On the spectral gap of higher-dimensional Schrödinger operators on large domains

Abstract

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.

Keywords

Bessel functions fundamental gap spectral theory spectral gap Schrödinger operator

1. Introduction

We investigate the spectral gap, that is the difference between the two lowest eigenvalues of two- and higher-dimensional Schrödinger operators for certain (non-negative and bounded) potentials in a limit where the volume of the underlying domain tends to infinity. In one dimension, investigations of the spectral gap have a long history, see [1,1,4,9,10] and references therein. In higher dimensions, a milestone has been the proof of the fundamental gap conjecture by Andrews and Clutterbuck [3]. This conjecture stated that on any bounded convex domain $Ω \subset R^{n}$ , the spectral gap of the Dirichlet Laplacian with a (weakly) convex potential is larger or equal than the spectral gap of the Laplacian without potential on an interval with length equal to the diameter of Ω.

Our paper is motivated by a recent observation on the behaviour of the spectral gap of one-dimensional Schrödinger operators on intervals in the limit where the interval length tends to infinity. More explicitly, in [7] it was shown that the spectral gap of Schrödinger operators converges to zero strictly faster than of order $L^{- 2}$ – the decay rate of the gap of the free Laplacian – as soon as a non-zero and sufficiently fast decaying potential is added. This class includes, in particular, compactly supported potentials. This somewhat surprising effect is a consequence of the effective degeneracy of the ground state in the infinite-volume limit. Loosely speaking, on large intervals, even the smallest potential splits the interval into two congruent subintervals with an extra Dirichlet condition in the middle. Hence, the corresponding operators have an effectively degenerate ground state which will make the spectral gap close faster than the gap of the free Laplacian. An analogous phenomenon has since been observed in [8] for Schrödinger operators on discrete graphs. There are still open questions related to this phenomenon: for example, one would like to determine the exact decay rate of the spectral gap. A conjecture that this decay rate universally is always $L^{- 3}$ was put forward in [7] and first results, proving a lower bound of order $L^{- 4}$ , were subsequently obtained in [6].

In this paper, our aim is to investigate if the effect found in one dimension persists in two and higher dimensions. More explicitly, we study Schrödinger operators on squares and hypercubes with side lengths going to infinity and with non-negative and bounded potentials that fulfil certain decay conditions. While it is relatively straightforward to see that in dimension three and larger, fast decaying (e.g. integrable) potentials cannot change the asymptotics of the spectral gap, the two-dimensional situation turns out to be more subtle. Even though it requires more technical effort, we also prove that fast decaying potentials cannot change the asymptotic decay rate of the spectral gap. We can push this result up to the borderline case of potentials that decay exactly like $| x |^{- 2}$ , assuming the potential is sufficiently small. However, if the potential of a two-dimensional Schrödinger operator does decay in one direction but remains constant in the other, we prove – again with a larger technical effort – that the ground state will be again effectively degenerate in the large volume limit. As a consequence, the faster decay rate of the spectral gap reappears.

The rest of the paper is structured as follows: At the beginning of Section 2 the setting and basic notation is introduced. Section 2.1 contains our results and proofs in two dimensions whereas Section 2.2 is about the higher-dimensional situation. Some technical aspects regarding Bessel functions are deferred to the appendix.

2. Notation and results

On $Λ_{L} = {(- \frac{L}{2}, + \frac{L}{2})}^{d} \subset R^{d}$ with $L > 0$ and $d ⩾ 2$ , we consider self-adjoint Schrödinger operators of the form $\begin{matrix} (1) & h_{L} : = - Δ + v \end{matrix}$ where Δ is the d-dimensional Laplace operator with Dirichlet boundary conditions on $\partial Λ_{L}$ , and v an external potential. We always assume that $v \in L^{\infty} (R^{d})$ and that v is non-negative.

The operator $h_{L}$ is defined on the domain $H_{0}^{1} (Λ_{L}) \cap H^{2} (Λ_{L})$ in the Hilbert space $L^{2} (Λ_{L})$ where $H^{k} (Ω)$ is the Sobolev space of all square integrable functions on Ω with square integrable weak derivatives up to order k, and $H_{0}^{1} (Ω)$ denotes the space of all functions in $H^{1} (Ω)$ with zero boundary trace. It is nonnegative, self-adjoint, and has purely discrete spectrum. We denote its eigenvalues by $λ_{0} (L) < λ_{1} (L) ⩽ \dots$ . The operator with zero potential, which is simply the Dirichlet Laplacian on $Λ_{L}$ , shall be denoted by $h_{L}^{0}$ and its eigenvalues by $μ_{0} (L) ⩽ μ_{1} (L) ⩽ \dots$ . The main object of interest in this paper is the spectral gap which is defined by $\begin{matrix} Γ_{v} (L) : = λ_{1} (L) - λ_{0} (L) . \end{matrix}$ For a measurable set $A \subset R^{d}$ , we write $1_{A}$ for the associated indicator function. We write $B_{r} = B_{r} (0)$ for the d-dimensional ball of radius $r > 0$ . We are also going to use Bessel functions, details on which we refer to the appendix. Here, we only remark that $J_{α}$ is the Bessel function of the first kind with parameter $α ⩾ 0$ , and its zeroes on the positive real line form a monotonously increasing sequence $0 < j_{α, 1} < j_{α, 2} < \dots$ .

2.1. The two-dimensional case

For $d = 2$ , we have $\begin{matrix} μ_{0} (L) = \frac{2 π^{2}}{L^{2}}, μ_{1} (L) = \frac{(1^{2} + 2^{2}) π^{2}}{L^{2}} = \frac{5 π^{2}}{L^{2}} \end{matrix}$ with associated normalized eigenfunctions $\begin{array}{l} φ_{0} (x_{1}, x_{2}) = \frac{2}{L} cos (\frac{π x_{1}}{L}) \cdot cos (\frac{π x_{2}}{L}), \\ φ_{1} (x_{1}, x_{2}) = \frac{2}{L} sin (\frac{2 π x_{1}}{L}) \cdot cos (\frac{π x_{2}}{L}) . \end{array}$ The eigenvalue $μ_{1} (L)$ has multiplicity two: An orthogonal eigenfunction to $φ_{1}$ with the same eigenvalue can be obtained by interchanging the roles of $x_{1}$ and $x_{2}$ .

Let us introduce an auxiliary operator which is unitarily equivalent to $h_{L}$ and which is defined on the fixed domain $Λ_{1}$ . The L-dependence then manifests itself in the potential and in an additional prefactor $L^{- 2}$ .

Proposition 1 (Unitary transformation).

In every dimension $d ⩾ 1$ , the operator $h_{L}$ is unitarily equivalent to the operator $\begin{matrix} L^{- 2} g_{L} where g_{L} = - Δ + L^{2} v (L x) \end{matrix}$ is defined in $L^{2} (Λ_{1})$ with Dirichlet boundary conditions.

Proof.
This follows immediately from a direct calculation, cf. [7] for the case $d = 1$ . □

In the following, ${\tilde{λ}}_{0} (L)$ and ${\tilde{λ}}_{1} (L)$ shall denote the first two eigenvalues of $g_{L}$ and ${\tilde{Γ}}_{v} (L)$ the associated spectral gap. In a first step we study potentials that decay sufficiently fast at infinity. Indeed, we will impose the same decay condition as in one dimension (see [7, Theorem 2.5]) where it was shown that the spectral gap of the one-dimensional analogue of $h_{L}$ decays faster than the gap of the free Laplacian. It turns out, however, that this effect is absent in two dimensions.
Theorem 2 (Gap for fast decaying potentials).

Let $0 ⩽ v \in L^{\infty} (R^{2})$ and assume $\begin{matrix} | v (x) | ⩽ \frac{C}{| x |^{2 + μ}} \end{matrix}$ for some $C, μ > 0$ . Then, there are $α, β > 0$ such that for sufficiently large L $\begin{matrix} \frac{α}{L^{2}} ⩽ Γ_{v} (L) ⩽ \frac{β}{L^{2}} . \end{matrix}$

Proof.
By Proposition 1 it suffices to find $\tilde{α}, \tilde{β} > 0$ such that $\begin{matrix} (2) & \tilde{α} < {\tilde{Γ}}_{v} (L) < \tilde{β} \end{matrix}$ for sufficiently large L. Since $v ⩾ 0$ , the eigenvalue ${\tilde{λ}}_{1} (L)$ is bounded from below by the corresponding eigenvalue with zero potential, that is $\begin{matrix} μ_{1} (1) = 5 π^{2} ⩽ {\tilde{λ}}_{1} (L) . \end{matrix}$ Thus, in order to establish the lower bound in (2), it suffices to show ${lim sup}_{L \to \infty} {\tilde{λ}}_{0} (L) < 5 π^{2}$ . Let $0 < δ < \frac{1}{2}$ such that $B_{δ} \subset B_{\frac{1}{2}} \subset Λ_{1}$ . Then, for any $ε > 0$ we have, for sufficiently large L, $\begin{matrix} (3) & {‖ L^{2} v (L \cdot) 1_{Λ_{1} ∖ B_{δ}} ‖}_{\infty} ⩽ \frac{C L^{2}}{L^{2 + μ} δ^{2 + μ}} < ε . \end{matrix}$ By standard operator bracketing arguments, (3) implies $\begin{matrix} (4) & {\tilde{λ}}_{0} (L) ⩽ {\tilde{μ}}_{0, δ} + ε \end{matrix}$ where ${\tilde{μ}}_{0, δ}$ is the ground-state eigenvalue of the Dirichlet Laplacian in $L^{2} (B_{\frac{1}{2}} ∖ B_{δ})$ . Now, a result by Ozawa [11] shows that ${\tilde{μ}}_{0, δ}$ converges to the ground-state eigenvalue of the Dirichlet Laplacian on $B_{\frac{1}{2}}$ . The latter is given by $4 j_{0}^{2}$ where $j_{0}$ is the first positive root of the Bessel function $J_{0}$ . Since ε in (4) was arbitrary and since $4 j_{0}^{2} < 5 π^{2}$ , the lower bound follows.

In the same way, one has $\begin{matrix} {\tilde{Γ}}_{v} (L) ⩽ {\tilde{λ}}_{1} (L) ⩽ {\tilde{μ}}_{1, δ} + ε \end{matrix}$ where ${\tilde{μ}}_{1, δ}$ is the second eigenvalue of the Dirichlet Laplacian in $L^{2} (B_{\frac{1}{2}} ∖ B_{δ})$ . This proves the upper bound. □

In the previous proof, $Λ_{L}$ has been replaced by the subdomain $B_{\frac{1}{2}} ∖ B_{δ}$ which is a rather crude step. This is due to [11] requiring a domain with a smooth boundary. It would suffice to work with a slightly regularized, punctured version of $Λ_{1}$ , but this would also not improve the statement of the theorem. It indicates however that one can generalize Theorem 2 to bounded domains which can be sufficiently well approximated from outside and inside by domains with smooth boundary. One example of such domains would be convex domains.
Remark 3.
There is also a heuristic explanation of Theorem 2 in terms of capacities, referring to the fact that single points in two and higher dimensional Euclidean space have harmonic (or Newtonian) capacity zero; see [12] for a reference. This means that the infimum of the Dirichlet form $\int | \nabla u |^{2} d x$ over all $u \in C_{0}^{\infty} (R^{d})$ which satisfy $u (0) = 1$ is zero if and only if $d ⩾ 2$ . Since, using the scaling of Proposition 1, compactly supported potentials are essentially reduced to a single point, it is unsurprising that they do not have the power to warp the asymptotics of the spectral gap whereas other objects which macroscopically look like objects with positive capacity, do, as we will see in Theorem 8 below.

We now turn to the extremal case, left open by Theorem 2, that is potentials which decay quadratically. Since $x \mapsto \frac{1}{| x |^{2}}$ is invariant under the unitary transformation of Proposition 1, the proof will have to be amended accordingly. We start with an upper bound.
Theorem 4.
Let $0 ⩽ v \in L^{\infty} (R^{2})$ be such that $\begin{matrix} | v (x) | ⩽ \frac{C}{| x |^{2}} \end{matrix}$ for some $C > 0$ and almost all $x \in R^{2}$ . Then, there is $β > 0$ such that, for sufficiently large L, one has $\begin{matrix} Γ_{v} (L) ⩽ \frac{β}{L^{2}} . \end{matrix}$
Proof.
We use the unitary transformation given in Proposition 1 and note that ${\tilde{Γ}}_{v} (L)$ is bounded by the second eigenvalue of the Schrödinger operator $\begin{matrix} h_{δ} : = - Δ + \frac{C}{| x |^{2}}, \end{matrix}$ defined in $L^{2} (B_{\frac{1}{2}} ∖ B_{δ})$ , $0 < δ < 1 / 2$ , subject to Dirichlet boundary conditions. □

Next, we complement Theorem 4 by a lower bound on the spectral gap for potentials that decay exactly quadratically at infinity. Recall that $J_{α}$ is the Bessel function of the first kind with parameter $α ⩾ 0$ and that it has zeroes $0 < j_{α, 1} < j_{α, 2} < \dots$ .
Theorem 5 (Regularized quadratic potentials).

Let $c > 0$ be such that $\begin{matrix} (5) & j_{\sqrt{c}, 1} < \frac{min {j_{\sqrt{1 + c}, 1}, j_{\sqrt{c}, 2}}}{\sqrt{2}}, \end{matrix}$ and let $\begin{matrix} v (x) = \frac{c 1_{| x | ⩾ 1}}{| x |^{2}} . \end{matrix}$ Then, there are constants $α, β > 0$ such that, for sufficiently large L, one has $\begin{matrix} \frac{α}{L^{2}} ⩽ Γ_{v} (L) ⩽ \frac{β}{L^{2}} . \end{matrix}$

Remark 6.
Condition (5) holds for sufficiently small c, that is for $c < 0.0468 \dots$ .
Proof.
The upper bound was established in Theorem 4. As for the lower bound, recall that ${\tilde{λ}}_{0} (L)$ and ${\tilde{λ}}_{1} (L)$ denote the eigenvalues of the operator $\begin{matrix} g_{L} = - Δ + \frac{c 1_{| x | ⩾ \frac{1}{L}}}{| x |^{2}} on L^{2} (Λ_{1}) with Dirichlet boundary conditions . \end{matrix}$ By Proposition 1, it then suffices to prove ${\tilde{Γ}}_{v} (L) = {\tilde{λ}}_{1} (L) - {\tilde{λ}}_{0} (L) ⩾ α > 0$ for some $α > 0$ and for sufficiently large L.

We compare these eigenvalues to eigenvalues on spherical domains. For this purpose, we apply Lemma 11 to the operator ${\tilde{g}}_{ρ, L, c} : = - Δ + \frac{c 1_{| x | ⩾ \frac{1}{L}}}{| x |^{2}}$ on balls of radius ρ with Dirichlet boundary conditions. For a lower bound on ${\tilde{λ}}_{1} (L)$ , we choose ρ to be the outer radius of $Λ_{1}$ , that is $B_{1 / \sqrt{2}}$ , whereas for an upper bound on ${\tilde{λ}}_{0} (L)$ , we choose ρ as the inradius, leading to $B_{1 / 2}$ . Since Dirichlet eigenvalues are monotonous with respect to the domain, this provides lower and upper bounds on eigenvalues, respectively.

The asymptotic expressions for the eigenvalues of ${\tilde{g}}_{ρ, L, c}$ in Lemma 11 then yields $\begin{array}{l} lim_{L \to \infty} {\tilde{λ}}_{1} (L) - {\tilde{λ}}_{0} (L) & ⩾ lim_{L \to \infty} ν_{1} (L, 1 / \sqrt{2}, c) - ν_{0} (L, 1 / 2, c) \\ = 2 min {j_{\sqrt{1 + c}, 1}, j_{\sqrt{c}, 2}}^{2} - 4 j_{\sqrt{c}, 1}^{2} > 0 \end{array}$ as $L \to \infty$ , where $ν_{k} (L, ρ, c)$ , $k = 0, 1$ , are the two lowest eigenvalues of ${\tilde{g}}_{ρ, L, c}$ , and the last inequality is due to (5). □
Remark 7.
Although we believe that Theorem 5 actually holds for all values $c > 0$ , let us comment in more detail on the origin of condition (5): Both, $j_{\sqrt{c}, 1}^{2}$ and $min {j_{\sqrt{1 + c}, 1}, j_{\sqrt{c}, 2}}^{2}$ , correspond to the two smallest λ solving the formal eigenvalue equation $- Δ u + \frac{c}{| x |^{2}} u = λ u$ on $B_{1}$ with Dirichlet boundary conditions. General properties of Bessel functions imply $min {j_{\sqrt{1 + c}, 1}, j_{\sqrt{c}, 2}} - j_{\sqrt{c}, 1} > 0$ , see also (8) in Appendix A. One could, however, improve on the range of admissible values of c if the divisor $\sqrt{2}$ in (5) was closer to one. It originates from the geometric argument in the proof of Theorem 5 where eigenvalues on $Λ_{1}$ are estimated by eigenvalues on inscribed and circumscribed balls.

Consequently, if $Λ_{L}$ was replaced by a scaled version $K_{L} : = {x \in R^{2} : x / L \in K}$ of another set $K \subset R^{2}$ containing $0 \in R^{2}$ , the factor ${\sqrt{2}}^{- 1}$ would be replaced by $\begin{matrix} R (K) : = \frac{sup {r > 0 : B_{r} \subset K}}{inf {r > 0 : K \subset B_{r}}} . \end{matrix}$ In Table 1 we sketch the consequences on the range of admissible c for some domains $K \subset R^{2}$ .

Table 1
Range of parameters c for which we can prove a quadratic lower bound on the spectral gap of the operator $- Δ + \frac{c 1_{| x | ⩾ 1 / L}}{| x |^{2}}$ on different domains

Shape K $R (K)$ Maximal c for quadratic lower bound on gap

Square $cos (π / 4)$ $0.0468 \dots$

Regular hexagon $cos (π / 6)$ $0.8719 \dots$

Regular octagon $cos (π / 8)$ $2.4379 \dots$

Ball 1 ∞

Our results so far have been negative: in contrast to the one-dimensional setting, in two dimensions the asymptotic decay rate of the gap seems robust. Fast decaying potentials are unable to warp the decay rate. But one can still reproduce the effective degeneracy of the ground state by using potentials on a strip in two dimensions. For such potentials, we prove an analogue of the one-dimensional case described in [7, Theorem 2.5] and show that the gap converges to zero strictly faster than the spectral gap of the free Laplacian.
Theorem 8 (One-sided decay).

Shape K	$R (K)$	Maximal c for quadratic lower bound on gap
Square	$cos (π / 4)$	$0.0468 \dots$
Regular hexagon	$cos (π / 6)$	$0.8719 \dots$
Regular octagon	$cos (π / 8)$	$2.4379 \dots$
Ball	1	∞

Assume that $0 ⩽ v \in L^{\infty} (R^{2})$ is a potential such that $v (x_{1}, x_{2}) ⩾ γ > 0$ for $- δ < x_{1} < + δ$ , $δ > 0$ , and which is zero elsewhere. Then, one has $\begin{matrix} lim_{L \to \infty} L^{2} Γ_{v} (L) = 0 . \end{matrix}$

Proof.
By Proposition 1, it suffices to show ${lim}_{L \to \infty} {\tilde{Γ}}_{v} (L) = 0$ for the gap ${\tilde{Γ}}_{v} (L)$ of the operator $g_{L}$ .

Let us first pick two test functions: $\begin{array}{l} φ_{1}^{(L, ϵ)} (x_{1}, x_{2}) = \{\begin{matrix} sin (\frac{π}{\frac{1}{2} - ϵ} (x_{1} - ϵ)) cos (π x_{2}) & if ϵ ⩽ x_{1} ⩽ \frac{1}{2}, \\ 0 & if - \frac{1}{2} ⩽ x_{1} < ϵ, \end{matrix} and \\ φ_{2}^{(L, ϵ)} (x_{1}, x_{2}) = φ_{1}^{(L, ϵ)} (- x_{1}, x_{2}) \end{array}$ which are both in $H_{0}^{1} (Λ_{1})$ and satisfy Dirichlet boundary conditions on $\partial Λ_{1}$ . Plugging $φ_{k}^{(L, ϵ)}$ , $k = 1, 2$ , into the Rayleigh quotient for the quadratic form corresponding to $g_{L}$ , we find, for every $\hat{δ} > 0$ , parameters $L_{0}, ϵ > 0$ such that for $L ⩾ L_{0}$ one has ${\tilde{λ}}_{1} (L) ⩽ 5 π^{2} + \hat{δ}$ . Since $\hat{δ} > 0$ is arbitrary, we conclude $\begin{matrix} \underset{L \to \infty}{lim sup} {\tilde{λ}}_{1} (L) ⩽ 5 π^{2} \end{matrix}$ where ${\tilde{λ}}_{1} (L)$ is the second eigenvalue of $g_{L}$ . It remains to prove $\begin{matrix} (6) & \underset{L \to \infty}{lim inf} {\tilde{λ}}_{0} (L) ⩾ 5 π^{2}, \end{matrix}$ where ${\tilde{λ}}_{0} (L)$ is the ground state eigenvalue of $g_{L}$ : Let $L_{k} \to \infty$ and take a sequence of corresponding normalized ground states $φ_{k} \in H_{0}^{1} (Λ_{1})$ of $g_{L_{k}}$ . Obviously, since ${\tilde{λ}}_{0} (L)$ is bounded as $L \to \infty$ , the sequence $(φ_{k})$ is bounded in $H^{1} (Λ_{1})$ . We extract a subsequence that converges weakly in $H_{0}^{1} (Λ_{1})$ and strongly in $L^{2} (Λ_{1})$ to a function $φ_{\infty}$ . Note here that, with a slight abuse of notation, in the following we may write $φ_{\infty}$ although one actually considers a restriction of it to a certain domain. Also note that we will, at various points and sometimes without further mentioning, restrict attention to subsequences; in this context, we also employ Cantor’s diagonalization argument.

For every $n \in N$ , on the rectangles $L_{n} : = (- \frac{1}{2}, - \frac{1}{n}) \times (- \frac{1}{2}, \frac{1}{2})$ , and $R_{n} : = (\frac{1}{n}, \frac{1}{2}) \times (- \frac{1}{2}, \frac{1}{2})$ the elliptic regularity estimate $‖ u ‖_{H^{2} (Ω)} ⩽ C (‖ Δ u ‖_{L^{2} (Ω)} + ‖ u ‖_{L^{2} (Ω)})$ , see, e.g., [5], implies $\begin{matrix} ‖ φ_{k} ‖_{H^{2} (R_{n})} ⩽ C and ‖ φ_{k} ‖_{H^{2} (L_{n})} ⩽ C \end{matrix}$ for sufficiently large k, where the constant $C > 0$ might depend on $n \in N$ .

Let us from now on focus on $R_{n}$ since the subsequent arguments for $L_{n}$ will be completely analogous. The uniform $H^{2}$ -boundedness of the $φ_{k}$ on $R_{n}$ for every fixed $n \in N$ implies – again after passing to a subsequence – that $\begin{matrix} φ_{\infty} \in H^{1} (R_{n}) for all n \in N \end{matrix}$ with an n-independent bound on $‖ φ_{\infty} ‖_{H^{1} (R_{n})}$ (this follows from the fact that the norm of $φ_{k}$ is uniformly bounded in $H^{1} (Λ_{1})$ in combination with Fatou’s lemma). Therefore, $\begin{matrix} φ_{\infty} \in H^{1} ((0, \frac{1}{2}) \times (- \frac{1}{2}, \frac{1}{2})) = : H^{1} (R) . \end{matrix}$ Next, we define $\begin{matrix} [0, \frac{1}{2}] ∋ x_{1} \mapsto f_{k} (x_{1}) : = \int_{- 1 / 2}^{1 / 2} {| φ_{k} (x_{1}, x_{2}) |}^{2} d x_{2} for k \in N \cup {\infty} . \end{matrix}$ By the trace theorem and uniform $H_{1} (R)$ -boundedness of the $φ_{k}$ , $\begin{matrix} | f_{k} (x_{1}) | ⩽ C_{1} ‖ φ_{k} ‖_{H^{1} (R)}^{2} ⩽ C_{2} \end{matrix}$ with constants $C_{1}$ , $C_{2}$ that are independent of k and $x_{1}$ . We estimate, for $x_{1} \in [0, 1 / 2)$ and sufficiently small $h > 0$ , $\begin{array}{l} | f_{k} (x_{1} + h) - f_{k} (x_{1}) | \\ = | \int_{- 1 / 2}^{1 / 2} (φ_{k} (x_{1} + h, x_{2}) - φ_{k} (x_{1}, x_{2})) \cdot (φ_{k} (x_{1} + h, x_{2}) + φ_{k} (x_{1}, x_{2})) d x_{2} | \\ = | \int_{- 1 / 2}^{1 / 2} (\int_{x_{1}}^{x_{1} + h} \partial_{x_{1}} φ_{k} (t, x_{2}) d t) \cdot (φ_{k} (x_{1} + h, x_{2}) + φ_{k} (x_{1}, x_{2})) d x_{2} | \\ ⩽ {(\int_{- 1 / 2}^{1 / 2} {[\int_{x_{1}}^{x_{1} + h} \partial_{x_{1}} φ_{k} (t, x_{2}) d t]}^{2} d x_{2})}^{1 / 2} {(2 \int_{- 1 / 2}^{1 / 2} {| φ_{k} (x_{1} + h, x_{2}) |}^{2} + {| φ_{k} (x_{1}, x_{2}) |}^{2} d x_{2})}^{1 / 2} \\ ⩽ C_{3} \cdot {(h \int_{- 1 / 2}^{1 / 2} \int_{x_{1}}^{x_{1} + h} {| \partial_{x_{1}} φ_{k} (t, x_{2}) |}^{2} d t d x_{2})}^{1 / 2} ⩽ C_{4} \sqrt{h} \end{array}$ where we used the Cauchy–Schwarz inequality and Jensen’s inequality. Furthermore, $C_{3}, C_{4} > 0$ are independent of k, $x_{1}$ , h. We conclude that the family ${(f_{k})}_{k \in N}$ is equicontinuous and pointwise bounded on a compact interval. By the Arzelá Ascoli theorem, and since $f_{k} \to f_{\infty}$ pointwise almost everywhere, we can therefore – again passing to a subsequence – assume that $f_{k} \to f_{\infty}$ uniformly in $x_{1}$ .

We now claim $f_{\infty} (0) = 0$ : Indeed, if $f_{\infty} (0) > 0$ , then continuity of $f_{\infty}$ would imply $f_{\infty} (x_{1}) > κ$ on an interval $[0, ϵ)$ , for some $ϵ, κ > 0$ . But then $\begin{array}{l} ⟨ φ_{k}, g_{L_{k}} φ_{k} ⟩ & ⩾ \int_{R} L_{k}^{2} v (L_{k} x_{1}, L_{k} x_{2}) {| φ_{k} (x_{1}, x_{2}) |}^{2} d x_{1} d x_{2} \\ ⩾ γ L_{k}^{2} \int_{0}^{δ / L_{k}} f_{k} (x_{1}) d x_{1} \\ ⩾ \frac{δ κ γ}{2} L_{k} \to \infty \end{array}$ for sufficiently large k by uniform convergence. However, this would contradict the boundedness of the ground state eigenvalue. As a consequence, $f_{\infty} (0) = 0$ which means that $φ_{\infty} \in H_{0}^{1} (R)$ and – by a completely analogous argument – $φ_{\infty} \in H_{0}^{1} (L)$ where $L : = (- 1 / 2, 0) \times (- 1 / 2, 1 / 2)$ . But since the ground state energy of the Dirichlet Laplacian on $L$ and $R$ is $5 π^{2}$ , this shows (6) and the proof is complete. □
2.2. The higher-dimensional case

We now consider the operator $h_{L}$ , defined in (1), on $Λ_{L} \subset R^{d}$ for $d ⩾ 3$ . If the potential v is identically zero, the least two eigenvalues are $\begin{matrix} μ_{0} (L) = \frac{d π^{2}}{L^{2}} and μ_{1} (L) = \frac{(d + 3) π^{2}}{L^{2}} \end{matrix}$ where $μ_{1}$ is d-fold degenerate. Also, the ground state eigenfunction of $h_{L}^{0}$ is $\begin{matrix} φ_{0} (x_{1}, \dots, x_{d}) = {(\frac{2}{L})}^{d / 2} \prod_{j = 1}^{d} cos (\frac{π x_{j}}{L}) . \end{matrix}$ It is easier than in the two-dimensional setting to prove that fast decaying potentials cannot affect the decay rate of the spectral gap:

Theorem 9.

Let $d ⩾ 3$ and $h_{L}$ be given with an arbitrary non-negative potential $v \in (L^{1} \cap L^{\infty}) (R^{d})$ . Then, $\begin{matrix} \frac{α}{L^{2}} ⩽ Γ_{v} (L) ⩽ \frac{β}{L^{2}} \end{matrix}$ for some constants $α, β > 0$ and all $L > 0$ large enough.

Proof.

Regarding the lower bound we first notice that $\begin{matrix} \frac{(d + 3) π^{2}}{L^{2}} = μ_{1} (L) ⩽ λ_{1} (L) . \end{matrix}$ On the other hand, $\begin{matrix} λ_{0} (L) ⩽ {⟨ φ_{0}, h_{L} φ_{0} ⟩}_{L^{2} (Λ_{L})} ⩽ \frac{d π^{2}}{L^{2}} + ‖ φ_{0} ‖_{\infty}^{2} \cdot ‖ v ‖_{L^{1} (R^{d})} = \frac{d π^{2}}{L^{2}} + \frac{2^{d} ‖ v ‖_{L^{1} (R^{3})}}{L^{d}} . \end{matrix}$ From this the statement follows immediately.

In a similar fashion, the upper bound follows from the minmax-principle by looking at the two-dimensional subspace generated by two eigenstates corresponding to the two lowest eigenvalues of $h_{L}^{0}$ . □

Footnotes

Acknowledgements

We shall thank the referee for a careful reading of the manuscript and various helpful comments.

Bessel functions and the operator − Δ + | x | − 2

Let us collect some facts on Bessel’s differential equation with parameter $α ⩾ 0$ , $\begin{matrix} (7) & r^{2} R^{″} (r) + r R^{'} (r) + [r^{2} - α^{2}] R (r) = 0, r \in [0, \infty) . \end{matrix}$ This one-dimensional ordinary differential equation naturally appears when the equation $- Δ u = λ u$ is rewritten in polar coordinates. There is a well-developed theory on solutions of (7). We collect some useful properties:

For $c ⩾ 0$ , let $\begin{matrix} Σ_{c} : = ⋃_{n = 0}^{\infty} ⋃_{k = 1}^{\infty} {\frac{j_{\sqrt{n^{2} + c}, k}^{2}}{ρ^{2}}} = : {ν_{m} (\infty, ρ, c) : m \in {0, 1, 2, \dots}}, \end{matrix}$ where the $ν_{m} (\infty, ρ, c)$ are the elements of the discrete set $Σ_{c}$ , enumerated increasingly and counting multiplicities. Proposition 10 implies $\begin{matrix} (8) & ν_{0} (\infty, 1, c) = j_{\sqrt{c}, 1}^{2} < min {j_{\sqrt{c}, 2}^{2}, j_{\sqrt{c + 1}, 1}^{2}} = ν_{1} (\infty, 1, c) . \end{matrix}$ The following lemma characterises $Σ_{c}$ as the limiting spectrum of the operator $- Δ + \frac{c 1_{| x | ⩾ \frac{1}{L}}}{| x |^{2}}$ in $L^{2} (B_{ρ})$ with Dirichlet boundary conditions as $L \to \infty$ .

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