Non-monotonicity of the first eigenvalue for the 3D magnetic Robin Laplacian

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer, Kachmar, and Raymond to provide one of the first counterexamples in 3D. Considering the magnetic Robin Laplacian on the unit ball with a constant magnetic field, we show the non-monotonicity of the lowest eigenvalue asymptotics when the Robin parameter tends to +∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+\infty $$\end{document}.


Main theorem.
In [10], the authors considered the magnetic Robin Laplacian in the unit ball Ω = {x ∈ R 3 : |x| < 1} with a constant magnetic field (see (6)). They established a precise asymptotics for the lowest eigenvalue of the operator when the Robin parameter tends to +∞ (see Section 1.2). In particular, if the magnetic field is uniform of strength b > 0, the asymptotics for the lowest eigenvalue λ(γ, b) is given by as γ → +∞, where e(b) = inf m∈Z λ m (b) and λ m (b) is the first eigenvalue, given by where H = L 2 ((0, π); sin θ dθ), and We denote by S m,b the self-adjoint operator associated with the closed quadratic form q m,b . Recall that Numerical computations stated in [10] suggest a non-monotonic behaviour. Our goal is to give a formal proof of this.

Discussion and motivation.
Let us put the result into context. The identification of domains for which the lowest eigenvalue for the magnetic Laplacian is monotone or not with respect to the strength of the magnetic field has been actively studied in the past years. For weak magnetic fields, the diamagnetic inequality gives a monotonic behaviour. Moreover, for strong magnetic fields, monotonicity, known as strong diamagnetism, has also been proved for a large variety of domains in R 2 [1,2,4] and also in R 3 [5,7]. A detailed discussion and summary of this can be found in [6].
For a magnetic field, whose strength lies in the middle region, less is known. Non-monotone phase transitions occur in domains having specific topological properties. A famous example of these phenomena is the Little-Parks effect for 2D annuli [3,8,14], where an oscillatory behaviour in the critical temperature of the superconductor appears as the magnetic field varies. Similar phenomena are observed for thin domains [9]. In the disk, counterexamples can be found applying a non-uniform magnetic field [8] or by imposing Robin boundary condition with a strong coupling parameter [13]. The topological defects can also be induced by an Aharonov-Bohm magnetic potential [11,12].
However, counterexamples in three dimensions are less studied. Using the asymptotics of the lowest eigenvalue for the magnetic Robin Laplacian when the Robin parameter goes to +∞, obtained in [10], we are able to provide one of the first counterexamples.
Vol. 120 (2023) Non-monotonicity for the 3D magnetic Robin Laplacian 645 The magnetic Robin Laplacian is given by with domain where γ > 0 is the Robin parameter, n the unit inward 1 pointing normal vector of ∂Ω, and a is any magnetic vector potential that generates the magnetic field (0, 0, b), for example take a(x 1 , x 2 , x 1 , 0). The asymptotics of the lower eigenvalue of P γ when γ → +∞ can be obtained using the associated quadratic form The relevant semiclassical parameter is h = γ −2 and we can rewrite Q γ in terms of this parameter. Introducing spherical coordinates (r, θ, ϕ) and decomposing into Fourier modes, with respect to ϕ ∈ [0, 2π], we get a family of quadratic forms indexed by m ∈ Z. Having this, q m,b can be used to bound the part depending on θ to obtain the asymptotic expansion of λ(γ, b) stated before.
The text is organized as follows. In Section 2, we prove an auxiliary result regarding the spectrum of the self-adjoint operator associated to q m,b . In Section 3, we prove Theorem 1.

Preliminary result.
The goal of this section is showing that the spectrum of the self-adjoint operator S m,b associated with the quadratic form q m,b (see (3)) is discrete. We show this by proving that D(q m,b ) is compactly embedded in H. The proof relies on a diagonal sequence argument after doing an exhaustion by compact sets of (0, π). This is needed because the weight vanishes at the endpoints of the interval.

Lemma 2.
If m ∈ Z and b ∈ R, then S m,b has purely discrete spectrum.