ω)}^{2} + h^{1 + α} γ \int_{\partial Ω}^{} {| u (x) |}^{2} d x \end{matrix}$ defined on the ‘magnetic’ Sobolev space $H_{h^{- 1} b A_{0}}^{1} (Ω)$ introduced in (1.4).

The parameters α and γ serve in controlling the ‘strength’ of the boundary condition in (1.12). 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 (1.10). Notice that $γ = 0$ corresponds to the extensively studied magnetic Laplacian with Neumann condition (cf. [4,7]), while $b = 0$ corresponds to the Laplacian without a magnetic field. That justifies the assumption $γ \neq 0$ and $b > 0$ .

The ground state energy (lowest eigenvalue) of the operator in (1.10) is: $\begin{matrix} (1.13) & μ_{1} (h; b, α, γ) = inf_{u \in H_{h^{- 1} b A_{0}}^{1} (Ω) ∖ {0}} \frac{Q_{h, b, Ω}^{α, γ} (u)}{‖ u ‖_{L^{2} (Ω)}^{2}} . \end{matrix}$ 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 $b > 0$ , $\begin{matrix} (1.14) & μ_{1} (h; b, α, γ) = b^{2} μ_{1} (b^{- 1} h; b = 1, α, b^{- 1 + α} γ), \end{matrix}$ and that as long as we assume b fixed, $b^{- 1} h \to 0$ when $h \to 0_{+}$ .

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 (1.13) is established; the ground state energy is in the discrete spectrum (cf. [12,17]); and the ground states are localized near the boundary points where the curvature is maximal.

The second situation corresponds to $α < \frac{1}{2}$ 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 $\begin{matrix} μ_{1} (h; b, α, γ) = b h + h o (1) (h \to 0_{+}) . \end{matrix}$ For $γ < 0$ , the behavior of the ground state energy is completely different and displayed as follows, $\begin{matrix} μ_{1} (h; b, α, γ) = - γ^{2} h^{2 α} + h^{2 α} o (1) (h \to 0_{+}) . \end{matrix}$ 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. [12]). In [10], it is proved that the ground states concentrate near the boundary (when $α < \frac{1}{2}$ and $γ < 0$ ). The natural question is then whether 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 1.2 below.

In the statement of Theorem 1.2, $κ_{max}$ is the maximum of the curvature along the boundary, $\begin{matrix} ζ = \frac{b}{γ^{2}} h^{1 - 2 α}, \end{matrix}$ $n \in N$ is the smallest positive integer satisfying $\begin{matrix} (1.15) & (n + 1) \frac{1 - 2 α}{1 - α} > \frac{1}{2}, \end{matrix}$ and $e_{n} (ζ)$ is the quantity that we will introduce in (2.9) below. As $ζ \to 0_{+}$ , $e_{n} (ζ)$ satisfies (1.9).

Now we are ready to state:

Theorem 1.2.
Suppose that $b > 0$ , $α < \frac{1}{2}$ and $γ < 0$ .
As $h \to 0_{+}$ , the ground state energy in ( 1.13 ) satisfies $\begin{matrix} μ_{1} (h; b, α, γ) = \{\begin{matrix} - γ^{2} h^{2 α} + e_{n} (ζ) γ^{2} h^{2 α} + γ κ_{max} h^{1 + α} + h^{1 + α} o (1) & if \frac{1}{3} < α < \frac{1}{2}, \\ - γ^{2} h^{2 / 3} + (\frac{b^{2}}{4 γ^{2}} + γ κ_{max}) h^{4 / 3} + h^{4 / 3} o (1) & if α = \frac{1}{3}, \\ - γ^{2} h^{2 α} + γ κ_{max} h^{1 + α} + h^{1 + α} o (1) & if α < \frac{1}{3} . \end{matrix} \end{matrix}$

Suppose that the boundary of Ω is $C^{4}$ smooth. There exist constants $ϵ > 0$ , $ρ > 0$ , $η > 0$ and $h_{0} > 0$ such that, for all $h \in (0, h_{0})$ , every $L^{2}$ -normalized ground state $u_{h}$ of ( 1.13 ) satisfies $\begin{matrix} \int_{Ω_{bnd}} | u_{h} |^{2} d x ⩽ exp (- h^{- ϵ}), \int_{Ω_{int}} | u_{h} |^{2} d x ⩽ exp (- h^{- ϵ}), \end{matrix}$ where $\begin{matrix} \begin{matrix} Ω_{int} = {x \in Ω : dist (x, \partial Ω) ⩾ h^{ρ}} and \\ Ω_{bnd} = {x \in Ω ∖ {\overline{Ω}}_{int} : κ_{max} - κ (p (x)) ⩾ h^{η}} . \end{matrix} \end{matrix}$

Notice that the asymptotic expansions for $μ_{1} (h; b, α, γ)$ are compatible in the cases $α = \frac{1}{3}$ and $α < \frac{1}{3}$ . Formally, we get the expansion for $α < \frac{1}{3}$ by taking $γ \to - \infty$ in the case $α = \frac{1}{3}$ .

Theorem 1.2 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 1.3.
The magnetic field is assumed constant in Theorem 1.2, but the methods in this paper should allow for dealing with non-constant $C^{1}$ magnetic fields.

1.3. The Robin-magnetic Laplacian and the semi-classical parameter

At the first glance, the presence of the semi-classical parameter h (through the term $h^{α} γ$ ) in the boundary condition (1.11) might seem artificial. Commonly, the semi-classical parameter appears in the expression of the operator only (cf. (1.10)).

However, for the analysis of the magnetic ground state energy $μ_{1} (β, H)$ in the limit $β \to - \infty$ , it is natural to consider the strong field limit $H \to \infty$ as well. The analysis of this limit is strongly related to the semi-classical analysis of the operator in (1.10) with the boundary condition in (1.11) where the term $h^{α} γ$ has an unusual appearance. This is due to a simple relationship between the eigenvalues in (1.13) and (1.5) that we will explain below.

Suppose that the hypothesis of Theorem 1.1 holds for some $σ > 0$ . Define the parameters h, b, α and γ as follows $\begin{matrix} h = H^{- 1}, b = 1, α = 1 - \frac{1}{σ}, γ = β H^{- 1 + α} . \end{matrix}$ The strong field limit $H \to \infty$ yields the semi-classical limit $h \to 0_{+}$ . The assumptions $β < 0$ and $H \approx | β |^{σ}$ yield that $γ < 0$ and remains uniformly bounded, i.e. $γ = O (1)$ .

Now, the eigenvalues in (1.13) and (1.5) satisfy the simple relation $\begin{matrix} {\tilde{μ}}_{1} (β; H) = H^{2} μ_{1} (h; b, α, γ) . \end{matrix}$ Besides this relationship, the semi-classical analysis of the eigenvalue $μ_{1} (h; b, α, γ)$ for $α \neq 0$ is motivated by the Ginzburg–Landau theory of superconductivity. In that context, both the magnetic field and the Robin condition should be present (to account for the applied magnetic field and the interaction with the material surrounding the superconductor, respectively). We refer to [10] and [11, Section 1.3] for a discussion of this point.

1.4. On the notation and organization of the paper

Through this paper, the following notation will be used. C, $\tilde{C}$ etc., denote constants independent from the semi-classical parameter h. $O (h^{\infty})$ is a quantity satisfying that, for all $N \in N$ , there exist two constants $h_{0} \in (0, 1)$ and $C_{N} > 0$ such that, for all $h \in (0, h_{0})$ , $| O (h^{\infty}) | ⩽ C_{N} h^{N}$ .

The paper is organized as follows. In Section 2, we analyze three auxiliary differential operators useful to prove Theorem 1.2. The proof of Theorem 1.2 occupies all of Section 3. Finally, in Section 4, we explain how to get the result in Theorem 1.1 from the existing results on the semi-classical magnetic Laplacian with a Robin condition, in particular those in Theorem 1.2.

2. Analysis of auxiliary operators

2.1. 1D Laplacian on the half line

Here we introduce a simple $1 D$ 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. [6,15]). The operator is $\begin{matrix} (2.1) & H_{0, 0} : = - \frac{d^{2}}{d τ^{2}} in L^{2} (R_{+}) \end{matrix}$ with domain $\begin{matrix} (2.2) & D (H_{0, 0}) = {u \in H^{2} (R_{+}) : u^{'} (0) = - u (0)} . \end{matrix}$ The spectrum of this operator is ${- 1} \cup [0, \infty)$ , and $- 1$ is a simple eigenvalue with the $L^{2}$ normalized eigenfunction $\begin{matrix} (2.3) & u_{0} (τ) = \sqrt{2} exp (- τ) . \end{matrix}$ Consequently, we may invert the operator $H_{0, 0} + 1$ on the orthogonal complement of the eigenfunction $u_{0}$ , thereby leading us to introduce the regularized resolvent $\begin{matrix} (2.4) & R_{0, 0} u = \{\begin{matrix} 0 & if u ∥ u_{0} \\ {(H_{0, 0} + 1)}^{- 1} u & if u ⊥ u_{0} . \end{matrix} \end{matrix}$ Through this paper, if $u ⊥ u_{0}$ , we will denote $R_{0, 0} u$ simply by ${(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1} u$ .

2.2. Harmonic oscillator on the half-line

The key element in the proof of Theorem 1.2 is the analysis of the harmonic oscillator $\begin{matrix} (2.5) & H_{harm} (ζ, ξ) = - \frac{d^{2}}{d τ^{2}} + {(ζ τ - ξ)}^{2} in L^{2} (R_{+}), \end{matrix}$ with domain $\begin{matrix} D (H_{harm} (ζ, ξ)) = {u \in H^{1} (R_{+}) : τ^{k} u \in L^{2} (R_{+}), k \in {1, 2}, u^{'} (0) = - u (0)} . \end{matrix}$ Here $ζ > 0$ and $ξ \in R$ 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 asymptotic behavior of the eigenvalue $\begin{matrix} (2.6) & λ_{1} (H_{harm} (ζ, ξ)) = inf (σ (H_{harm} (ζ, ξ))), \end{matrix}$ as $ζ \to 0$ .

By comparison with the operator in (2.1), we get:

Lemma 2.1.
For all $ζ > 0$ and $ξ \in R$ , it holds, $\begin{matrix} λ_{1} (H_{harm} (ζ, ξ)) ⩾ - 1 and λ_{2} (H_{harm} (ζ, ξ)) ⩾ 0 . \end{matrix}$

The lower bound in Lemma 2.1 can be improved as follows:
Lemma 2.2.
There exists a universal constant $A_{0} > 0$ such that, if $0 < ζ < 1$ and $| ξ | ⩾ A_{0} ζ$ , then $\begin{matrix} λ_{1} (H_{harm} (ζ, ξ)) ⩾ - 1 + \frac{3}{2} ζ^{2} . \end{matrix}$
Proof.
Let u be an $L^{2}$ normalized ground state of the operator $H_{harm} (ζ, ξ)$ . Let us write, $\begin{array}{l} λ_{1} (H_{harm} (ζ, ξ)) = & \int_{0}^{\infty} ({| u^{'} |}^{2} + {| (ζ τ - ξ) u |}^{2}) d τ - {| u (0) |}^{2} \\ = & (1 - ζ^{2}) (\int_{0}^{\infty} {| u^{'} |}^{2} d τ - {| u (0) |}^{2}) \\ + ζ^{2} (\int_{0}^{\infty} ({| u^{'} |}^{2} + {| (τ - ζ^{- 1} ξ) u |}^{2}) d τ - {| u (0) |}^{2}) . \end{array}$ We know that the lowest eigenvalue of the operator in (2.1) is $- 1$ . Let $μ (A)$ be the eigenvalue of the operator $\begin{matrix} - \frac{d^{2}}{d τ^{2}} + {(τ - A)}^{2} in L^{2} (R_{+}), \end{matrix}$ with the boundary condition $u^{'} (0) = - u (0)$ .

Now, it results from the min–max principle that $\begin{matrix} λ_{1} (H_{harm} (ζ, ξ)) ⩾ - (1 - ζ^{2}) + ζ^{2} μ (ζ^{- 1} ξ) . \end{matrix}$ When $| ζ^{- 1} ξ |$ is sufficiently large, we get $μ (ζ^{- 1} ξ) ⩾ \frac{1}{2}$ , which is a result of the following two facts proved in [10], $\begin{matrix} lim_{A \to - \infty} μ (A) = \infty and lim_{A \to \infty} μ (A) = 1 . \end{matrix}$ □

We will prove that:
Theorem 2.3.
Let $n \in N$ . There exist $C > 0$ , a collection of vectors $\begin{matrix} {μ_{j} = (μ_{j, 1}, μ_{j, 2}, \dots, μ_{j, 2 j + 1}) \in R^{2 j + 1}}_{j = 1}^{n}, \end{matrix}$ and a collection of vector functions, $\begin{matrix} {u_{j} = (u_{j, 1}, u_{j, 2}, \dots, u_{j, 2 j + 1}) \in {(S (R_{+}))}^{2 j + 1}}_{j = 1}^{n}, \end{matrix}$ such that, if $0 < ζ ⩽ 1$ and $| ξ | ⩽ 1$ , then $\begin{matrix} {‖ (H_{harm} (ζ, ξ) - λ_{n}) w_{n} ‖}_{L^{2} (R_{+})} ⩽ C (ζ^{2 n + 2} + ξ^{2 n + 2}), \end{matrix}$ where $\begin{matrix} w_{n} = u_{0} + \sum_{j = 1}^{n} \sum_{p = 0}^{2 j} ζ^{2 j - p} ξ^{p} u_{j, p + 1}, \end{matrix}$ $u_{0}$ is the eigenfunction in ( 2.3 ), and $\begin{matrix} λ_{n} = - 1 + \sum_{j = 1}^{n} \sum_{p = 0}^{2 j} μ_{j, p + 1} ζ^{2 j - p} ξ^{p} . \end{matrix}$ Furthermore, $\begin{matrix} μ_{1} = (μ_{1, 1} = \frac{1}{2}, μ_{1, 2} = - 1, μ_{1, 3} = 1) . \end{matrix}$
Proof.
Step 1 (Construction of $(μ_{1}, u_{1})$ ).

Here we construct $μ_{1} = (μ_{1, 1}, μ_{1, 2}, μ_{1, 3}) \in R^{3}$ and $u_{1} = (u_{1, 1}, u_{1, 2}, u_{1, 3})$ such that the conclusion of Theorem 2.3 is valid for $n = 1$ .

Let us define $\begin{matrix} λ_{1} = - 1 + μ_{1, 1} ζ^{2} + μ_{1, 2} ζ ξ + μ_{1, 3} ξ^{2} and w_{1} (τ) = u_{0} (τ) + ζ^{2} u_{1, 1} (τ) + ζ ξ u_{1, 2} (τ) + ξ^{2} u_{1, 3} (τ) . \end{matrix}$ For simplicity of the notation, we will write $H = H_{harm} (ζ, ξ)$ . Notice that, since $(- \frac{d^{2}}{d τ^{2}} + 1) u_{0} = 0$ , $\begin{array}{rcl} (H - λ_{1}) w_{1} & = & ζ^{2} [(- \frac{d^{2}}{d τ^{2}} + 1) u_{1, 1} + (τ^{2} - μ_{1, 1}) u_{0}] + ζ ξ [(- \frac{d^{2}}{d τ^{2}} + 1) u_{1, 2} - (2 τ + μ_{1, 2}) u_{0}] \\ (2.7) & + ξ^{2} [(- \frac{d^{2}}{d τ^{2}} + 1) u_{1, 3} + (1 - μ_{1, 3}) u_{0}] + R_{1} . \end{array}$ The remainder $R_{1}$ is $\begin{array}{rcl} R_{1} & = & ζ^{4} (τ^{2} - μ_{1, 1}) u_{1, 1} + ζ^{3} ξ [(τ^{2} - μ_{1, 1}) u_{1, 2} + (μ_{1, 2} - 2 τ) u_{1, 2}] \\ + ζ^{2} ξ^{2} [(1 - μ_{1, 1}) u_{1, 3} + (μ_{1, 2} - 2 τ) u_{1, 2} + (1 - μ_{1, 3}) u_{1, 1}] \\ (2.8) & + ζ ξ^{3} [(1 - μ_{1, 3}) u_{1, 2} + (μ_{1, 2} - 2 τ) u_{1, 3}] + ξ^{4} (1 - μ_{1, 3}) u_{1, 3} . \end{array}$ We choose the coefficients and the functions in (2.7) so that the terms with coefficients $ζ^{2}$ , $ζ ξ$ and $ξ^{2}$ vanish. This is possible since the operator $- \frac{d^{2}}{d τ^{2}} + 1$ can be inverted in the orthogonal complement of the eigenfunction $u_{0}$ (cf. (2.3) and (2.4)). That way we choose, $\begin{array}{l} μ_{1, 1} = \int_{0}^{\infty} τ^{2} {| u_{0} (τ) |}^{2} d τ = \frac{1}{2}, u_{1, 1} = - {(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1} {(τ^{2} - μ_{1, 1}) u_{0}}, \\ μ_{1, 2} = - \int_{0}^{\infty} 2 τ {| u_{0} (τ) |}^{2} d τ = - 1, u_{1, 2} = {(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1} {(2 τ + μ_{1, 1}) u_{0}}, \\ μ_{1, 3} = \int_{0}^{\infty} {| u_{0} (τ) |}^{2} d τ = 1, u_{1, 3} = - {(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1} {(1 - μ_{1, 3}) u_{0}} . \end{array}$ The operator ${(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1}$ respects the Schwartz space $S (R_{+})$ . The proof of this is standard (cf. [3, Lemma A.5]). Now we infer from (2.7) and (2.8), $\begin{matrix} {‖ (H - λ_{1}) w_{1} ‖}_{L^{2} (R_{+})} ⩽ C (ζ^{4} + | ζ |^{3} | ξ | + ζ^{2} ξ^{2} + | ζ | | ξ |^{3} + ξ^{4}) . \end{matrix}$ Using Young’s inequality (for $a, b > 0$ and $\frac{1}{p} + \frac{1}{q} = 1$ , $a b ⩽ \frac{1}{p} a^{p} + \frac{1}{q} b^{q}$ ), we get, $\begin{matrix} | ζ |^{3} | ξ | + ζ^{2} ξ^{2} + | ζ | | ξ |^{3} ⩽ \frac{3}{2} (ζ^{4} + ξ^{4}) . \end{matrix}$

Step 2 (The iteration process).

Suppose that we have constructed ${(μ_{j})}_{j = 1}^{n}$ and ${(u_{j})}_{j = 1}^{n}$ such that $\begin{matrix} (H - λ_{n}) w_{n} = R_{n} + f_{n, ζ, ξ}, \end{matrix}$ $R_{n}$ has the form $\begin{matrix} R_{n} = \sum_{p = 0}^{2 n + 2} ζ^{2 n + 2 - p} ξ^{p} v_{n, p}, \end{matrix}$ 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, $\begin{matrix} ‖ f_{n, ζ, ξ} ‖_{L^{2} (R_{+})} ⩽ C (ζ^{2 n + 4} + ξ^{2 n + 4}), \end{matrix}$ where C is a constant independent of ζ and ξ.

We outline the construction of $\begin{matrix} μ_{n + 1} = (μ_{n + 1, 1}, μ_{n + 1, 2}, \dots, μ_{n + 1, 2 n + 3}) \in R^{2 n + 3} and u_{n + 1} = (u_{n + 1, 1}, u_{n + 1, 2}, \dots, u_{n + 1, 2 n + 3}), \end{matrix}$ such that $\begin{matrix} (H - λ_{n + 1}) w_{n + 1} = R_{n + 1} + f_{n + 1, ζ, ξ}, \end{matrix}$ $R_{n + 1}$ has the form $\begin{matrix} R_{n + 1} = \sum_{p = 0}^{2 n + 4} ζ^{2 n + 4 - p} ξ^{p} v_{n + 1, p}, \end{matrix}$ for a collection $(v_{n + 1, p})$ of Schwartz functions that do not depend on ζ and ξ, and $f_{n + 1, ζ, ξ}$ satisfies, $\begin{matrix} ‖ f_{n + 1, ζ, ξ} ‖_{L^{2} (R_{+})} ⩽ C (ζ^{2 n + 6} + ξ^{2 n + 6}), \end{matrix}$ 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, $\begin{array}{l} (H - λ_{n + 1}) w_{n + 1} = & \sum_{p = 0}^{2 n + 2} ζ^{2 n + 2 - p} ξ^{p} {(- \frac{d^{2}}{d τ^{2}} + 1) u_{n + 1, p + 1} + v_{n, p} - μ_{n + 1, p + 1} u_{0}} \\ + \sum_{p = 0}^{2 n + 4} ζ^{2 n + 4 - p} ξ^{p} v_{n + 1, p} + \sum_{p = 0}^{2 n + 6} ζ^{2 n + 6 - p} ξ^{p} g_{n + 1, p}, \end{array}$ 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 $\begin{matrix} \sum_{p = 0}^{2 n + 2} ζ^{2 n + 2 - p} ξ^{p} {(- \frac{d^{2}}{d τ^{2}} + 1) u_{n + 1, p + 1} + v_{n, p} - μ_{n + 1, p + 1} u_{0}} = 0 . \end{matrix}$ To that end, we select $μ_{n + 1, p + 1}$ such that, $\begin{matrix} μ_{n + 1, p + 1} = \int_{0}^{\infty} v_{n, p} u_{0} d τ, \end{matrix}$ so that $\begin{matrix} v_{n, p} - μ_{n + 1, p + 1} u_{0} ⊥ u_{0} in L^{2} (R_{+}) . \end{matrix}$ Finally, we define the function $u_{n + 1, p + 1}$ as follows, $\begin{matrix} u_{n + 1, p + 1} = - {(- \frac{d^{2}}{d τ^{2}} + 1)}^{- 1} (v_{n, p} - μ_{n + 1, p + 1} u_{0}) . \end{matrix}$

□

As a consequence of Theorem 2.3, Lemma 2.1 and the spectral theorem, we get:

Corollary 2.4.
Let $n \in N$ and $A > 0$ . If $| ξ | ⩽ A ζ$ , then as $ζ \to 0_{+}$ , the eigenvalue $λ_{1} (H_{harm} (ζ, ξ))$ satisfies, $\begin{matrix} λ_{1} (H_{harm} (ζ, ξ)) = - 1 + \sum_{j = 1}^{n} \sum_{p = 0}^{2 j} μ_{j, p + 1} ζ^{2 j - p} ξ^{p} + O (ζ^{2 n + 2}) . \end{matrix}$
Definition 2.5.
Let $n \in N$ and $A_{0}$ be the universal constant in Lemma 2.2. For all $ζ \in (0, 1)$ , we define the following quantity $\begin{matrix} (2.9) & e_{n} (ζ) = inf {f_{n} (ζ, ξ) : | ξ | ⩽ ζ max (A_{0}, 1)}, \end{matrix}$ where $\begin{matrix} (2.10) & f_{n} (ζ, ξ) = \sum_{j = 2}^{n} \sum_{p = 0}^{2 n} ζ^{2 n - p} ξ^{p} μ_{j, p + 1}, \end{matrix}$ and $(μ_{j, p + 1})$ are the constants in Theorem 2.3.
Remark 2.6.
Note that for $n = 1$ , $f_{1} (ζ, ξ) = \frac{1}{2} ζ^{2} - ζ ξ + ξ^{2}$ , and $\begin{matrix} min f_{1} (ζ, ξ) = f_{1} (ζ, \frac{1}{2} ζ) = \frac{1}{4} ζ^{2} . \end{matrix}$ Consequently, for all $n \in N$ , as $ζ \to 0_{+}$ , $\begin{matrix} e_{n} (ζ) = \frac{1}{4} ζ^{2} + O (ζ^{4}) . \end{matrix}$

2.3. A family of operators in a weighted space

Here we will study an operator that arises in many papers concerned with the semi-classical magnetic Laplacian (cf. [7,10]). Let $h \in (0, 1)$ , $ζ \in (0, 1)$ , $ξ \in R$ , $β \in R$ , $δ \in (0, 1 / 2)$ , $m ⩾ 0$ , $σ \in (0, 1)$ , $M > 0$ and $| β | h^{\frac{1}{2} - δ} < \frac{1}{3}$ .

Consider the Hilbert-space $\begin{matrix} L^{2} ((0, h^{- δ}), \tilde{a} d τ), \tilde{a} = 1 - (β + m h^{σ}) h^{1 / 2} τ, ‖ \cdot ‖_{L^{2} ((0, h^{- δ}), \tilde{a} d τ)} = {(\int_{0}^{h^{- δ}} | \cdot |^{2} \tilde{a} d τ)}^{1 / 2}, \end{matrix}$ and the self-adjoint operator $\begin{array}{l} H_{ζ, β, ξ, h} = & - {\tilde{a}}^{- 1} \partial_{τ} (\tilde{a} \partial_{τ}) + (1 + h^{1 / 2} Δ_{β, τ}) {(ζ τ (1 - \frac{1}{2} β h^{1 / 2} τ) - ξ)}^{2} \\ = & - \frac{d^{2}}{d τ^{2}} + {(ζ τ - ξ)}^{2} + (β + m h^{σ}) h^{1 / 2} {(1 - (β + m h^{σ}) h^{1 / 2} τ)}^{- 1} \frac{d}{d τ} \\ (2.11) & + h^{1 / 2} Δ_{β, τ} {(ζ τ - ξ)}^{2} + β h^{1 / 2} ζ τ^{2} (1 + h^{1 / 2} Δ_{β, τ}) (- (ζ τ - ξ) + \frac{1}{4} β h^{1 / 2} ζ τ^{2}), \end{array}$ in $L^{2} ((0, h^{- δ}), \tilde{a} d τ)$ . Here $Δ_{β, τ}$ is a function of $(β, τ)$ and satisfies, for all $h \in (0, 1)$ , $\begin{matrix} | Δ_{β, τ} | ⩽ M (| β | + 1) τ . \end{matrix}$ The domain of the operator $H_{ζ, β, ξ, h}$ is $\begin{matrix} (2.12) & D (H_{ζ, β, ξ, h}) = {u \in H^{2} ((0, h^{- δ})) : u^{'} (0) = - u (0) and u (h^{- δ}) = 0} . \end{matrix}$ The operator $H_{ζ, β, ξ, h}$ is the Friedrich’s extension in $L^{2} ((0, h^{- δ}); \tilde{a} d τ)$ associated with the quadratic form $q_{ζ, β, h, ξ} (u)$ defined on ${u \in H^{1} ((0, h^{- δ})), u (h^{- δ}) = 0}$ as follows $\begin{array}{l} q_{ζ, β, h, ξ} (u) \\ = \int_{0}^{h^{- δ}} ({| u^{'} (τ) |}^{2} + (1 + h^{1 / 2} Δ_{β, τ}) {| (ζ τ (1 - \frac{1}{2} β h^{1 / 2} τ) - ξ) u |}^{2}) (1 - (β + m h^{σ}) h^{1 / 2} τ) d τ \\ - {| u (0) |}^{2} . \end{array}$ The operator $H_{ζ, β, ξ, h}$ is with compact resolvent. The increasing sequence of the eigenvalues of $H_{ζ, β, ξ, h}$ is denoted by ${(λ_{n} (H_{ζ, β, ξ, h}))}_{n \in N}$ .

2.3.1. Harmonic oscillator on an interval

Here we study the operator in (2.11) for $β = 0$ , $m = 0$ and $Δ_{β, τ} = 0$ which becomes the harmonic oscillator $\begin{matrix} (2.13) & H_{ζ, 0, ξ, h} = - \frac{d^{2}}{d τ^{2}} + {(ζ τ - ξ)}^{2} in L^{2} ((0, h^{- δ}); d τ), \end{matrix}$ and with the boundary conditions $u^{'} (0) = - u (0)$ and $u (h^{- δ}) = 0$ .

By comparison of the quadratic forms of the operators $H_{ζ, β, ξ, h}$ and $H_{ζ, 0, ξ, h}$ , we get that the spectrum of $H_{ζ, β, ξ, h}$ is localized near that of $H_{ζ, β, ξ, h}$ as h goes to 0. This gives us a rough information about the spectrum of the operator $H_{ζ, β, ξ, h}$ precisely stated in:

Lemma 2.7.
Let $0 < c_{1} < c_{2}$ , $0 < ϵ ⩽ \frac{1}{4}$ and $δ \in (0, 1)$ . There exist two constants $C > 0$ and $h_{0} \in (0, 1)$ such that, for all $\begin{matrix} h \in (0, h_{0}), c_{1} h^{ϵ} ⩽ ζ ⩽ c_{2} h^{ϵ} and | β | + m ⩽ c_{2}, \end{matrix}$ it holds the following.
$λ_{2} (H_{ζ, β, ξ, h}) ⩾ - C | β | h^{\frac{1}{2} - δ}$ .

If $ϵ = \frac{1}{4}$ , $δ < \frac{1}{4}$ and $| ξ | ⩾ (3 c_{2} + 2) h^{\frac{1}{4} - δ}$ , then $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) ⩾ - 1 + h^{\frac{1}{2} - 2 δ} . \end{matrix}$

Let $A_{0}$ be the universal constant in Lemma 2.2 . If $ϵ < \frac{1}{4}$ , $δ < \frac{1}{2} - 2 ϵ$ and $| ξ | ⩾ A_{0} ζ$ , then $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) ⩾ - 1 + ζ^{2} . \end{matrix}$

Proof.
Step 1.
There exists a constant $C > 0$ such that, for all $u \in H^{1} ((0, h^{- δ}))$ , $\begin{array}{l} | q_{ζ, β, h, ξ} (u) - q_{ζ, β = 0, h, ξ} (u) | ⩽ C | β | h^{\frac{1}{2} - δ} (q_{0, h} (u) + ‖ u ‖_{L^{2} ((0, h^{- δ}); d τ)}^{2}) \\ | ‖ u ‖_{L^{2} ((0, h^{- δ}); (1 - β h^{1 / 2} τ) d τ)}^{2} - ‖ u ‖_{L^{2} ((0, h^{- δ}); d τ)}^{2} | ⩽ | β | h^{\frac{1}{2} - δ} ‖ u ‖_{L^{2} ((0, h^{- δ}); d τ)}^{2} . \end{array}$ The min–max principle yields, for all $n \in N$ , $\begin{matrix} (2.14) & | λ_{n} (H_{ζ, β, ξ, h}) - λ_{n} (H_{ζ, 0, ξ, h}) | ⩽ C | β | h^{\frac{1}{2} - δ} (| λ_{n} (H_{ζ, 0, ξ, h}) | + 1) . \end{matrix}$ Since the form domain of the operator $H_{harm} (ζ, ξ)$ contains that of the operator $H_{ζ, β, ξ, h}$ (cf. (2.5)), then the min–max principle yields $\begin{matrix} (2.15) & λ_{n} (H_{ζ, 0, ξ, h}) ⩾ λ_{n} (H_{harm} (ζ, ξ)) . \end{matrix}$ In particular, for $n = 2$ , Lemma 2.1 gives us the statement in the first item of Lemma 2.7.
Step 2.
We estimate the quadratic form for the operator $H_{ζ, 0, ξ, h}$ as follows, for all $u \in H^{1} (0, h^{- δ})$ , $\begin{matrix} q_{ζ, 0, ξ, h} (u) ⩾ \int_{0}^{h^{- δ}} ({| u^{'} (t) |}^{2} + (- 2 c_{2} h^{\frac{1}{4} - δ} ξ + ξ^{2}) | u |^{2}) d τ - {| u (0) |}^{2} . \end{matrix}$ The min–max principle and Lemma 2.1 yield, $\begin{matrix} λ_{1} (H_{ζ, 0, ξ, h}) ⩾ - 1 + (| ξ | - 2 c_{2} h^{\frac{1}{4} - δ}) | ξ | . \end{matrix}$ We insert this into (2.14). That way, for $| ξ | ⩾ (3 c_{2} + 2) h^{\frac{1}{4} - δ}$ , we get the conclusion in the second item of Lemma 2.7.
Step 3.
Using (2.14) and (2.15) for $n = 1$ , we get, $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) ⩾ λ_{1} (H_{ζ, 0, ξ, h}) - C h^{\frac{1}{2} - δ} . \end{matrix}$ Now, we assume that $| ξ | ⩾ A_{0} ζ$ , $ϵ < \frac{1}{4}$ and $δ < \frac{1}{2} - 2 ϵ$ . By applying Lemma 2.2 we get, for h sufficiently small, the statement in the third item in Lemma 2.7.
□

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 $h \to 0_{+}$ and is uniform with respect to the parameters ξ, ζ and β.

Proposition 2.8.
Let $0 < c_{1} < c_{2}$ , $σ \in (0, 1)$ , $ϵ \in (0, \frac{1}{4})$ , $δ \in (0, \frac{1}{2} - 2 ϵ)$ and $n \in N$ be the smallest positive integer such that $\begin{matrix} (2 n + 2) ϵ > \frac{1}{2} . \end{matrix}$ There exist constants $C > 0$ and $h_{0} \in (0, 1)$ such that, for all $\begin{matrix} h \in (0, h_{0}), c_{1} h^{ϵ} < ζ ⩽ c_{2} h^{ϵ}, ξ \in R, | β | h^{δ} < \frac{1}{3}, | β | + m ⩽ c_{2}, \end{matrix}$ it holds, $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) ⩾ - 1 + e_{n} (ζ) - β h^{1 / 2} - C h^{r}, \end{matrix}$ where $\begin{matrix} r = min ((2 n + 2) ϵ, \frac{1}{2} + 2 ϵ, \frac{1}{2} + σ) . \end{matrix}$
Proof.
Let $A_{0}$ be the universal constant in Lemma 2.2. In light of the results in Remark 2.6 and Lemma 2.7, the lower bound in Lemma 2.8 holds true for $| ξ | ⩾ A_{0} ζ$ . It remains to prove the lower bound for $| ξ | ⩽ A_{0} ζ$ .

Consider the function $\begin{matrix} f (τ) = χ (τ h^{δ}) w_{n} (τ), \end{matrix}$ where $w_{n} (τ)$ is the function in Theorem 2.3 and $χ \in C_{c}^{\infty} ([0, \infty))$ satisfies $\begin{matrix} 0 ⩽ χ ⩽ 1 in [0, \infty), χ = 1 in [0, 1 / 2) and χ = 0 in [1, \infty) . \end{matrix}$ Clearly, the function f is in the domain of the operator $H_{ζ, β, ξ, h}$ . It is easy to check that, $\begin{matrix} | ‖ f ‖_{L^{2} ((0, h^{- ρ}); (1 - β h^{1 / 2} τ) d τ)}^{2} - 1 | ⩽ C ζ^{2} . \end{matrix}$ In light of Theorem 2.3 and the expression of $H_{ζ, β, ξ, h} f$ in (2.11), we may write, $\begin{array}{l} {‖ {H_{ζ, β, ξ, h} - (- 1 + f_{n} (ζ, ξ) - β h^{1 / 2})} f ‖}_{L^{2} ((0, h^{- ρ}); (1 - β h^{1 / 2} τ) d τ)} & ⩽ C (ζ^{2 n + 2} + h^{\frac{1}{2} + 2 ϵ} + h^{\frac{1}{2} + σ}) \\ (2.16) & ⩽ C h^{r} . \end{array}$ By the spectral theorem, we deduce that there exists an eigenvalue $λ (H_{ζ, β, ξ, h})$ of $H_{ζ, β, ξ, h}$ such that $\begin{matrix} | λ (H_{ζ, β, ξ, h}) - (- 1 + f_{n} (ζ, ξ)) | ⩽ C h^{r} . \end{matrix}$ Now, Lemma 2.7 tells us that $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) = λ (H_{ζ, β, ξ, h}) . \end{matrix}$ Finally, by definition of $e_{n} (ζ)$ in (2.9), we have $f_{n} (ζ, ξ) ⩾ e_{n} (ζ)$ . □
Proposition 2.9.
Let $0 < c_{1} < c_{2}$ , $σ \in (0, 1)$ and $δ \in (0, \frac{1}{8})$ . There exist constants $C > 0$ and $h_{0} \in (0, 1)$ such that, for all $\begin{matrix} h \in (0, h_{0}), c_{1} h^{\frac{1}{4}} ⩽ ζ ⩽ c_{2} h^{\frac{1}{4}}, ξ \in R, | β | h^{δ} < \frac{1}{3}, | β | + m ⩽ c_{2}, \end{matrix}$ it holds, $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) ⩾ - 1 + \frac{1}{4} ζ^{2} - β h^{1 / 2} - C h^{r}, \end{matrix}$ where $\begin{matrix} r = min (1 - 4 δ, \frac{1}{2} + σ) . \end{matrix}$
Proof.
The lower bound in Lemma 2.9 trivially holds when $| ξ | ⩾ (3 c_{2} + 2) h^{\frac{1}{4} - δ}$ thanks to Lemma 2.7.

Now we handle the case where $| ξ | ⩽ (3 c_{2} + 2) h^{\frac{1}{4} - δ}$ . Let $w_{1}$ be the function constructed in Theorem 2.3 and choose $χ \in C_{c}^{\infty} ([0, \infty))$ such that $\begin{matrix} 0 ⩽ χ ⩽ 1 in [0, \infty), χ = 1 in [0, 1 / 2) and χ = 0 in [1, \infty) . \end{matrix}$ Consider the function $\begin{matrix} f (τ) = χ (τ h^{δ}) w_{n} (τ) . \end{matrix}$ Clearly, the function f is in the domain of the operator $H_{ζ, β, ξ, h}$ and $\begin{matrix} | ‖ f ‖_{L^{2} ((0, h^{- ρ}); (1 - β h^{1 / 2} τ) d τ)}^{2} - 1 | ⩽ C ζ^{2} . \end{matrix}$ Inserting the estimates in Theorem 2.3 into the expression of $H_{ζ, β, ξ, h} f$ in (2.11), and using that $ζ = O (h^{1 / 4})$ and $ξ = O (h^{\frac{1}{4} - δ})$ , we may write, $\begin{array}{l} {‖ {H_{ζ, β, ξ, h} - (- 1 + f_{1} (ζ, ξ) - β h^{1 / 2})} f ‖}_{L^{2} ((0, h^{- ρ}); (1 - β h^{1 / 2} τ) d τ)} \\ ⩽ C (ζ^{4} + ξ^{4} + (ζ^{2} + ξ^{2} + h^{σ}) h^{\frac{1}{2}} + h) \\ (2.17) & ⩽ C h^{r} . \end{array}$ Now, the spectral theorem and Lemma 2.7 yield $\begin{matrix} λ_{1} (H_{ζ, β, ξ, h}) = - 1 + f_{1} (ζ, ξ) - β h^{1 / 2} + O (h^{r}) . \end{matrix}$ Noticing that ${min}_{| ξ | ⩽ (3 c_{2} + 2) h^{\frac{1}{4} - δ}} f_{1} (ζ, ξ) = \frac{1}{4} ζ^{2}$ , we finish the proof of Lemma 2.9. □

3. 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 semi-classical 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 $\begin{matrix} (3.1) & P_{h, ζ} = - {(h \nabla - i ζ A_{0})}^{2} in L^{2} (Ω), \end{matrix}$ whose domain is (cf. (1.4) for the definition of the magnetic Sobolev space) $\begin{matrix} (3.2) & D (P_{h, ζ}) = {u \in H_{h^{- 1} ζ A_{0}}^{1} (Ω) : - {(h \nabla - i ζ A_{0})}^{2} \in L^{2} (Ω) and ν \cdot (h \nabla - i A_{0}) u = - h^{1 / 2} u on \partial Ω} . \end{matrix}$ This operator is defined via the quadratic form $\begin{matrix} (3.3) & H_{h^{- 1} ζ A_{0}}^{1} (Ω) ∋ u \mapsto q_{h, ζ} (u) = \int_{Ω} {| (h \nabla - i ζ A_{0}) u |}^{2} d x - h^{3 / 2} \int_{\partial Ω} | u |^{2} d s (x) . \end{matrix}$ Let $σ (P_{h, ζ})$ be the spectrum of the operator $P_{h, ζ}$ . We introduce the ground state energy, $\begin{matrix} (3.4) & λ_{1} (h, ζ) = inf σ (P_{h, ζ}) . \end{matrix}$ There is a relationship between the ground state energies in (1.13) and (3.4) displayed as follows: $\begin{matrix} (3.5) & μ_{1} (h; b, α, γ) = \frac{γ^{4}}{h^{2 - 4 α}} λ_{1} (\frac{h^{2 - 2 α}}{γ^{2}}, b \frac{h^{1 - 2 α}}{γ^{2}}) . \end{matrix}$

Now Theorem 1.2 follows from:

Theorem 3.1.
Let $0 < c_{1} < c_{2}$ and $ϵ > 0$ . Suppose that $\begin{matrix} 0 < h < 1 and c_{1} h^{ϵ} ⩽ ζ ⩽ c_{2} h^{ϵ} . \end{matrix}$ It holds the following.
If $ϵ < \frac{1}{4}$ , then as $h \to 0_{+}$ , the ground state energy in ( 3.4 ) satisfies $\begin{matrix} λ_{1} (h, ζ) = - h + e_{n} (ζ) h - κ_{max} h^{3 / 2} + h^{3 / 2} o (1), \end{matrix}$ where $e_{n} (ζ)$ is introduced in ( 2.9 ) and $n \in N$ is the smallest positive integer such that $(2 n + 2) ϵ > \frac{1}{2}$ .

If $ϵ = \frac{1}{4}$ , then as $h \to 0_{+}$ , the ground state energy in ( 3.4 ) satisfies $\begin{matrix} λ_{1} (h, ζ) = - h + \frac{1}{4} ζ^{2} h - κ_{max} h^{3 / 2} + h^{3 / 2} o (1) . \end{matrix}$

If $ϵ > \frac{1}{4}$ , then as $h \to 0_{+}$ , the ground state energy in ( 3.4 ) satisfies $\begin{matrix} λ_{1} (h, ζ) = - h - κ_{max} h^{3 / 2} + h^{3 / 2} o (1) . \end{matrix}$

There exist constants $ρ \in (0, \frac{1}{2})$ , $η^{} \in (0, \frac{1}{4})$ , $C > 0$ and* $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ , every ground state $u_{h}$ of $λ_{1} (h, ζ)$ satisfies, $\begin{matrix} ‖ u_{h, ζ} ‖_{L^{2} (Ω_{bnd})} ⩽ C exp (- \frac{1}{2} h^{η^{} - \frac{1}{4}}), ‖ u_{h, ζ} ‖_{L^{2} (Ω_{int})} ⩽ C exp (- \frac{1}{2} h^{ρ - \frac{1}{2}}), \end{matrix}$ where* $\begin{matrix} \begin{matrix} Ω_{int} = {x \in Ω : dist (x, \partial Ω) ⩾ h^{ρ}} and \\ Ω_{bnd} = {x \in Ω ∖ {\overline{Ω}}_{int} : κ_{max} - κ (p (x)) ⩾ h^{η^{}}} . \end{matrix} \end{matrix}$

The proof of the items (1)–(3) in Theorem 3.1 follows from Propositions 3.2 and 3.6. We will give explicit bounds to the remainder $h^{3 / 2} o (1)$ in the form $O (h^{r})$ where r depends on ϵ and satisfies $r > \frac{3}{2}$ . More specifically, we find that $\begin{matrix} r = min (r_{}, r^{}), \end{matrix}$ where $r_{}$ and $r^{*}$ are introduced in (3.14) and (3.31) respectively.

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 Laplacian (cf. [4,7]).

The boundary coordinates are valid in every connected component of the boundary. For simplicity, we will suppose that $\partial Ω$ has one connected component; if more than one connected component exists, then we use the coordinates in each connected component independently. Let $\begin{matrix} R / (| \partial Ω | Z) ∋ s \mapsto M (s) \in \partial Ω \end{matrix}$ be the arc-length parametrization of $\partial Ω$ . At the point $M (s) \in \partial Ω$ , $T (s) = M^{'} (s)$ is the unit tangent vector and $ν (s)$ is the unit outward normal vector. We choose the orientation such that $\begin{matrix} \forall s \in R / (| \partial Ω | Z), det (T (s), ν (s)) = 1 . \end{matrix}$ The curvature $κ (s)$ is then defined as follows $\begin{matrix} T^{'} (s) = κ (s) ν (s) . \end{matrix}$ Note that our definition yields that the curvature is signed, for instance, if Ω is the exterior of the unit disc, the curvature is negative, while it is positive if Ω is the interior of the disc.

The smoothness of the boundary yields the existence of a constant $t_{0} > 0$ such that, upon defining $\begin{matrix} V_{t_{0}} = {x \in Ω : dist (x, \partial Ω) < t_{0}}, \end{matrix}$ the map $\begin{matrix} Φ : R / (| \partial Ω | Z) \times (0, t_{0}) ∋ (s, t) \mapsto x = M (s) - t ν (s) \in V_{t_{0}} \end{matrix}$ becomes a diffeomorphism. Let us note that, for $x \in V_{t_{0}}$ , one can write $\begin{matrix} (3.6) & Φ^{- 1} (x) : = (s (x), t (x)) \in R / (| \partial Ω | Z) \times (0, t_{0}), \end{matrix}$ where $t (x) = dist (x, \partial Ω)$ and $s (x) \in R / (| \partial Ω | Z)$ is (uniquely) defined via the relation $dist (x, \partial Ω) = | x - M (s (x)) |$ .

Now we express various integrals in the new coordinates $(s, t)$ . First, note that the Jacobian determinant of the transformation $Φ^{- 1}$ is given by: $\begin{matrix} a (s, t) = 1 - t κ (s) . \end{matrix}$ In the new coordinates, the components of the vector field $A_{0}$ are given as follows, $\begin{matrix} (3.7) & \begin{matrix} {\tilde{A}}_{1} (s, t) & = A_{0} \cdot \frac{\partial x}{\partial s} = (1 - t κ (s)) A_{0} (Φ (s, t)) \cdot M^{'} (s), \\ {\tilde{A}}_{2} (s, t) & = A_{0} \cdot \frac{\partial x}{\partial t} = A_{0} (Φ (s, t)) \cdot ν (s) . \end{matrix} \end{matrix}$ The new magnetic potential ${\tilde{A}}_{0} = ({\tilde{A}}_{1}, {\tilde{A}}_{2})$ satisfies, $\begin{matrix} [\frac{\partial {\tilde{A}}_{2}}{\partial s} (s, t) - \frac{\partial {\tilde{A}}_{1}}{\partial t} (s, t)] d s \land d t = curl A_{0} (Φ^{- 1} (s, t)) d x \land d y = (1 - t κ (s)) d s \land d t . \end{matrix}$

For all $u \in L^{2} (V_{δ})$ , we assign the function $\tilde{u}$ defined in the new coordinates as follows $\begin{matrix} (3.8) & \tilde{u} (s, t) : = u (Φ (s, t)) . \end{matrix}$ Consequently, for all $u \in H^{1} (V_{t_{0}})$ , we have, with $\tilde{u} = u \circ Φ$ , $\begin{array}{l} \int_{V_{t_{0}}} {| (h \nabla - i ζ A_{0}) u |}^{2} d x \\ (3.9) & = \int [{(1 - t κ (s))}^{- 2} {| (h \partial_{s} - i ζ {\tilde{A}}_{1}) \tilde{u} |}^{2} + {| (h \partial_{t} - i ζ {\tilde{A}}_{2}) \tilde{u} |}^{2}] (1 - t κ (s)) d s d t, \\ (3.10) & \int_{V_{t_{0}}} | u |^{2} d x = \int {| \tilde{u} (s, t) |}^{2} (1 - t κ (s)) d s d t, \end{array}$ and $\begin{matrix} (3.11) & \int_{V_{t_{0}} \cap \partial Ω} | u |^{2} d x = \int {| \tilde{u} (s, t = 0) |}^{2} d s . \end{matrix}$ Finally, we recall a useful gauge transformation that we borrow from [4,7]. Let $x_{0} \in \partial Ω$ and $V_{x_{0}}$ be a neighborhood of $x_{0}$ in $\overline{Ω}$ . There exists a smooth function $φ_{x_{0}}$ in $Φ^{- 1} (V_{x_{0}})$ such that, in the boundary coordinates, $\begin{matrix} (3.12) & \tilde{A} - \nabla_{(s, t)} φ_{x_{0}} = (- t + \frac{t^{2}}{2} κ (s), 0) . \end{matrix}$

3.3. Upper bound for the principal eigenvalue

In the rest of this paper, we will use the following notation. For all $ϵ > 0$ and $ζ \in (0, 1)$ , define $\begin{matrix} (3.13) & b_{ϵ} (ζ) = \{\begin{matrix} e_{n} (ζ) & if ϵ < 1 / 4, \\ \frac{1}{4} ζ^{2} & if ϵ = 1 / 4, \\ 0 & if ϵ > 1 / 4, \end{matrix} \end{matrix}$ where $n \in N$ is the smallest positive integer satisfying $(2 n + 2) ϵ > \frac{1}{2}$ .

Proposition 3.2.
Under the assumptions in Theorem 3.1 , there exist two constants $C > 0$ and $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ , the ground state energy in ( 3.4 ) satisfies, $\begin{matrix} λ_{1} (h, ζ) ⩽ - h + b_{ϵ} (ζ) h - κ_{max} h^{3 / 2} + C h^{r_{}}, \end{matrix}$ where* $\begin{matrix} (3.14) & r_{} = \{\begin{matrix} 1 + min ((2 n + 2) ϵ, \frac{1}{2} + ϵ) & if ϵ < 1 / 4, \\ 13 / 8 & if ϵ ⩾ 1 / 4 . \end{matrix} \end{matrix}$ Here* $b_{ϵ} (ζ)$ is as in ( 3.13 ).
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 (3.6). Select $x_{0} \in \partial Ω$ such that $\begin{matrix} κ (s (x_{0})) = κ_{max} \end{matrix}$ is equal to the maximal curvature. We may choose the coordinates $(s, t)$ in (3.6) such that $s (x_{0}) = 0$ . Let $V_{x_{0}}$ be a neighborhood of the point $x_{0}$ in $\overline{Ω}$ , and $φ_{0} = φ_{x_{0}}$ be the function defined in $V_{x_{0}}$ and satisfying (3.12).

The construction of the trial function $v_{h}$ and the computation of its energy will be done for the cases $ϵ < \frac{1}{4}$ , $ϵ = \frac{1}{4}$ and $ϵ > \frac{1}{4}$ independently. The case $ϵ < \frac{1}{4}$ .
Let $n \in N$ be the largest positive integer such that $(2 n + 2) ϵ > \frac{1}{2}$ . Recall the definition of $f_{n} (ζ, ξ)$ in (2.10). Select $ξ_{n} = ξ_{n} (ζ)$ such that $\begin{matrix} f_{n} (ζ, ξ_{n}) = e_{n} (ζ) = min {f_{n} (ζ, ξ) : | ξ | ⩽ ζ max (A_{0}, 1)} and | ξ_{n} | ⩽ ζ max (A_{0}, 1) . \end{matrix}$

The trial function $v_{h}$ is defined using the $(s, t)$ -coordinates and the relation in (3.8) as follows, $\begin{matrix} (3.15) & {\tilde{v}}_{h} (s, t) = c h^{- \frac{1 + ϵ}{4}} χ_{1} (\frac{t}{h^{ρ}}) χ_{1} (\frac{s}{h^{ϵ / 2}}) e^{- i φ_{0}} w_{n} (h^{1 / 2} t) exp (\frac{i ξ_{n}}{h^{1 / 2}}) . \end{matrix}$ Several objects appear in the definition of ${\tilde{v}}_{h}$ :
$χ_{1} \in C_{c}^{\infty} (R)$ satisfies $0 ⩽ χ ⩽ 1$ in $R$ , $χ_{1} = 1$ in $[- 1 / 2, 1 / 2]$ and $supp χ_{1} \subset [- 1, 1]$ ;

$c = ‖ χ_{1} ‖_{L^{2} (R)}^{- 1}$ ;

$w_{n} (τ)$ is the function in Theorem 2.3;

$ρ = \frac{1}{4} + ϵ$ .
The upper bound in Proposition 3.2 follows from the min–max principle and the following two estimates: $\begin{array}{l} (3.16) & | ‖ v_{h} ‖_{L^{2} (Ω)} - 1 | ⩽ C h^{1 / 2}, \\ (3.17) & {‖ (P_{h, ζ} + h - e_{n} (ζ) + κ_{max} h^{3 / 2}) v_{h} ‖}_{L^{2} (Ω)} ⩽ h^{r_{}}, \end{array}$ where $r_{} = 1 + min ((2 n + 2) ϵ, \frac{1}{2} + \frac{ϵ}{2})$ is given in (3.14).

The estimate in (3.16) is easy to obtain in light of the expression of ${\tilde{v}}_{h}$ and the formula in (3.10). For the estimate in (3.17), notice that, after expressing the operator $P_{ζ, h}$ in the boundary coordinates $(s, t)$ , we get (compare with (3.9)) $\begin{matrix} {‖ (P_{h, ζ} + h - e_{n} (ζ) + κ_{max} h^{3 / 2}) v_{h} ‖}_{L^{2} (Ω)} = {‖ (L_{h, ζ} + h - e_{n} (ζ) + κ_{max} h^{3 / 2}) u_{h} ‖}_{L^{2} (a d s d t)}, \end{matrix}$ where $\begin{array}{rcl} a (s, t) = 1 - t κ (s), \\ u_{h} (s, t) = c h^{- \frac{1 + ϵ}{4}} χ_{1} (\frac{t}{h^{ρ}}) χ_{1} (\frac{s}{h^{ϵ / 2}}) w_{n} (h^{1 / 2} t), \end{array}$ and $\begin{matrix} L_{h, ζ} = - h^{2} a^{- 1} \partial_{t} (a \partial_{t}) - a^{- 2} {(h \partial_{s} - i ζ t (1 - \frac{t}{2} κ (s)))}^{2} . \end{matrix}$ Note that, $\begin{matrix} (3.18) & L_{h, ζ} u_{h} = c h^{- \frac{1 + ϵ}{4}} χ_{1} (\frac{s}{h^{ϵ / 2}}) P L_{h, ζ} χ_{1} (\frac{t}{h^{ρ}}) w_{n} (h^{1 / 2} t) + R_{h}, \end{matrix}$ where $\begin{matrix} (3.19) & P L_{h, ζ} = - h^{2} a^{- 1} \partial_{t} (a \partial_{t}) + a^{- 2} {(ζ t (1 - \frac{t}{2} κ (s)) - h^{1 / 2} ξ_{n})}^{2}, \end{matrix}$ and $\begin{array}{rcl} R_{h} & = & c h^{- \frac{1 + ϵ}{4}} χ_{1} (\frac{t}{h^{ρ}}) w_{n} (h^{1 / 2} t) [h^{2} \partial_{s}^{2} χ_{1} (\frac{s}{h^{ϵ / 2}})] \\ (3.20) & + 2 i c h^{- \frac{1 + ϵ}{4}} χ_{1} (\frac{t}{h^{ρ}}) w_{n} (h^{1 / 2} t) [(h^{1 / 2} ξ_{n} - ζ t (1 - \frac{t}{2} κ (s))) h \partial_{s}] χ_{1} (\frac{s}{h^{ϵ / 2}}) . \end{array}$ It is easy to check that $\begin{matrix} (3.21) & ‖ R_{h} ‖_{L^{2} (a d s d t)} ⩽ C ζ h^{\frac{3}{2} + \frac{ϵ}{2}} + C h^{2 - ϵ} ⩽ C h^{\frac{3}{2} + \frac{ϵ}{2}} . \end{matrix}$ Now we perform the change of variable $t = h^{1 / 2} τ$ and get ( $\tilde{a} = 1 - h^{1 / 2} τ κ (s)$ ), $\begin{matrix} \begin{matrix} P L_{h, ζ} & = h [- {\tilde{a}}^{- 1} \partial_{τ} (a \partial_{τ}) + {\tilde{a}}^{- 2} {(ζ τ (1 - \frac{h^{1 / 2} τ}{2} κ (s)) - ξ_{n})}^{2}] \\ = h H_{ζ, κ (s), ξ_{n}, h}, \end{matrix} \end{matrix}$ where the operator $H_{ζ, κ (s), ξ_{n}, h}$ is introduced in (2.11) with $β = κ (s)$ , $Δ_{β, τ} = h^{- 1 / 2} ({\tilde{a}}^{- 2} - 1)$ , $m = 0$ and $δ = \frac{1}{2} - ρ = \frac{1}{2} (\frac{1}{2} - 2 ϵ) < \frac{1}{2} - 2 ϵ$ . Now, it is easy to prove that (compare with (2.16)), $\begin{matrix} {‖ (h^{- 1} P L_{h, ζ} - (- 1 + f_{n} (ζ, ξ_{n}) - κ (s) h^{1 / 2})) χ_{1} (h^{\frac{1}{2} - ρ} τ) w_{n} (τ) ‖}_{L^{2} (\tilde{a} d τ)} ⩽ C h^{r}, \end{matrix}$ where $\begin{matrix} r = min ((2 n + 2) ϵ, \frac{1}{2} + 2 ϵ) . \end{matrix}$ Returning back to the t variable then integrating with respect to the s variable, we get, $\begin{matrix} {‖ (P L_{h, ζ} - (- h + h^{1 / 2} f_{n} (ζ, ξ_{n}) - κ (s) h^{1 / 2})) χ_{1} (h^{- ρ} t) w_{n} (h^{1 / 2} t) ‖}_{L^{2} (a d s d τ)} ⩽ C h^{\frac{3}{4} + r} . \end{matrix}$ Now, we insert this and (3.21) into (3.18) to get, $\begin{matrix} {‖ (L_{h, ζ} + h - e_{n} (ζ) + κ (s) h^{3 / 2}) u_{h} ‖}_{L^{2} (a d s d t)} ⩽ C h^{min (r, \frac{1}{2} + \frac{ϵ}{2})} = C h^{r_{*}} . \end{matrix}$ To finish the proof, we notice that $κ_{max} = κ (0)$ and in the support of the function $u_{h}$ , we have since $κ (\cdot)$ is a smooth function (it inherits this from the smoothness of the boundary) $\begin{matrix} | κ (s) - κ (0) | ⩽ C h^{1 / 8} . \end{matrix}$
The case $ϵ = \frac{1}{4}$ .
Now the trial function $v_{h}$ is defined using the $(s, t)$ -coordinates and the relation in (3.8) as follows, $\begin{matrix} (3.22) & {\tilde{v}}_{h} (s, t) = c h^{- 5 / 16} χ_{1} (\frac{t}{h^{7 / 16}}) χ_{1} (\frac{s}{h^{1 / 8}}) e^{- i φ_{0}} w_{1} (h^{1 / 2} t) exp (\frac{i ζ}{2 h^{1 / 2}}), \end{matrix}$ where the constant c and the function $χ_{1}$ are as in (3.15), and $w_{1}$ is the function defined in Theorem 2.3.

Performing a calculation similar to the one done for the case $ϵ < \frac{1}{4}$ (in particular, using (2.17) for $δ = \frac{1}{16}$ and $β = κ (s)$ ) we get, $\begin{array}{l} | ‖ v_{h} ‖_{L^{2} (Ω)} - 1 | ⩽ C h^{1 / 2}, \\ {‖ (P_{h, ζ} + h - \frac{1}{4} ζ^{2} + κ_{max} h^{3 / 2}) v_{h} ‖}_{L^{2} (Ω)} ⩽ C h^{13 / 8} . \end{array}$ The min–max principle now yields the desired upper bound for $λ_{1} (h, ζ)$ .
The case $ϵ > \frac{1}{4}$ .
Here we simply take the same trial state for the case without a magnetic field but times a phase (cf. [6,15]). Precisely, we define $v_{h}$ as follows, $\begin{matrix} (3.23) & {\tilde{v}}_{h} (s, t) = c h^{- 5 / 16} χ_{1} (\frac{t}{h^{3 / 8}}) χ_{1} (\frac{s}{h^{1 / 8}}) e^{- i φ_{0}} u_{0} (h^{1 / 2} t), \end{matrix}$ where the constant c and the function $χ_{1}$ are as in (3.15) and $u_{0} (τ) = \sqrt{2} exp (- τ)$ . Easy calculations similar to those done for $ϵ < \frac{1}{4}$ give us (cf. [6,15]) $\begin{array}{l} | ‖ v_{h} ‖_{L^{2} (Ω)} - 1 | ⩽ C h^{1 / 2}, \\ {‖ (P_{h, ζ} + h + κ_{max} h^{3 / 2}) v_{h} ‖}_{L^{2} (Ω)} ⩽ h^{13 / 8} . \end{array}$ The min–max principle now yields the upper bound for $λ_{1} (h, ζ)$ .
□

3.4. Concentration of bound states near the boundary

Theorem 3.3.
Let $0 < c_{1} < c_{2}$ , $ϵ > 0$ and $0 < α < 1$ . There exist constants $C > 0$ and $h_{0} \in (0, 1)$ such that, if $h \in (0, h_{0})$ , $ζ \in (c_{1} h^{ϵ}, c_{2} h^{ϵ})$ and $u_{h, ζ}$ is a $L^{2}$ -normalized ground state of $P_{h, ζ}$ , then, $\begin{matrix} \int_{Ω} ({| u_{h, ζ} (x) |}^{2} + h^{- 1} {| (h \nabla - i ζ A_{0}) u_{h, ζ} (x) |}^{2}) exp (\frac{2 α dist (x, \partial Ω)}{h^{1 / 2}}) d x ⩽ C . \end{matrix}$

The proof of Theorem 3.3 makes use of the result in:
Lemma 3.4.
Let $0 < c < 1$ and $0 < ρ < \frac{1}{2}$ be two constants. Under the assumptions in Theorem 3.3 , there exists $h_{0} \in (0, 1)$ such that, if $h \in (0, h_{0})$ , $w \in H^{1} (Ω)$ and $supp w \subset {dist (x, \partial Ω) ⩽ 2 h^{ρ}}$ , then $\begin{matrix} \int_{Ω} {| (h \nabla - i ζ A_{0}) w |}^{2} d x - h^{3 / 2} \int_{\partial Ω} | w |^{2} d x ⩾ - c h \int_{Ω} | w |^{2} d x . \end{matrix}$
Proof.
Let $q_{h, ζ} (\cdot)$ be the quadratic form in (3.3). The diamagnetic inequality yields, $\begin{matrix} q_{h, ζ} (w) ⩾ \int_{Ω} {| h \nabla | w | |}^{2} d x - h^{3 / 2} \int_{\partial Ω} | w |^{2} d s (x) . \end{matrix}$ In boundary coordinates, the inequality reads (cf. (3.9)), $\begin{matrix} q_{h, ζ} (w) ⩾ (1 - C h^{ρ}) \iint | h \nabla v |^{2} d s d t - h^{3 / 2} \int {| v (s, t = 0) |}^{2} d s, \end{matrix}$ where $v = | w \circ Φ (s, t) |$ (cf. (3.8)). Applying the change of the variable $t = h^{1 / 2} τ$ and comparing with the operator in (2.1), we get, $\begin{matrix} q_{h, ζ} (w) ⩾ - (1 - C h^{ρ}) h λ_{1} (H_{0, 0}) \iint | v (s, t) | d s d t = - (1 - C h^{ρ}) h \iint | v (s, t) | d s d t . \end{matrix}$ Returning back to Cartesian coordinates, we get the inequality in Lemma 3.4. □
Proof of Theorem 3.3.
The proof is similar to that of Theorem 5.1 in [6]. Let $t (x) = dist (x, \partial Ω)$ and $Φ (x) = exp (\frac{α t (x)}{h^{1 / 2}})$ . Also, for each $N ⩾ 1$ , we define the function $Φ_{N} (x) = g (\frac{t (x)}{N}) Φ (x)$ , where g is a smooth positive valued function satisfying $‖ g ‖_{\infty} ⩽ 1$ , $g = 1$ in $[0, 1]$ and $g = 0$ in $[2, \infty)$ . Note that $0 ⩽ Φ_{N} ⩽ Φ$ and when the domain Ω is bounded, we can find $N_{0} ⩾ 1$ such that, for all $N ⩾ N_{0}$ , $Φ_{N} = Φ$ in Ω.

We perform an integration by parts to write the following identity, $\begin{array}{l} q_{h}^{Φ_{N}} (u_{h, ζ}) : = & \int_{Ω} ({| (h \nabla - i ζ A_{0}) (Φ_{N} u_{h, ζ}) |}^{2} - h^{2} | \nabla Φ_{N} |^{2} | u_{h, ζ} |^{2}) d x - h^{3 / 2} \int_{\partial Ω} | Φ_{N} u_{h, ζ} |^{2} d s (x) \\ (3.24) & = & λ_{1} (h, ζ) ‖ Φ_{N} u_{h, ζ} ‖_{L^{2} (Ω)}^{2} . \end{array}$ Consider a partition of unity of $R$ $\begin{matrix} χ_{1}^{2} + χ_{2}^{2} = 1, \end{matrix}$ such that $χ_{1} = 1$ in $(- \infty, 1)$ , $supp χ_{1} \subset (- \infty, 2)$ , $χ_{1} ⩾ 0$ and $χ_{2} ⩾ 0$ in $R$ .

Define $\begin{matrix} χ_{j, h} (x) = χ_{j} (\frac{t (x)}{h^{1 / 2}}), j \in {1, 2} . \end{matrix}$ Associated with this partition of unity, we have the simple standard decomposition $\begin{matrix} (3.25) & q_{h, ζ}^{Φ_{N}} (u_{h, ζ}) = \sum_{j = 1}^{2} q_{j, h, ζ}^{Φ_{N}} (u_{h, ζ}), \end{matrix}$ where (by Lemma 3.4) $\begin{array}{l} q_{1, h, ζ}^{Φ_{N}} (u_{h, ζ}) = & \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{1, h} Φ_{N} u_{h, ζ}) |}^{2} - h^{2} {| \nabla (χ_{1, h} Φ_{N}) |}^{2} | u_{h, ζ} |^{2}) d x \\ - h^{3 / 2} \int_{\partial Ω} | χ_{1, h} Φ_{N} u_{h, ζ} |^{2} d s (x) \\ (3.26) & ⩾ & - \frac{h}{2} \int_{Ω} | χ_{1, h} Φ_{N} u_{h, ζ} |^{2} d x - \tilde{C} h \int_{Ω} | u_{h, ζ} |^{2} d x ⩾ - C h, \end{array}$ and $\begin{matrix} (3.27) & q_{2, h, ζ}^{Φ_{N}} (u_{h, ζ}) = \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ_{N} u_{h, ζ}) |}^{2} - h^{2} {| \nabla (χ_{2, h} Φ_{N}) |}^{2} | u_{h, ζ} |^{2}) d x . \end{matrix}$ The definition of $Φ_{N}$ and the fact $| \nabla t (x) | ⩽ 1$ a.e. together yield $\begin{matrix} h^{2} \int_{Ω} {| \nabla (χ_{2, h} Φ_{N}) |}^{2} | u_{h, ζ} |^{2} d x ⩽ (α^{2} h + \frac{C h^{2}}{N^{2}}) \int_{Ω} | χ_{2, h} Φ_{N} u_{h, ζ} |^{2} d x + C h \int_{Ω} | u_{h, ζ} |^{2} d x . \end{matrix}$ Note that the constant C above is independent of h and $N ⩾ 1$ . We insert this and (3.26) into (3.24), write $λ_{1} (h, ζ) ⩽ \frac{1}{2} (- 1 - α^{2})$ by Proposition 3.2 and rearrange the terms to obtain, $\begin{matrix} \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ_{N} u_{h, ζ}) |}^{2} + \frac{1}{2} (1 - α^{2}) h | χ_{2, h} Φ_{N} u_{h, ζ} |^{2}) d x ⩽ C h . \end{matrix}$ Since $Φ_{N} (x) = Φ (x)$ for $t (x) ⩽ N$ , we get $\begin{matrix} \forall N ⩾ 1, \int_{Ω \cap {t (x) ⩽ N}} ({| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ u_{h, ζ}) |}^{2} + \frac{1}{2} (1 - α^{2}) h | χ_{2, h} Φ u_{h, ζ} |^{2}) d x ⩽ C h . \end{matrix}$ Now we take $N \to \infty$ and get by monotone convergence $\begin{matrix} (3.28) & \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ u_{h, ζ}) |}^{2} + \frac{1}{2} (1 - α^{2}) h | χ_{2, h} Φ u_{h, ζ} |^{2}) d x ⩽ C h . \end{matrix}$ This is enough to deduce the estimate in Theorem 3.3. □

We record the following simple corollary of Theorem 3.3. Corollary 3.5.
Let $ρ \in (0, \frac{1}{2})$ , $ϵ > 0$ and $0 < c_{1} < c_{2}$ . There exists $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ and $ζ \in (c_{1} h^{ϵ}, c_{2} h^{ϵ})$ , every $L^{2}$ -normalized ground state $u_{h, ζ}$ of the operator $P_{h, ζ}$ satisfies, $\begin{matrix} \int_{c_{1} h^{ρ} ⩽ dist (x, \partial Ω) ⩽ c_{2} h^{ρ}} | u_{h, ζ} |^{2} d x ⩽ exp (- c_{1} h^{ρ - \frac{1}{2}}) . \end{matrix}$

3.5. Lower bound for the principal eigenvalue

Proposition 3.6.
Let $ϵ > 0$ and $0 < c_{1} < c_{2}$ . There exist constants $C > 0$ , $h_{0} \in (0, 1)$ and $r^{} > \frac{3}{2}$ such that, for all* $h \in (0, h_{0})$ and $ζ \in (c_{1} h^{ϵ}, c_{2} h^{ϵ})$ , the ground state energy in ( 3.4 ) satisfies, $\begin{matrix} λ_{1} (h, ζ) ⩾ - h + b_{ϵ} (ζ) h - κ_{max} h^{3 / 2} - C h^{r^{}}, \end{matrix}$ where* $b_{ϵ} (ζ)$ is introduced in ( 3.13 ).

For $ϵ > \frac{1}{4}$ , Proposition 3.6 follows from:
Lemma 3.7.
Suppose that $ϵ > \frac{1}{4}$ . Under the assumption in Proposition 3.6 , for all u in the form domain of the operator $P_{h, ζ}$ , $\begin{matrix} q_{h, ζ} (u) ⩾ \int_{Ω} U_{h, ζ} (x) | u |^{2} d x, \end{matrix}$ where $\begin{matrix} U_{h, ζ} (x) = \{\begin{matrix} - h - κ (s (x)) h^{3 / 2} - C h^{7 / 4} & if dist (x, \partial Ω) < 2 h^{1 / 8}, \\ 0 & if dist (x, \partial Ω) ⩾ 2 h^{1 / 8}, \end{matrix} \end{matrix}$ and $q_{h, ζ} (\cdot)$ is the quadratic form in ( 3.3 ).
Proof.
This is a consequence of the diamagnetic inequality and [6, Theorem 5.2]. □

In the case $ϵ ⩽ \frac{1}{4}$ , Proposition 3.6 is a consequence of Lemma 3.8 below (applied with $w = u_{h, ζ}$ and $u_{h, ζ}$ a $L^{2}$ normalized ground state of the operator $P_{h, ζ}$ ) and the variational min–max principle.

The constant $r^{}$ in Proposition 3.6 depends on ϵ. It is introduced as follows. For all $ϵ > 0$ , let $\begin{array}{l} (3.29) & σ = \{\begin{matrix} \frac{1}{5} min (2 ϵ, 1 - 4 ϵ) & if ϵ < \frac{1}{4}, \\ 1 / 8 & if ϵ ⩾ \frac{1}{4}, \end{matrix} \\ (3.30) & ρ = \{\begin{matrix} \frac{1}{2} - \frac{1}{4} min (2 ϵ, 1 - 4 ϵ) & if ϵ < 1 / 4, \\ 7 / 16 & if ϵ = 1 / 4, \\ 1 / 8 & if ϵ > 1 / 4, \end{matrix} \\ (3.31) & r^{} = \{\begin{matrix} min (1 + (2 n + 2) ϵ, \frac{3}{2} + 2 ϵ, \frac{3}{2} + σ, 2 ϵ + 4 ρ + 2 σ - \frac{1}{2}, 1 + σ + ρ, 2 - 2 σ) & if ϵ ⩽ \frac{1}{4}, \\ \frac{7}{4} & if ϵ > \frac{1}{4}, \end{matrix} \end{array}$ where $n \in N$ is the smallest positive integer satisfying $(2 n + 2) ϵ > \frac{1}{2}$ .

Note that $0 < ρ < \frac{1}{2}$ , $0 < σ < 1$ and when $ϵ ⩽ \frac{1}{4}$ , the following three conditions are satisfied $\begin{matrix} (3.32) & \{\begin{matrix} 2 ϵ + 4 ρ + 2 σ - \frac{1}{2} > \frac{3}{2}, \\ 2 - 2 σ > \frac{3}{2}, \\ ρ + σ > \frac{1}{2} . \end{matrix} \end{matrix}$ Consequently, for all $ϵ > 0$ , the number $r^{}$ satisfies $\begin{matrix} r^{} > \frac{3}{2} . \end{matrix}$ Lemma 3.8.
Let $M > 0$ , $0 < ϵ ⩽ \frac{1}{4}$ and $0 < c_{1} < c_{2}$ . There exist two constants $C > 0$ and $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ and $ζ \in (c_{1} h^{ϵ}, c_{2} h^{ϵ})$ , if w is a $L^{2}$ normalized function in the form domain of the operator $P_{h, ζ}$ and $\begin{matrix} (3.33) & {‖ exp (\frac{dist (x, \partial Ω)}{2 h^{ρ^{'}}}) w ‖}_{L^{2} (Ω)} ⩽ M, \end{matrix}$ for some $ρ < ρ^{'} < \frac{1}{2}$ , then it holds the following: $\begin{matrix} (3.34) & q_{h, ζ} (w) ⩾ \int_{Ω} U_{h, ζ} (x) | w |^{2} d x - C h^{2} . \end{matrix}$ Here
$U_{h, ζ} (x) = \{\begin{matrix} - h + b_{ϵ} (ζ) - κ (s (x)) h^{3 / 2} - C h^{r^{}} & if dist (x, \partial Ω) < 2 h^{ρ}, \\ 0 & if dist (x, \partial Ω) ⩾ 2 h^{ρ}; \end{matrix}$

σ, ρ and* $r^{}$ are introduced in (* 3.29 ), ( 3.30 ) and ( 3.31 ) respectively;

$b_{ϵ} (ζ)$ is introduced in ( 3.13 );

$q_{h, ζ} (\cdot)$ is the quadratic form introduced in ( 3.3 ).

Proof of Lemma 3.8.
The lengthy proof of Lemma 3.8 is divided into four steps. Step 1 (Localization near the boundary).

Consider a partition of unity of $R$ , $\begin{matrix} χ_{1}^{2} + χ_{2}^{2} = 1 \end{matrix}$ with $χ_{1} = 1$ in $(- \infty, 1]$ , $supp χ_{1} \subset (- \infty, 2]$ and $supp χ_{2} \subset [1, \infty)$ . For $j \in {1, 2}$ , put, $\begin{matrix} χ_{j, h} (x) = χ_{j} (\frac{dist (x, \partial Ω)}{h^{ρ}}) . \end{matrix}$ We have the decomposition $\begin{matrix} q_{h, ζ} (w) = q_{h, ζ} (χ_{1, h} w) + q_{h, ζ} (χ_{2, h} w) - h^{2} \sum_{j = 1}^{2} {‖ | \nabla χ_{j, h} | w ‖}_{L^{2} (Ω)}^{2}, \end{matrix}$ where $\begin{matrix} q_{h, ζ} (χ_{2, h} w) = \int_{Ω} {| h \nabla (χ_{2, h} w) |}^{2} d x ⩾ 0, \end{matrix}$ and by (3.33), $\begin{matrix} h^{2} {‖ | \nabla χ_{j, h} | w ‖}_{L^{2} (Ω)}^{2} ⩽ h^{2 - 2 ρ} exp (- \frac{1}{4} h^{ρ - ρ^{'}}) = O (h^{\infty}) . \end{matrix}$ Thus, $\begin{matrix} (3.35) & q_{h, ζ} (w) ⩾ q_{h, ζ} (χ_{1, h} w) + O (h^{\infty}) . \end{matrix}$

Step 2 (Analysis near the boundary).

Let us cover the boundary $\partial Ω$ by a family of open disks $(B (x_{j}, h^{1 / 8}))$ . Let $(f_{j}) \subset C^{\infty} (\partial Ω)$ be a partition of unity in $\partial Ω$ such that, for all j, $\begin{matrix} supp f_{j} \subset B (x_{j}, h^{σ}) \cap \partial Ω, and | \nabla f_{j} | ⩽ C h^{- σ} . \end{matrix}$ We extend $f_{j}$ in the tubular neighborhood ${dist (x, \partial Ω) < 2 h^{ρ}}$ of the boundary via the formula $\begin{matrix} f_{j} (x) = f_{j} (s (x)) . \end{matrix}$ We decompose the boundary term in (3.35) as follows, $\begin{array}{l} q_{h, ζ} (χ_{1, h} w) & = \sum_{j} q_{h, ζ} (f_{j} χ_{1, h} w) - h^{2} \sum_{j} {‖ | \nabla f_{j} | χ_{1, h} w ‖}_{L^{2} (Ω)}^{2} \\ (3.36) & ⩾ \sum_{j} q_{h, ζ} (f_{j} χ_{1, h} w) - C h^{2 - 2 σ} ‖ χ_{1, h} w ‖_{L^{2} (Ω)}^{2} . \end{array}$ 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 $\begin{matrix} κ_{j} = κ (s (x_{j})) . \end{matrix}$ By smoothness of the curvature and boundedness of the boundary, we know that $\begin{matrix} | κ (s (x)) - κ_{j} | ⩽ m h^{σ} in B (x_{j}, h^{σ}), \end{matrix}$ where $\begin{matrix} m = sup_{x \in \partial Ω} | κ^{'} (s (x)) | . \end{matrix}$ That way we get the following pointwise lower bound in every $B (x_{j}, h^{σ})$ , $\begin{matrix} {| (h \partial_{s} - i ζ t (1 - \frac{1}{2} t κ (s))) \tilde{w} |}^{2} ⩾ (1 - h^{1 / 2}) {| (h \partial_{s} - i ζ t (1 - \frac{1}{2} t κ_{j})) \tilde{w} |}^{2} - C ζ^{2} t^{4} h^{2 σ - \frac{1}{2}} | \tilde{w} |^{2} . \end{matrix}$ Let $φ_{j} = φ_{x_{j}}$ be the function satisfying (3.12) in $B (x_{j}, 2 h^{σ} + 2 h^{ρ})$ . Define the function $v_{j} = {\tilde{f}}_{j} {\tilde{χ}}_{1, h} \times \tilde{w} e^{- i φ_{j}}$ . 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, $\begin{array}{l} q_{h, ζ} (f_{j} χ_{1, h} w) \\ ⩾ \iint (| h \partial_{t} v_{j} |^{2} \\ + (1 - h^{1 / 2}) {(1 - t κ_{j} - C h^{σ} t)}^{- 2} {| (h \partial_{s} - i ζ t (1 - \frac{1}{2} t κ_{j})) \tilde{w} |}^{2}) (1 - κ_{j} t - m h^{σ} t) d s d t \\ (3.37) & - \int {| v_{j} (s, t = 0) |}^{2} d s - C \iint ζ^{2} t^{4} h^{2 σ - \frac{1}{2}} | v_{j} |^{2} (1 - κ_{j} t - m h^{σ} t) d s d t . \end{array}$

Let $\begin{matrix} (3.38) & δ = \frac{1}{2} - ρ, β = κ_{j}, Δ_{β, τ} = h^{- 1 / 2} [(1 - h^{1 / 2}) {(1 - (κ_{j} + C h^{σ}) h^{1 / 2} τ)}^{- 2} - 1] . \end{matrix}$ Note that, for $t \in (0, h^{\frac{1}{2} - ρ})$ and $τ = h^{- \frac{1}{2}} t$ , $| Δ_{β, τ} | ⩽ C (| β | + 1) τ$ . Thus, we can apply the results in Section 2.3.

We return back to (3.37). Note that $ζ = O (h^{ϵ})$ and in the support of $v_{j}$ , the term $t^{4}$ is of order $O (h^{4 ρ})$ . We apply the change of variable $t = h^{1 / 2} τ$ then the Fourier transform with respect to the variable s to obtain, $\begin{matrix} (3.39) & q_{h, ζ} (f_{j} χ_{1, h} w) ⩾ \iint {(h inf_{ξ \in R} λ_{1} (H_{ζ, β, ξ, h})) - C h^{2 ϵ + 4 ρ + 2 σ - \frac{1}{2}}} | f_{j} χ_{1, h} w |^{2} (1 - κ_{j} t - m h^{σ} t) d s d t . \end{matrix}$

Step 3 (Lower bound in the case $ϵ < \frac{1}{4}$ ).

By the assumption on ρ and σ, we find that $\begin{matrix} 0 < δ = \frac{1}{2} - ρ < \frac{1}{2} - 2 ϵ \end{matrix}$ so that we can apply Proposition 2.8. Let $r^{*} > \frac{3}{2}$ be the constant introduced in (3.31). We infer from (3.39), $\begin{matrix} (3.40) & q_{h, ζ} (f_{j} χ_{1, h} w) ⩾ \iint {- h + h e_{n} (ζ) - κ_{j} h^{1 / 2} - C h^{r_{*}}} | f_{j} χ_{1, h} w |^{2} (1 - t κ_{j} - m h^{σ} t) d s d t . \end{matrix}$ Now, in (3.39), we replace $κ_{j}$ by $κ (s) + O (h^{σ})$ and use that $σ + ρ > \frac{1}{2}$ to replace the term $1 - t κ_{j} - m h^{σ} t$ by $1 - t κ (s)$ and get, $\begin{matrix} (3.41) & q_{h, ζ} (f_{j} χ_{1, h} w) ⩾ \iint {- h + h e_{n} (ζ) - κ (s) h^{1 / 2} - C h^{r_{*}}} | f_{j} χ_{1, h} w |^{2} (1 - t κ (s)) d s d t . \end{matrix}$ We insert (3.41) into (3.36) and obtain $\begin{matrix} \begin{matrix} q_{h, ζ} (χ_{1, h} w) & ⩾ \sum_{j} \iint {- h + h e_{n} (ζ) - κ (s) h^{1 / 2} - C h^{r_{*}}} | f_{j} χ_{1, h} w |^{2} (1 - t κ (s)) d s d t - C h^{2} \\ ⩾ \int_{Ω} {- h + h e_{n} (ζ) - κ (s (x)) h^{1 / 2} - C h^{r_{*}}} | χ_{1, h} w |^{2} d x - C h^{2} . \end{matrix} \end{matrix}$ Using that $| χ_{1, h} w | ⩽ | w |$ and that $- h + h e_{n} (ζ) - κ (s) h^{1 / 2} - C h^{r_{*}} < 0$ , we get further, $\begin{matrix} q_{h, ζ} (χ_{1, h} w) = \int_{Ω} {- h + h e_{n} (ζ) - κ (s) h^{1 / 2} - C h^{r_{*}}} | w |^{2} d x - C h^{2} . \end{matrix}$ Finally, we insert this into (3.35) to get (3.34).

Step 4 (Lower bound in the case $ϵ = \frac{1}{4}$ ).

The analysis here is similar to that in Step 3 and we will be rather succinct. Note that the assumption on δ and ρ ensure that $0 < δ = \frac{1}{2} - ρ = \frac{1}{16} < \frac{1}{8}$ so that we can apply Proposition 2.9 and infer from (3.39), $\begin{array}{l} q_{h, ζ} (f_{j} χ_{1, h} w) & ⩾ \iint {- h + \frac{1}{4} ζ^{2} h - κ_{j} h^{3 / 2} - C h^{r^{*}}} | f_{j} χ_{1, h} w |^{2} (1 - κ_{j} t - m h^{σ} t) d s d t \\ (3.42) & ⩾ \iint {- h + \frac{1}{4} ζ^{2} h - κ (s) h^{3 / 2} - C h^{r^{*}}} | f_{j} χ_{1, h} w |^{2} (1 - κ (s) t) d s d t . \end{array}$ We insert this into (3.36) and (3.35) to get (3.34) for $ϵ = \frac{1}{4}$ . □

3.6. Concentration of ground states near the points of maximal curvature

Theorem 3.9.
Suppose that the boundary of Ω is $C^{4}$ smooth. Let $0 < c_{1} < c_{2}$ and $ϵ > 0$ . There exist constants $ρ \in (0, \frac{1}{2})$ , $η^{} \in (0, \frac{1}{4})$ , $C > 0$ and* $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ , $ζ \in (c_{1} h^{ϵ}, c_{2} h^{ϵ})$ and $u_{h, ζ}$ a normalized ground state of the operator $P_{h, ζ}$ , $\begin{matrix} \int_{{dist (x, \partial Ω) ⩽ h^{ς}} \cap {κ_{max} - κ (s (x)) ⩾ h^{η^{}}}} | u_{h, ζ} |^{2} d x ⩽ C exp (- h^{η^{} - \frac{1}{4}}) . \end{matrix}$

We will prove Theorem 3.9 in the case $0 < ϵ ⩽ \frac{1}{4}$ . The case $ϵ > \frac{1}{4}$ is a standard consequence of the inequality in Lemma 3.7 (cf. [4, Theorem 8.3.4]).

An important ingredient in the proof of Theorem 3.9 is: Lemma 3.10.
Let $ϵ > 0$ be a fixed constant, ρ and $r^{}$ be as in (* 3.30 ) and ( 3.31 ) respectively. There exist two constants $\bar{C} > 0$ and $h_{0} \in (0, 1)$ such that, for all $h \in (0, h_{0})$ and u in the form domain of the operator $P_{h, ζ}$ , $\begin{matrix} q_{h, ζ} (u) ⩾ \int_{Ω} V_{h, ζ} (x) | u |^{2} d x, \end{matrix}$ where $\begin{matrix} V_{h, ζ} (x) = \{\begin{matrix} - h / 2 & if dist (x, \partial Ω) ⩾ 2 h^{ρ}, \\ - h + b_{ϵ} (ζ) h - κ (s (x)) h^{3 / 2} - \bar{C} h^{r^{}} & if dist (x, \partial Ω) < 2 h^{ρ}, \end{matrix} \end{matrix}$ and* $q_{h, ζ}$ is the quadratic form in ( 3.3 ).
Proof.
Let $\tilde{μ}$ be the ground state energy of the operator $P_{h, ζ} - V_{h, ζ}$ , where $V_{h, ζ}$ is defined for some constant $\bar{C}$ . We will prove that we can choose the constant $\bar{C}$ such that $\tilde{μ} > 0$ (for sufficiently small values of h).

The min–max principle and Theorem 3.3 together yield $\begin{matrix} \tilde{μ} ⩽ {⟨ (P_{h, ζ} - V_{h, ζ}) u_{h, ζ}, u_{h, ζ} ⟩}_{L^{2} (Ω)} = λ_{1} (h, ζ) - \int_{Ω} V_{h, ζ} | u_{h, ζ} |^{2} d x ⩽ \tilde{C} h^{r^{}} . \end{matrix}$

Let $ρ^{'} \in (ρ, \frac{1}{2})$ and w be a $L^{2}$ normalized ground state of the operator $P_{h, ζ} - V_{h, ζ}$ . We will prove that, $\begin{matrix} (3.43) & {‖ exp (\frac{t (x)}{h^{ρ^{'}}}) w ‖}_{L^{2} (Ω)}^{2} ⩽ C, \end{matrix}$ where $t (x) = dist (x, \partial Ω)$ . We will write the proof of (3.43) for the case of a bounded domain. If Ω is unbounded, we argue as in Lemma 3.4. We perform an integration by parts to write the following identity, $\begin{array}{l} {\tilde{q}}_{h}^{Φ} (u_{h, ζ}) : = & \int_{Ω} ({| (h \nabla - i ζ A_{0}) (Φ w) |}^{2} - V_{h} | Φ w |^{2} - h^{2} | \nabla Φ |^{2} | u_{h, ζ} |^{2}) d x - h^{3 / 2} \int_{\partial Ω} | Φ w |^{2} d s (x) \\ (3.44) & = & \tilde{μ} ‖ Φ w ‖_{L^{2} (Ω)}^{2} . \end{array}$ Here $Φ (x) = exp (\frac{t (x)}{h^{ρ^{'}}})$ . (Note that, if Ω is unbounded, we have to work with the function $Φ_{N} (x)$ instead of $Φ (x)$ and take $N \to \infty$ , cf. the proof of Lemma 3.4.) Consider a partition of unity of $R$ $\begin{matrix} χ_{1}^{2} + χ_{2}^{2} = 1, \end{matrix}$ such that $χ_{1} = 1$ in $(- \infty, 1)$ , $supp χ_{1} \subset (- \infty, 2)$ , $χ_{1} ⩾ 0$ and $χ_{2} ⩾ 0$ in $R$ .

Define $\begin{matrix} χ_{j, h} (x) = χ_{j} (\frac{t (x)}{h^{ρ^{'}}}), j \in {1, 2} . \end{matrix}$ Associated with this partition of unity, we have the simple standard decomposition $\begin{matrix} (3.45) & {\tilde{q}}_{h, ζ}^{Φ} (w) = \sum_{j = 1}^{2} {\tilde{q}}_{j, h, ζ}^{Φ} (w), \end{matrix}$ where $\begin{array}{rcl} {\tilde{q}}_{1, h, ζ}^{Φ} (w) & = & \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{1, h} Φ w) |}^{2} - V_{h, ζ} | χ_{1, h} Φ w |^{2} - h^{2} {| \nabla (χ_{1, h} Φ) |}^{2} | w |^{2}) d x \\ (3.46) & - h^{3 / 2} \int_{\partial Ω} | χ_{1, h} Φ w |^{2} d s (x), \end{array}$ and $\begin{matrix} (3.47) & {\tilde{q}}_{2, h, ζ}^{Φ} (w) = \int_{Ω} ({| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ w) |}^{2} - V_{h, ζ} | χ_{2, h} Φ w |^{2} - h^{2} {| \nabla (χ_{2, h} Φ) |}^{2} | w |^{2}) d x . \end{matrix}$ Lemma 3.4, the bound $V_{h, ζ} ⩽ 0$ and the normalization of $u_{h, ζ}$ together yield $\begin{matrix} {\tilde{q}}_{1, h, ζ}^{Φ} (u_{h, ζ}) ⩾ - C h . \end{matrix}$ We insert this into (3.45), then we insert the obtained inequality into (3.44), use the bounds $- V_{h, ζ} ⩾ \frac{1}{2} h$ , $\tilde{μ} ⩽ \tilde{C} h^{r^{}}$ and then rearrange the terms to obtain, $\begin{array}{rcl} (3.48) & \int_{Ω} ({| (h \nabla + i ζ A_{0}) (χ_{2, h} Φ u_{h, ζ}) |}^{2} + h (\frac{1}{2} - C h^{1 - 2 ρ^{'}} - \tilde{C} h^{r^{} - 1}) | χ_{2, h} Φ u_{h, ζ} |^{2}) d x ⩽ C h . \end{array}$ Using the inequality $\frac{1}{2} - C h^{1 - 2 ρ^{'}} - C h^{r^{} - 1} ⩾ \frac{1}{4}$ then dividing by h, we get $\begin{matrix} \int_{Ω} (h^{- 1} {| (h \nabla - i ζ A_{0}) (χ_{2, h} Φ w) |}^{2} + \frac{1}{4} | χ_{2, h} Φ w |^{2}) d x ⩽ C . \end{matrix}$ This is enough to deduce the estimate in (3.43).

Having proved (3.43), we may use the result in Lemma 3.8 and write, $\begin{matrix} q_{h, ζ} (w) ⩾ \int_{Ω} U_{h, ζ} (x) | w |^{2} d x - C h^{2} . \end{matrix}$ Note that, by selecting $\bar{C} > 0$ sufficiently large (in the definition of $V_{h, ζ}$ ), we may write $\begin{matrix} U_{h, ζ} (x) - C h^{2} ⩾ V_{h, ζ} (x) in Ω . \end{matrix}$ This finishes the proof of Lemma 3.10. □
Proof of Theorem 3.9.
Define the function $ϕ (s) = κ_{max} - κ (s)$ . The function ϕ is positive valued and periodic with period $| \partial Ω |$ . Furthermore, since $\partial Ω$ is $C^{4}$ smooth, the function ϕ is $C^{2}$ smooth and the boundedness of $\partial Ω$ yields the existence of a constant $M > 0$ such that $| ϕ^{″} | ⩽ M$ . Applying Taylor’s formula to order 2, we get for all s, θ, $\begin{matrix} 0 ⩽ ϕ (s + θ) = ϕ (s) + θ ϕ^{'} (s) + \frac{θ^{2}}{2} ϕ^{″} (a) ⩽ ϕ (s) + θ ϕ^{'} (s) + \frac{θ^{2}}{2} M . \end{matrix}$ We choose $θ = \frac{- ϕ^{'} (s)}{M}$ . In this way, we get for $C_{0} : = 2 M$ , $\begin{matrix} \forall s \in (0, | \partial Ω |), {| ϕ^{'} (s) |}^{2} ⩽ C_{0} ϕ (s) . \end{matrix}$ Let σ and ρ be as in (3.29) and (3.30). Choose $χ \in C_{c}^{\infty} (R)$ such that $0 ⩽ χ ⩽ 1$ in $R$ , $χ = 1$ in $[- t_{0}, t_{0}]$ and $supp χ \subset [- 1, 1]$ . Here $t_{0} < 1$ is a geometric constant such that the boundary coordinates $(s, t)$ are valid in the tubular neighborhood ${dist (x, \partial Ω) < t_{0}}$ (cf. Section 3.2).

Define the function $\begin{matrix} Ψ (x) = exp (\frac{δ χ (t (x)) ϕ (s (x))}{h^{1 / 4}}), \end{matrix}$ where $t (x) = dist (x, \partial Ω)$ and $δ \in (0, 1)$ . We will fix a choice for δ at a later point.

Let us write the following decomposition formula $\begin{matrix} λ_{1} (h, ζ) ‖ Ψ u_{h, ζ} ‖_{L^{2} (Ω)}^{2} + h^{3 / 2} δ^{2} {‖ | \nabla ψ | Ψ u_{h, ζ} ‖}_{L^{2} (Ω)}^{2} = q_{h, ζ} (Ψ u_{h, ζ}), \end{matrix}$ where $\begin{matrix} ψ (x) = χ (t (x)) ϕ (s (x)) . \end{matrix}$ Using Theorem 3.3 and the bound $| ϕ^{'} |^{2} ⩽ C_{0} ϕ$ , we may write, for a new constant $C > 0$ , $\begin{matrix} {‖ | \nabla ψ | Ψ u_{h, ζ} ‖}_{L^{2} (Ω)}^{2} ⩽ O (h^{\infty}) + C \iint_{t < 2 t_{0}} | ϕ (s) | | Ψ u_{h, ζ} |^{2} d s d t . \end{matrix}$ We use the upper bound for $λ_{1} (h, ζ)$ in Proposition 3.2, the lower bound for $q_{h, ζ} (\cdot)$ in Lemma 3.10 and the simple lower bound $V_{h, ζ} - λ_{1} (h, ζ) ⩾ \frac{1}{4} h$ in ${t (x) ⩾ 2 h^{ρ}}$ , we get, $\begin{matrix} \iint_{t < 2 h^{ρ}} (h^{3 / 2} ϕ (s) - 2 C h^{3 / 2} δ^{2} ϕ (s) - C h^{r^{}}) | Ψ u_{h, ζ} |^{2} d s d t ⩽ C h^{r^{}} . \end{matrix}$ We choose $δ = \frac{1}{2 \sqrt{C}}$ and get $\begin{matrix} \iint_{t < 2 t_{0}} (\frac{1}{2} ϕ (s) - C h^{r^{} - \frac{3}{2}}) | Ψ u_{h, ζ} |^{2} d s d t ⩽ C h^{r^{}} . \end{matrix}$ In particular, setting $η = max (r^{} - \frac{3}{2}, \frac{1}{4})$ , we write $\begin{matrix} \iint_{\begin{array}{c} t < 2 h^{2 ρ} \\ ϕ (s) ⩾ 4 C h^{η} \end{array}} ϕ (s) | Ψ u_{h, ζ} |^{2} d s d t ⩽ C . \end{matrix}$ Selecting $η^{} \in (0, η)$ finishes the proof of Theorem 3.9. □

4. Analysis of diamagnetism

Proof of Theorem 1.1

The assumption in Theorem 1.1 roughly says that $H \approx | β |^{σ}$ for $σ > 0$ . There exists a simple relationship between the eigenvalues in (1.13) and (1.5). This relationship is displayed as follows $\begin{matrix} {\tilde{μ}}_{1} (β; H) = H^{2} μ_{1} (h; b, α, γ), \end{matrix}$ where $h = H^{- 1}$ , $b = 1$ , $γ = β H^{- 1 + α}$ and $α = 1 - \frac{1}{σ}$ .

The assumption in Theorem 1.1 ensures that

As $β \to - \infty$ , the semi-classical parameter $h \to 0_{+}$ ;

The parameter γ is uniformly bounded, i.e. $γ = O (1)$ as $β \to - \infty$ .

The ground state energy

μ_{1} (h; b, α, γ)

is estimated in Theorem 1.2 for

α < \frac{1}{2}

(more precisely, this is a consequence of Theorem 3.1). Since

α = 1 - \frac{1}{σ}

, this yields the estimates announced for

{\tilde{μ}}_{1} (β; H)

in Theorem 1.1 for

σ < 2

For $α ⩾ \frac{1}{2}$ , the following result is proved in [10] $\begin{matrix} μ_{1} (h; b = 1, α, γ) = \{\begin{matrix} Θ (0) h + h o (1) & if α > \frac{1}{2}, \\ Θ (γ) h + h o (1) & if α = \frac{1}{2} . \end{matrix} \end{matrix}$ Consequently, we get the asymptotic expansions in Theorem 1.1 for $σ ⩾ 2$ .

Footnotes

Acknowledgement

The author is supported by a grant from the Lebanese University through ‘Equipe de Modelisation, Analyse et Applications’.

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Diamagnetism versus Robin condition and concentration of ground states

Abstract

Keywords

1. Introduction

1.1. Diamagnetism

1.4. On the notation and organization of the paper

2. Analysis of auxiliary operators

2.1. 1D Laplacian on the half line

2.2. Harmonic oscillator on the half-line

Step 2 (The iteration process).

2.3.1. Harmonic oscillator on an interval

3.1. Semi-classical Laplacian with weak magnetic field

3.3. Upper bound for the principal eigenvalue

Step 2 (Analysis near the boundary).

Step 3 (Lower bound in the case ϵ < 1 4 ).

Step 4 (Lower bound in the case ϵ = 1 4 ).

3.6. Concentration of ground states near the points of maximal curvature

Proof of Theorem 1.1

Footnotes

Acknowledgement

References

Step 3 (Lower bound in the case $ϵ < \frac{1}{4}$ ).

Step 4 (Lower bound in the case $ϵ = \frac{1}{4}$ ).