Inequalities à la Pólya for the Aharonov–Bohm eigenvalues of the disk

We prove an analogue of Pólya’s conjecture for the eigenvalues of the magnetic Schrödinger operator with Aharonov–Bohm potential on the disk, for Dirichlet and magnetic Neumann boundary conditions. This answers a question posed by R. L. Frank and A. M. Hansson in 2008.


§1. Introduction and main results
Consider a magnetic Schrödinger operator on R 2 with an Aharonov-Bohm potential, A p,α := −i∇ − αA p 2 , (1.1) where p = (p 1 , p 2 ) ∈ R 2 , α ∈ R, and A p = |x − p| −2 p 2 − x 2 x 1 − p 1 , x = (x 1 , x 2 ) ∈ R 2 .We refer to A p,α for brevity as the Aharonov-Bohm operator [FH08].It can be rewritten as where ∆ = ∂ 2 x 2 is the Laplacian and u is a complex-valued function.The point p is called the pole of the potential, and the number α is called the flux, cf.(1.2) below.If α − α ′ ∈ Z, one can show that the operator A p,α is unitarily equivalent to A p,α ′ (this is also known as gauge invariance, see, for instance, [CPS22,Appendix A.4]).Moreover, changing the sign of the flux is equivalent to choosing a coordinate system centred at p and interchanging the coordinates.Therefore, we may assume without loss of generality that α ∈ 0, 1 2 .
The operator A p,α arises in the study of an important phenomenon in quantum physics called the Aharonov-Bohm effect [AB59], which, roughly speaking, shows that a magnetic potential may affect a charged particle even if the corresponding magnetic field vanishes.Let be the dual 1-form to the vector potential A p .Recall that a 1-form A is called closed if dA = 0, and it is called exact if there is a smooth function f such that A = d f ; it is well-known that adding a magnetic potential corresponding to an exact 1-form does not change the magnetic Laplacian up to unitary equivalence.One can check that A p is a closed form on R 2 \ {p}, and hence the magnetic field dA p generated by the Aharonov-Bohm potential on R 2 \ {p} is equal to zero.At the same time, a direct calculation yields that A p = dθ, where θ is the angular coordinate in a polar coordinate system centred at p. Therefore, for any simple closed curve Γ containing p in its interior, and thus the form A p is not exact.Note that (1.2) implies that the (normalised) flux of the potential αA p is equal to α, which motivates the terminology.As shown below, the spectral properties of the operator A p,α depend on the position of the pole p and on the value of α, despite the fact that the magnetic field is zero.This can be viewed as a manifestation of the Aharonov-Bohm effect.There is a vast literature in spectral theory on the subject, see, for example, [H88,FH08,L15,BNNT14,CPS22] and references therein.More specifically, let Ω ⊂ R 2 be a bounded planar domain.Consider the eigenvalue problem A p,α u = λu in Ω for the Aharonov Applying a coordinate change if necessary, we may assume without loss of generality that the pole p coincides with the origin o, and we set A α := A o,α , α ∈ 0, 1 2 .Moreover, we may assume that o lies in the simply-connected hull of Ω, i.e., the smallest simply-connected set containing Ω. Indeed, otherwise the flux of the potential over any closed curve contained in Ω is equal to zero, hence the form A p is exact in Ω, and the magnetic potential vanishes after an application of a gauge transformation.
We will denote the Dirichlet eigenvalues of A α on Ω by and the magnetic Neumann eigenvalues by For α = 0 they coincide, respectively, with the Dirichlet and Neumann eigenvalues of the usual Laplacian on Ω.
Consider the Dirichlet problem first.Let be the eigenvalue counting function.It is known to satisfy Weyl's law ([F09,Theorem A.1], see also [F23a,Theorem 6]) as λ → ∞.If α = 0, the celebrated Pólya's eigenvalue conjecture [P54] states that for any bounded domain (1.4) for all λ ≥ 0. It was proved by Pólya for tiling domains [P61], and has been recently obtained for the disk in [FLPS23]; we refer to [LMP23,§3.3.3] for an overview of this topic.One may ask if Pólya's conjecture can be extended to magnetic Schrödinger operators.It is known to be false in the case of constant magnetic field [FLW09].At the same time, numerical experiments presented in [FH08] suggested that it holds for the Aharonov-Bohm operator on certain domains, in particular, for the disk.The goal of this paper is to confirm the latter, developing the approach proposed in [FLPS23].
From now on, let Ω be the unit disk D centred at the origin.The (non-ordered) eigenvalues of the Aharonov-Bohm operator A α on D are given by [FH08, §5.2] where j ν,k is the kth positive root of the Bessel function J ν (x).
Remark 1.2.We note that there is no uniform (in n) monotonicity relations in α between λ D n (D, α) and λ D n (D, 0), and hence (1.6) cannot be easily deduced from the already known case α = 0. Indeed, for α ∈ 0, 1 2 and m > 0, each double eigenvalue j 2 m,k of the Dirichlet Laplacian splits into a smaller eigenvalue j 2 m−α,k and a bigger eigenvalue j 2 m+α,k of the Aharonov-Bohm operator, whereas the single eigenvalues j 2 0,k of the Dirichlet Laplacian all move up to become the eigenvalues j 2 α,k of A α , see Figure 1.
Consider now the magnetic Neumann problem.Let For α = 0 we recover the counting function of the Neumann Laplacian.In this case, Pólya's conjecture states that (1.7) Similarly to (1.4), inequality (1.7) is known to hold for tiling domains [P61, K66] and for the disk [FLPS23].We would like to extend it for the Aharonov-Bohm operator on the disk with an arbitrary flux α.We note, first of all, that since we assume that the potential has the pole at the origin, the magnetic Neumann boundary conditions simplify to the ordinary Neumann conditions 〈 ∇u| ∂D , n〉 = 0.The (non-ordered) eigenvalues of the Neumann Aharonov-Bohm operator A α on the disk are given by [CPS22, §B.3] where j ′ ν,k is the kth positive root of the derivative of the Bessel function J ′ ν (x), with an exception j ′ 0,1 = 0. We immediately note that unlike the case α = 0 for which (1.9) holds for all λ > 0, we need to impose some restrictions when α > 0. Indeed, in this case the first eigenvalue of the Neumann Aharonov-Bohm operator is no longer zero, Therefore N N Ω (λ; 0) = 0 for all λ ∈ 0, j ′ α,1 , and (1.7) cannot hold in this interval.However, outside of this interval Pólya's conjecture for the Aharonov-Bohm operator on the disk with magnetic Neumann boundary condition still holds.
Equivalently, we have for all n ∈ N.
(1.10) Theorem 1.3 is proved in §3.We note that in the interval λ ∈ 5 2 , 9 we use a rigorous computerassisted argument based on integer arithmetic, see Proposition 3.8.
Remark 1.4.In fact, apart from the ground state [CS21,CPS22], the magnetic Neumann eigenvalues of the Aharonov-Bohm operator appear to be less explored than the Dirichlet ones.In particular, even the one-term Weyl's law, as λ → +∞, (1.11) has not been worked out in the literature in full detail.We note that the standard smooth methods do not apply due the presence of a singularity at the pole.To fill in this gap, we present an argument that was communicated to us by R. L. Frank [F23b].As before, assume that Ω has Lipschitz boundary and consider the Aharonov-Bohm operator A α with a pole at the origin o ∈ Ω.Since A o ∈ L 2 loc (Ω \ {o}), it follows from [HS04,Theorem 1.3] that the pointwise difference between the Neumann and Dirichlet heat kernels for A α on the diagonal satisfies Integrating both sides over x, multiplying by t d /2 and using the heat trace asymptotics for the Dirichlet and Neumann Laplacian, we get (1.12) By an Abelian theorem, the Weyl asymptotics (1.3) implies the one-term asymptotics for the Dirichlet heat trace for the Aharonov-Bohm operator, which by (1.12) implies the same asymptotics for the Neumann heat trace.The standard Tauberian argument then implies (1.11).
Remark 1.5.It would be interesting to know if analogues of Theorems 1.1 and 1.3 hold for other domains, even numerically.As we have mentioned previously, in the Dirichlet case it was checked computationally in [FH08] for some annuli and the square with the centred potential.We have conducted some rough numerical experiments for the disk, with both the Dirichlet and the Neumann conditions, and an off-centred pole position, which seem to indicate that experimentally the analogues of Pólya's conjecture still hold in these cases (with an appropriate modification for the first Neumann eigenvalue as above) for all α and all pole placements, but we do not have a proof of this claim.§2.Proof of Theorem 1.1 Recall formula (1.5).With an alternative parametrisation, we can split the eigenvalues into two doublyindexed sequences, Therefore, the counting function of eigenvalues which are less than or equal to a given λ 2 ∈ [0, +∞) is where in the last line we used the fact that j ν,1 > ν, and therefore # k ∈ N : j ν,k ≤ λ = 0 whenever ν > λ.Here and further on, ⌊x⌋ and ⌈x⌉ stand, as usual, for the floor and the ceiling of a real number x, respectively.The main ideas of the proof are similar to those in the case of the Dirichlet Laplacian [FLPS23], with some notable changes.The first important ingredient is a uniform upper bound on the number of zeros of a Bessel function in a given interval, which is directly borrowed from [FLPS23] and [S23].For λ ≥ 0, we define the function G λ : [0, +∞) → 0, λ π as We note that G λ (z) is convex on its domain, strictly monotone decreasing on 0 everywhere, and that (2.3) see [FLPS23,Lemmas 4.5 and 4.6].
Applying now Lemma 2.1 to the right-hand side of equation (2.1), we immediately get (2.4) Remark 2.2.It may happen that for m = ⌈λ⌉ − 1, the argument m + (1 − α), either on its own or together with m + α, exceeds λ, in which case the corresponding terms do not contribute to according to the definition of G λ .
We note that the quantity P D α (λ) appearing in the right-hand side of (2.4) is exactly the number of those points of shifted lattices (Z ± α)× Z − 1 4 which lie in the first quadrant under or on the graph of the function G λ .
Some novel methods for estimating the number of shifted lattice points under a graph of a convex decreasing function have been developed in [FLPS23]; they are not, however, directly applicable to the sum P D α (λ) for α > 0. The case α = 0 has been dealt with in [FLPS23, Theorem 5.1]; the following result is its extension for α ∈ 0, 1 2 .This is the second main ingredient of the proof of Theorem 1.1. (2.6) The proof of Theorem 2.3 is based on the following Lemma 2.4.Let i , j ∈ Z, i < j , let α ∈ 0, 1 2 , and let g be a decreasing convex function on [i , j + 1] satisfying (2.5) for all z, w ∈ [i , j + 1].We assume that and that (2.8) (2.9) Proof of Lemma 2.4.The validity of the statement does not change if an integer constant is added to g , so without loss of generality we can take n = 0, in which case (2.7) becomes and (2.8) becomes which means that we exclude the situation when x = j + 1 is the first and only zero of g (x) in the interval [i , j + 1].Consider the sequence We will distinguish five cases depending on which element of this sequence first drops below 3 4 .
We can now proceed to the Proof of Theorem 2.3.We extend the function g by zero onto the interval (b, b +1].Let N = g (0) ≥ 0. We have two cases.
Case 2: N ≥ 1.We set, for k = 0, 1, . . ., N , L k := max{m ∈ {0, . . ., b} : g (m) ≥ k}, see [FLPS23,Figure 5] for an illustration.Since, by the conditions of the theorem, the function g is strictly monotone decreasing at all points where it takes positive values, the inverse function g −1 is well-defined on (0, g (0)], and L k = g −1 (k) for k > 0; also, by definition, L 0 = b.We also set and in the latter case set L N +1 := 0. In any case, we have where the strict inequalities follow from (2.5).
The result of Theorem 1.1 now follows immediately from applying Theorem 2.3 to g = G λ with b = ⌈λ⌉, using (2.3) and (2.4), and recalling that G λ (z) ≡ 0 for z ∈ [λ, ⌈λ⌉].The strict inequality follows from Remark 2.5 and the fact that G λ is not (piecewise) linear.§3.Proof of Theorem 1.3 As in §2, in view of (1.8) the counting function of Neumann eigenvalues which are less than or equal to λ 2 ∈ [0, +∞) is given by where we used the fact that j ′ ν,1 ≥ ν, and therefore The graphs of the first few zeros of Bessel derivatives appearing in (3.1) are shown in Figure 2. We split the proof of Theorem 1.3 into three parts, for different intervals of λ.The proofs for very small and sufficiently large λs are analytic, and a small remaining gap is closed with the help of a rigorous computer-assisted algorithm similar to that in [FLPS23,§8].
We start with the case of small λs.
Proof of Proposition 3.1.It is sufficient to show that for all α ∈ 0, 1 2 , (1.10) holds with n = 1 and n = 2, namely, that we have It follows immediately from the monotonicity with respect to ν of the zero j ′ ν,k [DLMF, §10.21(iv)], and from the interlacing properties of these zeros [PA10,formula (10)] that ,1 for all α ∈ 0, 1 2 , we, in principle, could finish the proof of Proposition 3.1 here by an elementary observation that j ′ 1,1 ≈ 1.8412 and j ′ ≈ 2.4605.However we deliberately avoid the use of floating point approximations of zeros of Bessel functions and their derivatives anywhere in this paper, so we continue to give a fully analytic justification of (3.2) and (3.3).
The Bessel functions of half-integer order can be expressed explicitly in terms of trigonometric functions and powers of their argument, and as a result j ′ 3 2 ,1 coincides with the first positive root of the function Elementary calculations show that and therefore S(z) has a single root in (0, π).As S 5π 6 = 3 + 6 5π − 5π 9 > 0, we conclude that thus proving (3.3).
The proof of Proposition 3.2 is based on several results from [FLPS23], the first of which complements Lemma 2.1.
Applying Lemma 3.3 to the right-hand side of (3.1), we get (3.4) In order to obtain a lower bound on P N α (λ), we recall Remark 3.6.The bound in [FLPS23,Theorem 6.1] was stated slightly differently compared to (3.5), giving a marginally weaker result, however inequality (3.5) also appeared there towards the end of the proof.
Proof of Proposition 3.2.We want to apply Lemma 3.4 to each of the sums in the right-hand side of (3.4).
For the second sum, we take g (z Adding the two bounds (3.7) and (3.8) yields 4 . (3.9) We note that we additionally require (3.10) for (3.7)-(3.9) to be valid.We now check if we need to restrict the range of λ further to ensure that (3.10) holds.We can replace (3.10) by a stronger condition , it is sufficient to assume from now on that λ ≥ 3.
To make the bound in the right-hand side of (3.9) more explicit in terms of λ, we first estimate the integrals there.For integrals near zero, an obvious bound Proof of Lemma 3.7.We fix λ > 0, change the variable β = λγ, γ ∈ [0, 1], and intend to show that We have, by an explicit calculation, We again have χ(1) = 0, and the result follows from the fact that We now return to the previously obtained inequality (3.9) and estimate the integrals in its righthand side using (3.11).This gives 4 .
(3.12)  Proposition 3.8 is proved using the same strategy of applying a rigorous computer-assisted algorithm for estimating Q(λ) (which requires a finite number of steps and uses only integer arithmetic) as was done in the case of the Neumann Laplacian for the count P N 0 (λ) in [FLPS23,§8].Let, for x ∈ R, x ≤ x ≤ x, where x, x ∈ Q are some lower and upper rational approximations of x.For λ ∈ Q we compute a verified lower rational approximation Q(λ) using the techniques of [FLPS23,§8].We now use the analogue of [FLPS23,Lemma 8.The results of the calculations are shown in Table 1, and the script we used is available at https://michaellevitin.net/polya.html#AB. Step -Bohm operator with either Dirichlet u| ∂Ω = 0 or magnetic Neumann ∇u − iαA p u ∂Ω , n = 0 boundary conditions.Here n denotes the external unit normal at the boundary.It is well-known that the Dirichlet problem has discrete spectrum for any bounded domain Ω, and the same is true for the magnetic Neumann problem under the standard regularity assumptions, e.g., if ∂Ω is Lipschitz, see [CPS22, Appendix A].

Figure 1 :
Figure 1: Some zeros of Bessel functions as functions of α.

Figure 2 :
Figure 2: Some zeros of derivatives of Bessel functions as functions of α.

Figure 4 :
Figure 4: The difference Q(λ) − λ 24 plotted as a function of λ ∈ 5 2 , 9 .The plot has been produced using floating-point arithmetic and is included for illustration only: it does not constitute a part of the proof.

Table 1 :
Detailed output of the computer-assisted algorithm.