1 Introduction

This paper is about how little we know and how much the current mathematical tools are limited.

Among all membranes of a given area, the circular one produces the lowest fundamental tone. This is a well-known interpretation of the celebrated Faber–Krahn inequality stating that

$$\begin{aligned} \lambda _1(\Omega ) \ge \lambda _1(\Omega ^*) \,, \end{aligned}$$
(1)

where \(\Omega \) is any bounded open planar set, \(\Omega ^*\) is the disk of the same area and \(\lambda _1(\Omega )\) is the lowest eigenvalue of the boundary value problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&= \lambda u & \text{ in } & \Omega \,, \\ u&= 0 & \text{ on } & \partial \Omega \,. \end{aligned} \right. \end{aligned}$$
(2)

Restricting to rectangles, it is also true that the square is the optimal geometry for the Dirichlet Laplacian. More specifically, defining

$$\begin{aligned} \Omega _{a,b} := \left( -\frac{a}{2},\frac{a}{2}\right) \times \left( -\frac{b}{2},\frac{b}{2}\right) , \end{aligned}$$
(3)

where a, b are any positive numbers, the inequality (1) remains true for any rectangle \(\Omega _{a,b}\) instead of the arbitrary domain \(\Omega \) and a square \(\Omega _{a,b}^*:=\Omega _{a,a}\) instead of the disk \(\Omega ^*\) whenever the area \(|\Omega _{a,b}|=ab\) is fixed. While the general proof based on symmetrisation techniques applies to arbitrary quadrilaterals [1, Sec. 3.3], the case of rectangles can alternatively be established in an elementary way just by using the well-known fact that the problem (2) is explicitly solvable by separation of variables in terms of sine and cosine functions. We refer to [2] for a recent spectral optimisation of the Laplacian eigenvalues in the larger generality of rectangular boxes with Robin boundary conditions.

Interpreting the Laplacian as the free Schrödinger operator and the Dirichlet condition as hard-wall boundaries, the result (1) equally says that the ground-state energy of a non-relativistic quantum particle constrained to nanostructures of a given material is minimised by the disk. It has been conjectured recently that (1) also holds in the relativistic setting of the Dirac operator with infinite-mass boundary conditions instead of the Dirichlet Laplacian, see [3] (respectively, [4]) for arbitrary domains (respectively, rectangles). Despite a strong numerical evidence [3] (respectively, [5]) and partial analytical results [3, 6, 7] (respectively, [4, 5]), however, the proof of the relativistic conjectures seems to be beyond the reach of current mathematical apparatus. Indeed, no symmetrisation techniques are available for the Dirac operator. What is more, the case of rectangles is just illusively simpler, for the Dirac problem cannot be solved by separation of variables.

The objective of this paper is to point out yet another spectral optimisation problem which remains a mystery in the case of rectangles. Physically, it is still about the non-relativistic quantum particle constrained to \(\Omega \), but now subject to homogeneous magnetic fields. The ground-state energy of this system coincides with the lowest eigenvalue \(\lambda _1^B(\Omega )\) of the boundary value problem

$$\begin{aligned} \left\{ \begin{aligned} (-i\nabla -A)^2 u&= \lambda u & \text{ in } & \Omega \,, \\ u&= 0 & \text{ on } & \partial \Omega \,, \end{aligned} \right. \end{aligned}$$
(4)

where \(A:\overline{\Omega }\rightarrow \mathbb {R}^2\) is any smooth vector potential generating a constant magnetic field \({\textrm{curl}}A =: B \in \mathbb {R}\). By the Erdős–Faber–Krahn inequality [8] (see also [9]), one still has the magnetic variant of (1):

$$\begin{aligned} \lambda _1^B(\Omega ) \ge \lambda _1^B(\Omega ^*) \,, \end{aligned}$$
(5)

valid for every open set \(\Omega \subset \mathbb {R}^2\) of fixed area \(|\Omega | = |\Omega ^*|\) and any \(B \in \mathbb {R}\). Analogous (reversed) inequalities have recently been studied for the principal eigenvalue of the magnetic Neumann Laplacian [10, 11]. In the case of rectangles, the following conjecture remains open.

Conjecture 1

For every \(a>0\) and \(B \in \mathbb {R}\),

$$\begin{aligned} \lambda _1^B(\Omega _{a,a^{-1}}) \ge \lambda _1^B(\Omega _{1,1}) \,. \end{aligned}$$
(6)

Here we consider the class of rectangles of area equal to 1, but there is no loss of generality in this restriction, for other values can be recovered by scaling (see (10) below).

Remark 1

Following [4], it is also possible to state an analogous conjecture with a fixed perimeter instead of the fixed area. However, the isoperimetric constraint is known to be easier. Indeed, it is implied by Conjecture 1 via scaling and the classical (geometric) isoperimetric inequality.

We emphasise that the general validity of Conjecture 1 is far from being obvious. Indeed, one gets easily convinced that the problem (4) with \(\Omega _{a,b}\) instead of \(\Omega \) cannot be solved by separation of variables. At the same time, no suitable symmetrisation techniques are available for complex-valued functions. In this paper, we show that Conjecture 1 holds in the regime of weak magnetic fields.

Theorem 1

There exists a positive constant C (independent of a) such that (6) holds for every \(|B| \le C\).

We leave as an open problem to show that (6) holds for large values of |B|, too. This regime is semiclassical in nature [12], but the available techniques do not apply in the present setting of non-smooth boundary. It seems also reasonable to conjecture that a reverse inequality will hold for the magnetic Neumann Laplacian. Finally, apart from the homogeneous magnetic field, an interesting optimisation problem consists in considering the Aharonov–Bohm vector potential in the spirit of [13].

The organisation of this paper is as follows. In Section 2 we settle the problem in terms of a spectral problem of a self-adjoint operator in a fixed (i.e. a-independent) Hilbert space. Section 3 is not needed for the proof of Theorem 1, but it offers a robust way how to establish Conjecture 1 as a consequence of a symmetry of a non-convex optimisation. In Section 4 we show that the square is a local minimiser for weak magnetic fields, establishing thus Conjecture 1 for rectangles close to the square. In Section 5 we establish upper and lower bounds to the eigenvalue \(\lambda _1^B(\Omega _{a,a^{-1}})\), obtaining thus Conjecture 1 for rectangles far from the square.

2 Preliminaries

The boundary value problem (4) in the rectangle \(\Omega _{a,a^{-1}} =: \Omega _a\) is understood as the spectral problem for the self-adjoint operator \(H_a^A\) in \(L^2(\Omega _a)\) associated with the form

$$\begin{aligned} h_a^A[u] := \Vert \partial _1^A u\Vert ^2 + \Vert \partial _2^A u\Vert ^2 \,, \qquad u \in \textsf{D}(h_a^A) := W_0^{1,2}(\Omega _a) \,, \end{aligned}$$
(7)

where \(\Vert \cdot \Vert \) denotes the norm of \(L^2(\Omega _a)\) and we abbreviate \(\partial _k^A:= \partial _k-iA_k\) with \(k \in \{1,2\}\).

The spectrum of \(H_a^A\) is independent of the choice of the vector potential A giving the same magnetic field \({\textrm{curl}}A = B\). Indeed, if \(\tilde{A}:\overline{\Omega _a} \rightarrow \mathbb {R}^2\) is another smooth potential satisfying \({\textrm{curl}}\tilde{A} = B\), then \({\textrm{curl}}(\tilde{A}-A)=0\), so there exists a smooth function \(\phi :\overline{\Omega _a} \rightarrow \mathbb {R}\) satisfying \(\tilde{A}-A = \nabla \phi \). Consequently, \(H_a^{\tilde{A}} = e^{i\phi } H_a^A e^{-i\phi }\), which means that the operators \(H_a^{\tilde{A}}\) and \(H_a^A\) are unitarily equivalent, therefore isospectral. This is the well-known gauge invariance. Without loss of generality, we choose the gauge

$$\begin{aligned} A(x) := \big (-\theta x_2,(1-\theta )x_1\big ) B \,, \end{aligned}$$
(8)

where \(x \in \Omega _{a}\) and \(\theta ,B \in \mathbb {R}\) are arbitrary.

A variational characterisation of the lowest eigenvalue \(\lambda _1^B(\Omega _a)\) reads

$$\begin{aligned} \lambda _1^B(\Omega _a) = \inf _{{\mathop {u}\limits ^{[}} \not = 0]{}{u \in W_0^{1,2}(\Omega _a) }} \frac{h_a^A[u]}{\, \Vert u\Vert ^2} \,, \end{aligned}$$
(9)

where the infimum can be replaced by a minimum. From this formula, it is straightforward to deduce the scaling behaviour

$$\begin{aligned} \lambda _1^B(\varepsilon \, \Omega _a) = \frac{1}{\varepsilon ^2} \, \lambda _1^{\varepsilon ^2 B}(\Omega _a) \,, \end{aligned}$$
(10)

where \( \varepsilon \, \Omega _a := \{\varepsilon x : x \in \Omega _a\} = \Omega _{\varepsilon a,\varepsilon a^{-1}} \), with any positive \(\varepsilon \). For rectangles of arbitrary area, Theorem 1 therefore implies that the lowest eigenvalue is minimised by the square of the same area, provided that the magnetic field is sufficiently small (the smaller the area, the larger the constant bounding the magnitude of the magnetic field from above).

The variational formula (9) also implies \(\lambda _1^B(\Omega _a) \ge \lambda _1^0(\Omega _a)\). This is a consequence of the diamagnetic inequality (see, e.g., [14, Thm. 7.21] or [15, Thm. 2.1.1])

$$\begin{aligned} \big |(\nabla - iA) u(x)\big | \ge \big |\nabla |u|(x)\big | \end{aligned}$$
(11)

valid pointwise for almost every \(x \in \Omega _a\) with \(u \in W_\textrm{loc}^{1,2}(\Omega _a)\).

It is useful to work in a Hilbert space independent of the parameter a. More specifically, we introduce the unitary transform \( U: L^2(\Omega _a) \rightarrow L^2(\Omega _1) \) by setting \((Uu)(x):=u(a x_1,a^{-1} x_2)\) and define a unitarily equivalent (therefore isospectral) operator \(\hat{H}_{a}^A:= U H_{a}^A U^{-1}\). It is associated with the form \(\hat{h}_a^A\) defined by \(\hat{h}_a^A[u]:= h_a^A[U^{-1}u]\) and \(\textsf{D}(\hat{h}_a^A):= U \textsf{D}(h_a^A)\). It is easily verified that

$$\begin{aligned} \hat{h}_a^A[u] = a^{-2} \, \Vert \partial _1^A u\Vert ^2 + a^2 \, \Vert \partial _2^A u\Vert ^2 \,, \qquad u \in \textsf{D}(\hat{h}_a^A) = W_0^{1,2}(\Omega _1) \,, \end{aligned}$$
(12)

where \(\Vert \cdot \Vert \) denotes the norm of \(L^2(\Omega _1)\) and A still means (8) but with \(x \in \Omega _1\) now. Clearly, \(\hat{H}_{1}^A = H_{1}^A\). Moreover, \(\textsf{D}(\hat{h}_{a}^A)=\textsf{D}(\hat{h}_{1}^A)\) for every \(a>0\), so we actually have \(\textsf{D}(\hat{h}_{a}^A) = \textsf{D}(h_{1}^A)\).

A variational characterisation of the lowest eigenvalue \(\lambda _1^B(\Omega _a)\) reads

$$\begin{aligned} \lambda _1^B(\Omega _a) = \inf _{{\mathop {u}\limits ^{[}} \not = 0]{}{u \in W_0^{1,2}(\Omega _1) }} \frac{\hat{h}_a^A[u]}{\, \Vert u\Vert ^2} \,, \end{aligned}$$
(13)

where the infimum can be replaced by a minimum.

3 From Symmetry to Optimality

In this section, we explain how the robust idea of [4] suggested to prove the optimality of a square among all Dirac rectangles can be adapted to the present magnetic setting.

Using the inequality between the arithmetic and geometric means, (13) implies

$$\begin{aligned} \lambda _1^B(\Omega _a) \ge \inf _{{\mathop {u}\limits ^{[}} \not = 0]{}{u \in W_0^{1,2}(\Omega _1) }} \frac{2 \, \Vert \partial _1^A u\Vert \, \Vert \partial _2^A u\Vert }{\, \Vert u\Vert ^2} =: \mu \,. \end{aligned}$$
(14)

The minimisation problem on the right-hand side of (14) does not involve a convex functional. In fact, the associated Euler equation is a non-linear problem. Recalling that we use the symbol \(\Vert \cdot \Vert \) for the norm of \(L^2(\Omega _1)\), let us denote by \((\cdot ,\cdot )\) the corresponding inner product.

Lemma 1

The infimum on the right-hand side of (14) is achieved. Any minimiser u satisfies the weak eigenvalue equation

$$\begin{aligned} \alpha ^{-2} \, (\partial _1^A v,\partial _1^A u) + \alpha ^2 \, (\partial _2^A v,\partial _2^A u) = \mu \, (v,u) \end{aligned}$$
(15)

for every \(v \in W_0^{1,2}(\Omega _1)\), where

$$ \alpha := \sqrt{\frac{\Vert \partial _1^A u\Vert }{\Vert \partial _2^A u\Vert }} \,. $$

Proof

First of all, let us argue that the infimum in (14) is indeed achieved. Define the functional

$$\begin{aligned} J[u] := 2\,\Vert \partial _1^A u\Vert \, \Vert \partial _2^A u\Vert \,, \qquad u \in \textsf{D}(J) := W_0^{1,2}(\Omega _1) \,. \end{aligned}$$

Then

$$\begin{aligned} \mu = \inf _{{\mathop {\Vert }\limits ^{[}}u\Vert = 1]{} {u \in W_0^{1,2}(\Omega _1)}} J[u] \,. \end{aligned}$$
(16)

The functional J is coercive and lower semicontinuous.

  • To see that J is coercive, we observe that the diamagnetic and Poincaré inequalities imply \( \Vert \partial _k^A u\Vert \ge \Vert \partial _k |u|\Vert \ge \pi \Vert u\Vert \) for every \(u \in W_0^{1,2}(\Omega _1)\) and \(k \in \{1,2\}\). At the same time, \( \Vert \partial _k^A u\Vert ^2 \ge \frac{1}{2} \Vert \partial _k u\Vert ^2 - \Vert A_k\Vert _\infty ^2 \Vert u\Vert ^2 \). These two inequalities can be combined in an elementary way to show that there exists a positive constant c such that \(J[u] \ge c \, \Vert u\Vert _{W^{1,2}(\Omega _1)}\).

  • To see that J is lower semicontinuous, we use the facts that the product of two lower semicontinuous functions is lower semicontinuous and that the square root of a lower semicontinuous function is lower semicontinuous. So, it remains to show that \(u \mapsto \Vert \partial _1^A u\Vert ^2\) and \(u \mapsto \Vert \partial _2^A u\Vert ^2\) are lower semicontinuous on \(L^2(\Omega _1)\), with the convention that the action is \(+\infty \) if \(u \not \in W_0^{1,2}(\Omega _1)\). Let us focus on the former, the proof for the latter is analogous. For every \(n \in \mathbb {N}\), define

    $$ h_n[u] := \left( u,n \, (-i\partial _1^A)^2 \, [n+(-i\partial _1^A)^2]^{-1} u \right) \,, \qquad \textsf{D}[h_n] := L^2(\Omega _1) \,, $$

    where \((-i\partial _1^A)^2\) is understood as a self-adjoint realisation of this operator in \(L^2((-\frac{1}{2}.\frac{1}{2}))\), subject to Dirichlet boundary conditions, and we denote by the same symbol the operator \((-i\partial _1^A)^2 \otimes 1\) in \(L^2(\Omega _1)\). Then \(h_n\) is bounded on the unit ball of \(L^2(\Omega _1)\) and continuous. By using the spectral theorem, \(h_n[u]\) increases monotonically to \(h[u]:= \Vert \partial _1^A u\Vert ^2\) for every \(u \in L^2(\Omega _1)\) (recall that \(h[u]=+\infty \) if \(u \not \in W_0^{1,2}(\Omega _1)\)). This implies that h is lower semicontinuous.

Let \(\{u_j\}_{j \in \mathbb {N}}\) be a minimising sequence, i.e. \(J[u_j] \rightarrow \mu \) as \(j \rightarrow \infty \) and \(\Vert u_j\Vert =1\) for every \(j \in \mathbb {N}\). Consequently,

$$\begin{aligned} c \, \Vert \nabla u_j\Vert \le J[u_j] = \mu \,. \end{aligned}$$

It follows that \(\{u_j\}_{j \in \mathbb {N}}\) is a bounded sequence in \(W^{1,2}(\Omega _1)\). Therefore, up to a subsequence, \(\{u_j\}_{j \in \mathbb {N}}\) converges weakly to some \(\psi \) in \(W^{1,2}(\Omega _1)\). By the compactness of the embedding \(W^{1,2}(\Omega _1)\) in \(L^2(\Omega _1)\), we may assume that \(\{u_j\}_{j \in \mathbb {N}}\) converges (strongly) to some u in \(L^2(\Omega _1)\) such that \(\Vert u\Vert =1\). By using u as a trial function in (16), we obviously have \(\mu \le J[u]\). On the other hand,

$$\begin{aligned} \mu = \liminf _{j\rightarrow \infty } J[u_j] \ge J[u] \,, \end{aligned}$$

where the inequality follows by the property that J is lower semicontinuous. In summary, \(\mu = J[u]\), so the infimum in (16) can be replaced by a minimum.

Now, let u be any minimiser of (16). Then u is a critical point of the functional J and the derivative

$$ \lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon } \left( \frac{J[u+\varepsilon v]}{\Vert u+\varepsilon v\Vert ^2} - \frac{J[u]}{\Vert u\Vert ^2} \right) $$

is necessarily equal to zero for any choice of the test function \(v \in W_0^{1,2}(\Omega _1)\). This leads to the equation

$$\begin{aligned} \alpha ^{-2} \, \Re (\partial _1^A v,\partial _1^A u) + \alpha ^2 \, \Re (\partial _2^A v,\partial _2^A u) = \mu \, \Re (v,u) \,. \end{aligned}$$

Combining this equation with its variant where v is replaced by iv, it is clear that the real part can be removed, so (15) follows.

Finally, let us argue that (15) is well defined, meaning that \(\alpha \) is positive and bounded. If \(\Vert \partial _1^A u\Vert =0\), then the diamagnetic inequality implies \(\Vert \partial _1 |u| \Vert =0\), so |u| is independent of the first variable, which is incompatible with \(|u| \in W_0^{1,2}(\Omega _1))\) unless \(|u| = 0\) identically. An analogous argument excludes the possibility \(\Vert \partial _2^A u\Vert =0\). \(\square \)

The following symmetry is naturally expected for the symmetric gauge (8) with \(\theta =\frac{1}{2}\).

Conjecture 2

If \(\theta =\frac{1}{2}\), then there exists a minimiser u of the right-hand side of (14) which satisfies

$$\begin{aligned} \Vert \partial _1^A u\Vert = \Vert \partial _2^A u\Vert \,. \end{aligned}$$
(17)

Note that this conjecture is true provided that the minimisation problem on the right-hand side of (14) admits a unique minimiser. Indeed, it is easy to see that if u is a minimiser, then so is the rotated function \(x \mapsto u(-x_2,x_1) =: v(x)\). The uniqueness of the minimiser together with the identities \(\Vert \partial _1^A v\Vert = \Vert \partial _2^A u\Vert \) and \(\Vert \partial _2^A v\Vert = \Vert \partial _1^A u\Vert \) then implies (17).

Because of the non-linear structure of the minimisation problem and the lack of positivity preserving property for the magnetic Laplacian, we have been able to establish neither the uniqueness of the minimiser nor Conjecture 2, respectively. This is unfortunate, because its validity immediately implies Conjecture 1.

Theorem 2

Conjecture 2 implies Conjecture 1.

Proof

Let us assume \(\theta =\frac{1}{2}\). As a consequence of (14) and Conjecture 2,

$$ \begin{aligned} \lambda _1^B(\Omega _a) \ge \inf _{{\mathop {u}\limits ^{[}} \not = 0]{} {u \in W_0^{1,2}(\Omega _1) \ {\mathrm{ \& }} \ (17) \text { holds}}} \frac{2\,\Vert \partial _1^A u\Vert \, \Vert \partial _2^A u\Vert }{\, \Vert u\Vert ^2} \end{aligned}$$
(18)

for every \(a>0\) and \(B \in \mathbb {R}\). At the same time, because of the rotational symmetry of the square \(\Omega _1\), it is easy to see that there exists an eigenfunction u of \(\hat{H}_1^A\) satisfying (17). Consequently,

$$ \begin{aligned} \begin{aligned} \lambda _1^B(\Omega _1)&= \inf _{{\mathop {u}\limits ^{[}} \not = 0]{} {\psi \in W_0^{1,2}(\Omega _1) \ {\mathrm{ \& }} \ (17) \text { holds}}} \frac{\Vert \partial _1^A u\Vert ^2 + \Vert \partial _2^A u\Vert ^2}{\, \Vert u\Vert ^2} \\&= \inf _{{\mathop {u}\limits ^{[}} \not = 0]{} {\psi \in W_0^{1,2}(\Omega _1) \ {\mathrm{ \& }} \ (17) \text { holds}}} \frac{2\,\Vert \partial _1^A u\Vert \, \Vert \partial _2^A u\Vert }{\, \Vert u\Vert ^2} \,. \end{aligned} \end{aligned}$$
(19)

Comparing (18) and (19), we get Conjecture 1. \(\square \)

4 From Simplicity to Local Optimality

Unable to prove Conjecture 1 in its full generality, in this section we focus on showing that the square \(\Omega _1\) is at least a local minimiser among all rectangles \(\Omega _a\) with \(a>0\).

For simplicity, we abbreviate \(\lambda _a:= \lambda _1^B(\Omega _a)\).

Lemma 2

Assume \(\theta =\frac{1}{2}\). Let \(B \in \mathbb {R}\) be such that the eigenvalue \(\lambda _1\) is simple. Then

$$\begin{aligned} \left. \frac{\partial \lambda _1^B(\Omega _a)}{\partial a} \right| _{a=1}&= 0 \,, \end{aligned}$$
(20)
$$\begin{aligned} \frac{1}{2} \left. \frac{\partial ^2 \lambda _1^B(\Omega _a)}{\partial a^2} \right| _{a=1}&= 2 \, \lambda _1^B(\Omega _1) + \lambda _1^B(\Omega _1) \, \Vert \dot{u}_1\Vert ^2 - \Vert \partial _1^A \dot{u}_1\Vert ^2 - \Vert \partial _2^A \dot{u}_1\Vert ^2 \,, \end{aligned}$$
(21)

where \(\dot{u}_1\) is the solution of the problem

$$\begin{aligned} (\partial _1^A v,\partial _1^A \dot{u}_1) + (\partial _2^A v,\partial _2^A \dot{u}_1) - \lambda _1^B(\Omega _1) \, (v,\dot{u}_1) = 2 \, (\partial _1^A v,\partial _1^A u_1) - 2 \, (\partial _2^A v,\partial _2^A u_1) \end{aligned}$$
(22)

for every \(v \in W_0^{1,2}(\Omega _1)\), with \(u_1\) being the eigenfunction of \(\hat{H}_1^A\) corresponding to \(\lambda _1^B(\Omega _1)\) and normalised by \(\Vert u_1\Vert =1\).

Proof

It is straightforward to verify that \(\{\hat{h}_a^A\}_{a > 0}\) is a holomorphic family of forms of type (a) in the sense of Kato [16, Sec. VII.4]. Indeed, one can directly use the criterion [16, Sec. VII.4.8]. Consequently, \(\{\hat{H}_a^A\}_{a > 0}\) is a holomorphic family of operators of type (B). Because of the simplicity hypothesis, \(a \mapsto \lambda _1^B(\Omega _a) =: \lambda _a\) is a real-analytic function on a neighbourhood of \(a=1\). At the same time, the corresponding eigenfunction \(u_a\) satisfying the normalisation \(\Vert u_a\Vert =1\) is a real-analytic function in the topology of \(W_0^{1,2}(\Omega _1)\) on a neighbourhood of \(a=1\). Then the claims follow by a routine differentiation of the weak formulation of the eigenvalue equation

$$\begin{aligned} a^{-2} \, (\partial _1^A v,\partial _1^A u_a) + a^2 \, (\partial _2^A v,\partial _2^A u_a) = \lambda _a \, (v,u_a) \end{aligned}$$
(23)

for every \(v \in W_0^{1,2}(\Omega _1)\).

In detail, differentiating (23) with respect to a and denoting the corresponding derivative by a dot, we find

$$\begin{aligned} \begin{array}{c} a^{-2} \, (\partial _1^A v,\partial _1^A \dot{u}_a) + a^2 \, (\partial _2^A v,\partial _2^A \dot{u}_a) - 2 a^{-3} \, (\partial _1^A v,\partial _1^A u_a) + 2a \, (\partial _2^A v,\partial _2^A u_a)\\ = \lambda _a \, (v,\dot{u}_a) + \dot{\lambda }_a \, (v,u_a) \,. \end{array}\end{aligned}$$
(24)

This equation reduces to (22) for \(a=1\). Taking \(v=u_\alpha \) in (24), \(v = \dot{u}_\alpha \) in (23) and combining these two identities evaluated at \(a=1\), we find

$$\begin{aligned} \dot{\lambda }_1 = -2 \, \Vert \partial _1^A u_1\Vert ^2 + 2 \, \Vert \partial _2^A u_1\Vert ^2 = 0 \,, \end{aligned}$$
(25)

where the second equality follows by the rotational symmetry of the square. Indeed, \(u_1\) necessarily satisfies (17) with \(u_1\) instead of u. This establishes (20).

Differentiating (24) with respect to a, we find

$$\begin{aligned} \begin{array}{c} a^{-2} \, (\partial _1^A v,\partial _1^A \ddot{u}_a) + a^2 \, (\partial _2^A v,\partial _2^A \ddot{u}_a) - 4 a^{-3} \, (\partial _1^A v,\partial _1^A \dot{u}_a) + 4a \, (\partial _2^A v,\partial _2^A \dot{u}_a) \\ + 6 a^{-4} \, (\partial _1^A v,\partial _1^A u_a) + 2 \, (\partial _2^A v,\partial _2^A u_a) = \lambda _a \, (v,\ddot{u}_a) + 2 \dot{\lambda }_a \, (v,\dot{u}_a) + \ddot{\lambda }_a \, (v,u_a) \,. \end{array}\end{aligned}$$
(26)

Taking \(v=u_\alpha \) in (26), \(v = \ddot{u}_\alpha \) in (23), combining these two identities evaluated at \(a=1\) and using (25), we find

$$ \begin{aligned} \ddot{\lambda }_1&= 6 \, \Vert \partial _1^A u_1\Vert ^2 + 2 \, \Vert \partial _2^A u_1\Vert ^2 - 4 \, (\partial _1^A u_1,\partial _1^A \dot{u}_1) + 4 \, (\partial _2^A u_1,\partial _2^A \dot{u}_1) \,. \end{aligned} $$

Using the symmetry and (22), we arrive at (21). \(\square \)

The desired local optimality of the square is equivalent to showing that the second derivative of \(\lambda _1^B(\Omega _a)\) with respect to a is positive. However, it is not clear whether the right-hand side of (21) is positive. We prove it for weak magnetic fields.

Theorem 3

There exists a positive constant C such that, for every \(|B| \le C\),

$$\begin{aligned} \left. \frac{\partial \lambda _1^B(\Omega _a)}{\partial a} \right| _{a=1} = 0 \qquad \text{ and } \qquad \left. \frac{\partial ^2 \lambda _1^B(\Omega _a)}{\partial a^2} \right| _{a=1} > 0 \,. \end{aligned}$$
(27)

Proof

Recall that we use the gauge (8) explicitly depending on B. It is straightforward to verify that \(\{\hat{h}_a^A\}_{B \in \mathbb {R}}\) is a holomorphic family of forms of type (a). Again, one can directly use the criterion [16, Sec. VII.4.8]. Consequently, \(\{\hat{H}_a^A\}_{B \in \mathbb {R}}\) is a holomorphic family of operators of type (B). Since \(\lambda _1^0(\Omega _a)\) is simple for every \(a>0\), there exists a positive constant C such that \(\lambda _1^B(\Omega _1)\) remains simple whenever \(|B| \le C\). It follows that \(B \mapsto \lambda _1^B(\Omega _1) =: \lambda ^B\) is a real-analytic function on a neighbourhood of \(B=0\). At the same time, the corresponding eigenfunction \(u^B\) satisfying the normalisation \(\Vert u^B\Vert =1\) is a real-analytic function in the topology of \(W_0^{1,2}(\Omega _1)\) on a neighbourhood of \(B=0\).

Assuming \(\theta =\frac{1}{2}\), Lemma 2 implies the first identity of (27). However, the hypothesis \(\theta =\frac{1}{2}\) is redundant, because \(\lambda _1^B(\Omega _a)\) is gauge invariant.

At the same time, under the hypothesis \(\theta =\frac{1}{2}\), Lemma 2 implies the identity (21). Here the first term on the right-hand side is positive for all sufficiently small |B|; explicitly, \(\lambda ^B \rightarrow 2\pi ^2\) as \(B \rightarrow 0\). At the same time, the right-hand side of (22) tends to zero as \(B \rightarrow 0\). Indeed, \(u^0(x_1,x_2) = \varphi (x_1) \varphi (x_2)\), where

$$\begin{aligned} \varphi (x) := \sqrt{2} \, \cos (\pi x_1) \end{aligned}$$
(28)

is the first eigenfunction of the Dirichlet Laplacian in \(L^2((-\frac{1}{2},\frac{1}{2}))\) normalised to 1, which ensures that \(\Vert \partial _1^A u^0\Vert = \Vert \partial _2^A u^0\Vert \). Consequently,

$$ \lambda _1^B(\Omega _1) \, \Vert \dot{u}_1\Vert ^2 - \Vert \partial _1^A \dot{u}_1\Vert ^2 - \Vert \partial _2^A \dot{u}_1\Vert ^2 \rightarrow 0 $$

as \(B \rightarrow 0\). In summary, the right-hand side of (21) converges to \(4\pi ^2\) as \(B \rightarrow 0\). It follows that there exists a positive constant C (possibly smaller than the previously chosen C) such that the right-hand side of (21) is positive for all \(|B| \le C\). Again, the smallness of |B| and the positivity must be independent of the choice of gauge. \(\square \)

5 Quantitative Bounds

To establish the global result of Theorem 1, in this section we establish explicit upper and lower bounds to the eigenvalue \(\lambda _1^B(\Omega _a)\).

Given \(\beta \in \mathbb {R}\), let \(\nu (\beta )\) denote the lowest eigenvalue of the operator \(T_\beta := -\partial _x^2 + \beta ^2 x^2\) in \(L^2((-\frac{1}{2},\frac{1}{2}))\), subject to Dirichlet boundary conditions. It is easy to see that

$$\begin{aligned} \pi ^2 \le \nu (\beta ) \le \pi ^2 + c \, \beta ^2 \qquad \text{ with } \qquad c := \int _{-\frac{1}{2}}^{\frac{1}{2}} x^2 \, \varphi (x)^2 \, \textrm{d}x = \frac{1}{12} - \frac{1}{2\pi ^2} \approx 0.03 \,, \end{aligned}$$
(29)

where \(\varphi \) is given in (28). While these estimates are good for small \(|\beta |\), we have \(\nu (\beta ) \sim |\beta |\) as \(|\beta | \rightarrow \infty \). See Figure 1 for the dependence of \(\nu (\beta )\) on \(\beta \). Let \(\varphi _\beta \) denote the positive eigenfunction of \(T_\beta \) corresponding to \(\nu (\beta )\) and normalised to 1 in \(L^2((-\frac{1}{2},\frac{1}{2}))\). Of course, \(\varphi _0 = \varphi \).

Fig. 1
figure 1

The eigenvalue \(\nu (\beta )\) as a function of \(\beta \) together with its asymptotics for small and large \(\beta \)

Full size image

We start with an upper bound.

Proposition 1

For every \(a>0\) and \(B \in \mathbb {R}\),

$$\begin{aligned} \lambda _1^B(\Omega _a) \le \pi ^2 \, (a^{-2}+a^2) + c \, \frac{a^2}{1+a^4} \, B^2 \,. \end{aligned}$$
(30)

Proof

Using \(x \mapsto \varphi _{\beta _1}(x_1) \varphi _{\beta _2}(x_2)\) with \(\beta _1:= a^2(1-\theta ) B\) and \(\beta _2:= a^{-2}\theta B\) as a trial function in (13), we get

$$\begin{aligned} \lambda _1^B(\Omega _a)&\le a^{-2} \, \nu \big (a^2(1-\theta ) B\big ) + a^2 \, \nu \big (a^{-2}\theta B\big ) \end{aligned}$$
(31)
$$\begin{aligned}&= (a^{-2} + a^2) \, \nu \left( \frac{a^2}{1+a^4} \, B \right) \end{aligned}$$
(32)

where equality follows by the special choice \(\theta := a^4/(1+a^4)\) (this is the best choice if the trial function \(x \mapsto \varphi _{0}(x_1) \varphi _{0}(x_2)\) were directly used instead). The announced bound (30) follows by (29). \(\square \)

As for the lower bound, we have a trivial result, which follows at once by the diamagnetic inequality.

Proposition 2

For every \(a>0\) and \(B \in \mathbb {R}\),

$$\begin{aligned} \lambda _1^B(\Omega _a) \ge \pi ^2 \, (a^{-2}+a^2) \,. \end{aligned}$$
(33)

Remark 2

The bounds (30) and (33) are enough for our purposes of weak magnetic fields. For strong fields, a better upper bound is given by (32). Indeed, the inequality becomes sharp as \(|B| \rightarrow \infty \). A better lower bound for strong fields is given by the uniform bound \(\lambda _1^B(\Omega _a) \ge |B|\). It follows at once by the domain monotonicity, by estimating \(H_a^A\) from below by the operator which acts in the same way but on the entire plane (the Landau Hamiltonian). Another lower bound, using the domain monotonicity \(\Omega _{a,a^{-1}} \subset (-\frac{a}{2},\frac{a}{2})\times \mathbb {R}\) with \(\theta =0\) or \(\Omega _{a,a^{-1}} \subset \mathbb {R}\times (-\frac{a}{2},\frac{a}{2})\) with \(\theta =1\), reads

$$ \lambda _1^B(\Omega _a) \ge \max \left\{ a^{-2} \nu (a^2 B),a^2 \nu (a^{-2} B) \right\} \,. $$

Now we are in a position to establish Theorem 1.

Proof of Theorem 1

Using Propositions 1 (with \(a=1\)) and 2 (with arbitrary \(a>0\)), we have

$$\begin{aligned} \lambda _1^B(\Omega _a) \ge \pi ^2 \, (a^{-2}+a^2) {\mathop {\ge }\limits ^{!}} 2 \pi ^2 + \frac{c}{2} \, B^2 \ge \lambda _1^B(\Omega _1) \,, \end{aligned}$$
(34)

where the central inequality (with !) holds provided that

$$ a^{-2}+a^2 \ge 2 + \frac{c}{2\pi ^2} \, B^2 \,. $$

In particular, this is satisfied if

$$ |a-1| \ge \sqrt{\frac{c}{2\pi ^2}} \, |B| \,. $$

In summary, Conjecture 1 holds provided that \(|a-1|\) is sufficiently large, with the largeness diminishing when \(B \rightarrow 0\). To complete this argument for small values of \(|a-1|\), we use Theorem 3. \(\square \)