Eigenvalue Clusters of the Landau Hamiltonian

in the Exterior of a Compact Domain

Alexander Pushnitski and Grigori Rozenblum

Received: August 30, 2007 Revised: October 20, 2007

Communicated by Heinz Siedentop

Abstract. We consider the Schrödinger operator with a constant magnetic field in the exterior of a compact domain on the plane. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We discuss the rate of accumulation of eigenvalues in a fixed cluster.

2000 Mathematics Subject Classification: Primary 35P20; Secondary 35Q40. Keywords and Phrases: Schrödinger operator, magnetic field, spectral asymptotics, exterior problem

1 Introduction

1.1 Preliminaries

The Landau Hamiltonian describes a charged particle confined to a plane in a constant magnetic field. The Landau Hamiltonian is one of the earliest explicitly solvable quantum mechanical models. Its spectrum consists of the Landau levels,^{1} infinitely degenerate eigenvalues placed at the points of an arithmetic progression. In [7], the Landau Hamiltonian was considered in the exterior of a compact obstacle. Introducing the obstacle produces clusters of eigenvalues of finite multiplicity around the Landau levels. Various asymptotics (high energy, semiclassical) of these eigenvalue clusters were studied in [7]. In this paper we focus on a different aspect of the spectral analysis of this model: for a fixed eigenvalue cluster, we consider the rate of accumulation of eigenvalues in this cluster to the Landau level. We describe this rate of accumulation rather precisely in terms of the logarithmic capacity of the obstacle. Our construction is motivated by the recent progress in the study of the Landau Hamiltonian on the whole plane perturbed by a compactly supported or fast decaying electric or magnetic field, see [15, 13, 3, 16]. In particular, we use some operator theoretic constructions from [15] and [13] and some concrete analysis (related to logarithmic capacity) from [3].

1It is a little known fact that this was worked out by Fock two years before Landau; see [4, 9].

Documenta Mathematica 12 (2007) 569–586

1.2 The Landau Hamiltonian

We will write x = (x_{1}, x_{2}) ∈ ℝ^{2} and identify ℝ^{2} with ℂ in the standard way, setting z = x_{1} + ix_{2} ∈ ℂ. The Lebesgue measure in ℝ^{2} will be denoted by dx and in ℂ by dm(z). The derivatives with respect to x_{1}, x_{2} are denoted by ∂_{k} = ∂_{x_{k}}; we set, as usual, ¯∂ = (∂_{1} + i∂_{2})/2, ∂ = (∂_{1} − i∂_{2})/2. We denote by B > 0 the magnitude of the constant magnetic field in ℝ^{2}. We choose the gauge A(x) = (A_{1}(x), A_{2}(x)) = (−\frac{1}{2}Bx_{2}, \frac{1}{2}Bx_{1}) for the magnetic vector potential associated with this field. The magnetic Hamiltonian on the whole plane is defined as

X_{0} = −(∇ − iA)^{2} in L^{2}(ℝ^{2}).  (1)

More precisely, for u ∈ C_{0}^{∞}(ℝ^{2}) we set

∥u∥_{H_{A}^{1}}^{2} = ∫_{ℝ^{2}} |i∇u(x) + A(x)u(x)|^{2} dx  (2)

and define X_{0} as the selfadjoint operator which corresponds to the closure of the quadratic form ∥u∥_{H_{A}^{1}}^{2}, u ∈ C_{0}^{∞}(ℝ^{2}). It is well known (see [4, 9] or [10]) that the spectrum of X_{0} consists of the eigenvalues Λ_{q} = (2q + 1)B, q = 0, 1, …, of infinite multiplicity. In particular, we have

∥u∥_{H_{A}^{1}}^{2} ≥ B∥u∥_{L^{2}}^{2}, u ∈ C_{0}^{∞}(ℝ^{2}).  (3)

We will denote by L_{q} the eigenspace of X_{0} corresponding to Λ_{q} and by P_{q} the operator of orthogonal projection onto L_{q} in L^{2}(ℝ^{2}). Later on, we will need an explicit description of L_{q}; this will be discussed in section 4.2. Let Ω ⊂ ℝ^{2} be an open set. In order to define the magnetic Hamiltonian in Ω, it is convenient to consider the associated quadratic form. Following [12], we denote by H_{A}^{1}(Ω) the closure of C_{0}^{∞}(Ω) with respect to the norm ∥u∥_{H_{A}^{1}}. The quadratic form ∥u∥_{H_{A}^{1}}^{2} is closed in L^{2}(Ω) and (by (3)) positively definite. This form defines a self-adjoint operator in ℝ^{2} which we denote by X(Ω). If Ω is bounded by a smooth curve, then the usual computations show that this definition of X(Ω) corresponds to setting the Dirichlet boundary condition on ∂Ω. The operator X_{0} corresponds to taking Ω = ℝ^{2} in the above definitions.

Note that for a bounded Ω, the norm in H_{A}^{1}(Ω) is equivalent to the standard Sobolev norm H^{1}(Ω); in particular, in this case the embedding H_{A}^{1}(Ω) ⊂ L^{2}(Ω) is compact.

1.3 Main results

Let K ⊂ ℝ^{2} be a compact set and K^{c} its complement. Our main results concern the spectrum of the operator X(K^{c}). First we state a preliminary result which gives a general description of the spectrum of X(K^{c}). This result is already known (see [7]) but as part of our construction, we provide a simple proof in Section 1.4.

Proposition 1.1. Let K ⊂ ℝ^{2} be a compact set. Then

σ_{ess}(X(K^{c})) = σ_{ess}(X_{0}) = ∪_{q=0}^{∞}{Λ_{q}}, Λ_{q} = (2q + 1)B.

Moreover, for all q and all λ ∈ (Λ_{q−1}, Λ_{q}) the number of eigenvalues of X(K^{c}) in (λ, Λ_{q}) is finite.

In other words, the last statement means that the eigenvalues of X(K^{c}) can accumulate to the Landau levels only from above. For all q ≥ 0, we enumerate the eigenvalues of X(K^{c}) in (Λ_{q}, Λ_{q+1}):

λ_{1}^{q} ≥ λ_{2}^{q} ≥ …

Proposition 1.1 ensures that λ_{n}^{q} → Λ_{q} as n → ∞. Below we describe the rate of this convergence. Roughly speaking, we will see that for large n,

\frac{a^{n}}{n!} ≤ λ_{n}^{q} − Λ_{q} ≤ \frac{b^{n}}{n!}  (4)

with some a, b depending on K. In order to discuss the dependence of a, b on the domain K, let us introduce the following notation:

Δ_{q}(K) = lim sup_{n→∞}[n!(λ_{n}^{q} − Λ_{q})]^{1/n},
δ_{q}(K) = lim inf_{n→∞}[n!(λ_{n}^{q} − Λ_{q})]^{1/n}.

(5)

The estimates for these spectral characteristics will be given in terms of the logarithmic capacity of K, which is denoted by Cap (K). Below we recall the definition and basic properties of logarithmic capacity; for the details, see e.g.

[11] or [18]. For a measure μ ≥ 0 in ℝ^{2}, its logarithmic energy is defined by

I(μ) = ∫ ∫ log \frac{1}{|z − t|}dμ(z)dμ(t).

For a compact set K ⊂ ℝ^{2}, its logarithmic capacity is defined by

Cap (K) = e^{−V (K)},

V (K) = inf{I(μ) | μ ≥ 0 is a measure in ℝ^{2}, supp μ ⊂ K, μ(K) = 1}.

The logarithmic capacity of compact sets in ℝ^{2} has the following properties:

(i) if K_{1} ⊂ K_{2} then Cap K_{1} ≤ Cap K_{2}; (ii) the logarithmic capacity of a disc of radius r is r; (iii) if K_{2} = {αx | x ∈ K_{1}}, α > 0, then Cap K_{2} = α Cap K_{1}; (iv) Cap K coincides with the logarithmic capacity of the polynomial convex hull Pc (K) of K (=the complement of the unbounded connected component of K^{c}).

(v) Continuity of capacity: if K_{ε} = {x ∈ ℝ^{2} | dist (x, K) ≤ ε}, then Cap K_{ε} → Cap K as ε → +0. To extend the notion of capacity to arbitrary Borel sets E, one defines the inner and outer capacities

Cap_{i}(E) = sup{Cap (K) | K ⊂ E, K compact}
Cap_{o}(E) = inf{Cap_{i}(U) | E ⊂ U, U open}.

Then every Borel set E is capacitable in the sense that Cap_{i}(E) = Cap_{o}(E) and one can simply write Cap (E) for this common value. We will also need another version of inner capacity, which we denote by Cap_{−}(K) and define by

sup{Cap S | S ⊂ K is a compact set with a Lipschitz boundary}.

Theorem 1.2. Let K ⊂ ℝ^{2} be a compact set; then for all q ≥ 0 one has

Δ_{q}(K) ≤ \frac{B}{2} (Cap (K))^{2},
δ_{q}(K) ≥ \frac{B}{2} (Cap_{−}(Pc (K)))^{2}.

In particular, if K is a compact set with a Lipschitz boundary, then Δ_{q}(K) = δ_{q}(K) = \frac{B}{2} (Cap (K))^{2} for all q ≥ 0.

The lower bound in the above theorem is strictly positive if and only if Pc (K) has a non-empty interior. In particular, for such compacts the number of eigenvalues λ_{1}^{q}, λ_{2}^{q}, … is infinite for each q. However, even for some compacts without interior points, lower spectral bounds can be obtained. In particular, this can be done for the compact K being a smooth (not necessarily closed) curve.

Theorem 1.3. Let K ⊂ ℝ^{2} be a C^{∞} smooth simple curve of a finite length. Then for all q ≥ 0, one has

Δ_{q}(K) = δ_{q}(K) = \frac{B}{2} (Cap (K))^{2}.

Remark. 1. One can prove that

if Cap (K) = 0, then C_{0}^{∞}(K^{c}) is dense in H_{A}^{1}(ℝ^{2}).  (6)

It follows that for K of zero capacity, H_{A}^{1}(K^{c}) = H_{A}^{1}(ℝ^{2}) and therefore X(K^{c}) = X_{0}. Thus, for such K the spectrum of X(K^{c}) consists of Landau levels Λ_{q}.

The statement (6) seems to be well known to the experts in the field although it is difficult to pinpoint the exact reference. One can use the argument of [1], Theorem 9.9.1; this argument applies to the usual H^{1} Sobolev norm, but it is very easy to modify it for the norm H_{A}^{1}. In this theorem the Bessel capacity rather than the logarithmic capacity is used; however, the Bessel capacity of a compact set vanishes if and only if its logarithmic capacity vanishes. In order to prove the latter fact (again, well known to experts) one has to combine Theorem 2.2.7 in [1] and Sect.II.4 in [11].

2. We do not know whether it possible for Λ_{q} to remain eigenvalues of X(K^{c}) of infinite multiplicity if Cap (K) > 0.

3. Following the proof of Theorem 1.2 and using the results of [3], it is easy to show that for q = 0, the lower bound in this theorem can be replaced by the following one:

δ_{0}(K) ≥ \frac{B}{2} (Cap (K_{−}))^{2},

K_{−} = \{z ∈ ℂ | lim sup_{r→+0} \frac{log m(P c(K) ∩ D_{r}(z))}{log r} < ∞\},

where D_{r}(z) = {ζ ∈ ℂ | |ζ − z| ≤ r}, and m(·) is the Lebesgue measure.

4. Analysing the proof of Theorem 1.3, it is easy to see that if we are only interested in its statement for finitely many q, it suffices to require some finite smoothness of the curve K.

1.4 Outline of the proof

Let us write L^{2}(ℝ^{2}) = L^{2}(K^{c})⊕L^{2}(K). (If the Lebesgue measure of K vanishes then, of course, L^{2}(K) = {0}.) With respect to this decomposition, let us define

R(K^{c}) = X(K^{c})^{−1} ⊕ 0 in L^{2}(ℝ^{2}) = L^{2}(K^{c}) ⊕ L^{2}(K).  (7)

Clearly, for any λ ≠ 0 we have

λ ∈ σ(X(K^{c})) ⇔ λ^{−1} ∈ σ(R(K^{c}))  (8)

with the same multiplicity. Thus, it suffices for our purposes to study the spectrum of the operator R(K^{c}). First note that in the “free” case K = ∅ we have R(ℝ^{2}) = X_{0}^{−1} and the spectrum of X_{0}^{−1} consists of the eigenvalues Λ_{q}^{−1} of infinite multiplicity and their point of accumulation, zero.

Next, it turns out (see section 3) that

R(K^{c}) = X_{0}^{−1} − W, where W ≥ 0 is compact.  (9)

Thus, the Weyl’s theorem on the invariance of the essential spectrum under compact perturbations ensures that σ_{ess}(R(K^{c})) = σ_{ess}(X_{0}^{−1}). Moreover, a simple operator theoretic argument (see e.g. [2, Theorem 9.4.7]) shows that the eigenvalues of R(K^{c}) do not accumulate to the inverse Landau levels Λ_{q}^{−1} from above. Thus, the spectrum of R(K^{c}) consists of zero and the eigenvalue clusters {(λ_{1}^{q})^{−1}, (λ_{2}^{q})^{−1}, … } with the eigenvalues in the q’th cluster accumulating to Λ_{q}^{−1}. In section 2.3 we show that the rate of accumulation of (λ_{n}^{q})^{−1} to Λ_{q}^{−1} can be described in terms of the spectral asymptotics of the Toeplitz type operator P_{q}W P_{q}; here W is defined by (9) and P_{q} is the projection onto L_{q} = Ker (X_{0} − Λ_{q}) = Ker (X_{0}^{−1} − Λ_{q}^{−1}). The spectrum of P_{q}W P_{q} is studied in sections 4 and 5, using the results of [3]. The fact that the logarithmic capacity of the domain appears in this problem probably deserves some explanation. In [3], the spectral asymptotics of P_{q}W P_{q} was related to the asymptotics of the singular numbers of the embedding of the Segal-Bargmann space F^{2} (see section 4.2 below) into an L^{2} space with the weight related to W. Following the technique of [14], in [3] the analysis of this asymptotics is then reduced to the analysis of the sequence of polynomials of a complex variable, orthogonal with respect to the relevant weight. After this, the results of [18] ensure that the asymptotics of these polynomials is determined by the logarithmic capacity of the support of the weight.

1.5 Acknowledgements

A large part of this work was completed during the authors’ stay at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) in the framework of the programme “Spectral Theory and Partial Differential Equations”. It is a pleasure to thank the Institute and the organisers of the programme for providing this opportunity. The authors are also grateful to Uzy Smilansky for useful discussions and to Nikolai Filonov for helpful remarks on the text of the manuscript.

2 Some abstract results

Here we collect some general operator theoretic statements that are used in the proof. The statements themselves, with the exception of the last one, are almost obvious, but spelling them out explicitly helps explain the main ideas of our construction.

2.1 Quadratic forms

Our arguments can be stated most succinctly if we are allowed to deal with quadratic forms whose domains are not necessarily dense in the Hilbert space.

Here is the corresponding abstract framework; related constructions appeared before in the literature; see e.g. [17]. Let a be a closed positive definite quadratic form in a Hilbert space H with the domain d[a]. Let the closure of d[a] in H be H_{a}. Then the form a defines a self-adjoint operator A in H_{a}. Let J_{a} : H_{a} → H be the natural embedding operator; its adjoint J_{a}^{∗} : H → H_{a} acts as the orthogonal projection onto the subspace H_{a} of H. The operator J_{a}A^{−1}J_{a}^{∗} in H can be considered as the direct sum

J_{a}A^{−1}J_{a}^{∗} = A^{−1} ⊕ 0 in the decomposition H = H_{a} ⊕ H_{a}^{⊥};

here we have in mind (7). Now let b be another closed positive definite form in H and let B, d[b], H_{b}, J_{b} be the corresponding objects constructed for this form.

Proposition 2.1. Suppose that d[b] ⊂ d[a] and b[x, y] = a[x, y] for all x, y ∈ d[b]. Then:

(i) J_{b}B^{−1}J_{b}^{∗} ≤ J_{a}A^{−1}J_{a}^{∗} on H; (ii) if x ∈ d[b] ∩ Dom (A), then x ∈ Dom (B), Bx = Ax, and J_{b}B^{−1}J_{b}^{∗}Ax = J_{a}A^{−1}J_{a}^{∗}Ax.

Proof. It suffices to consider the case H_{a} = H.

(i) The hypothesis implies

b[x, x] = a[J_{b}x, J_{b}x] for all x ∈ d[b].

This can be recast as ∥B^{1/2}x∥ = ∥A^{1/2}J_{b}x∥, x ∈ d[b]. It follows that the operator A^{1/2}J_{b}B^{−1/2} is an isometry on H_{b} and therefore A^{1/2}J_{b}B^{−1/2}J_{b}^{∗} is a contraction on H. By conjugation, we get that ∥J_{b}B^{−1/2}J_{b}^{∗}A^{1/2}z∥ ≤ ∥z∥ for all z ∈ d[a]. The last statement is equivalent to ∥J_{b}B^{−1/2}J_{b}^{∗}u∥ ≤ ∥A^{−1/2}u∥ for all u ∈ H, and so J_{b}B^{−1}J_{b}^{∗} ≤ A^{−1} as required. (ii) Let y ∈ d[b]; then

b[x, y] = a[x, y] = (Ax, y),

and so x ∈ Dom (B) and Bx = Ax. Next, J_{a}A^{−1}J_{a}^{∗}Ax = A^{−1}Ax = x, and

J_{b}B^{−1}J_{b}^{∗}Ax = J_{b}B^{−1}J_{b}^{∗}Bx = J_{b}B^{−1}Bx = J_{b}x = x,

which proves the required statement.

2.2 Shift in enumeration

The asymptotics of the type discussed in Theorems 1.2 and 1.3 is independent of a shift in the enumeration of eigenvalues. This is a consequence of the following elementary fact. Let b_{1} ≥ b_{2} ≥ … be a sequence of positive numbers such that lim sup_{n→∞}[n!b_{n}]^{1/n} < ∞. Then for all ℓ ∈ ℤ,

lim_{n→∞} \left\{{sup \atop inf}\right\}[n!b_{n+ℓ}]^{1/n} = lim_{n→∞} \left\{{sup \atop inf}\right\}[n!b_{n}]^{1/n}.  (10)

2.3 Accumulation of eigenvalues

Having in mind (9), let us consider the following general situation. Let T be a self-adjoint operator and let Λ be an isolated eigenvalue of T of infinite multiplicity with the corresponding eigenprojection P_{Λ}. Let τ > 0 be such that

((Λ − 2τ, Λ + 2τ) \ {Λ}) ∩ σ(T) = ∅.

Next, let W ≥ 0 be a compact operator; consider the spectrum of T − W. The Weyl’s theorem on the invariance of the essential spectrum under compact perturbations ensures that

((Λ − 2τ, Λ + 2τ) \ {Λ}) ∩ σ_{ess}(T − W) = ∅.

Moreover, a simple argument (see e.g. [2, Theorem 9.4.7]) shows that the eigenvalues of T − W do not accumulate to Λ from above (i.e. (Λ, Λ + ϵ) ∩ σ(T − W) = ∅ for some ϵ > 0). We will need a description of the eigenvalues of T − W below Λ in terms of the eigenvalues of the Toeplitz operator P_{Λ}W P_{Λ}. Let μ_{1} ≥ μ_{2} ≥ ⋯ be the eigenvalues of P_{Λ}W P_{Λ}; in order to exclude degenerate cases, let us assume that this operator has infinite rank. Let λ_{1} ≤ λ_{2} ≤ ⋯ be the eigenvalues of T − W in the interval (Λ − τ, Λ).

Proposition 2.2. Under the above assumptions, for any ϵ > 0 there exists ℓ ∈ ℤ such that for all sufficiently large n, one has

(1 − ϵ)μ_{n+ℓ} ≤ Λ − λ_{n} ≤ (1 + ϵ)μ_{n−ℓ}.

The proof borrows its key element from [8, Lemma 1.1]. An alternative proof can be found in [15, Proposition 4.1].

Proof. 1. We denote S = T − W and Q_{Λ} = I − P_{Λ} and consider the operators

R_{±} = ϵP_{Λ}W P_{Λ} + \frac{1}{ϵ}Q_{Λ}W Q_{Λ} ± (P_{Λ}W Q_{Λ} + Q_{Λ}W P_{Λ}).

and

S_{±} = P_{Λ}(T − (1 ± ϵ)W)P_{Λ} + Q_{Λ}(T − (1 ± \frac{1}{ϵ})W)Q_{Λ}.

We have

S = S_{+} + R_{−} = S_{−} − R_{+}.

2. Since W is compact, the operators R_{±} are also compact. Since R_{±} can be represented as

R_{±} = (\sqrt{ϵ}P_{Λ} ± \frac{1}{\sqrt{ϵ}}Q_{Λ})W (\sqrt{ϵ}P_{Λ} ± \frac{1}{\sqrt{ϵ}}Q_{Λ})

and W ≥ 0, we see that R_{±} ≥ 0.

3. Let us discuss the spectrum of S_{±} in (Λ − τ, Λ). Clearly, the spectrum of P_{Λ}(T − (1 ± ϵ)W)P_{Λ} = ΛP_{Λ} − (1 ± ϵ)P_{Λ}W P_{Λ} consists of the eigenvalues Λ − (1 ± ϵ)μ_{n}. Next, since by assumption, T |_{Ran Q_{Λ}} has no spectrum in (Λ − 2τ, Λ + 2τ) and W is compact, we see that Q_{Λ}(T − (1 ± \frac{1}{ϵ})W)Q_{Λ} |_{Ran Q_{Λ}} has only finitely many eigenvalues in the interval (Λ − τ, Λ + τ). Since the operators P_{Λ}(T − (1 ± ϵ)W)P_{Λ} and Q_{Λ}(T − (1 ± \frac{1}{ϵ})W)Q_{Λ} act in orthogonal subspaces of our Hilbert space, the spectrum of S_{±} is the union of the spectra of these operators. So we arrive at the following conclusion. Let ν_{1}^{±} ≤ ν_{2}^{±} ≤ ⋯ denote the eigenvalues of S_{±} in (Λ − τ, Λ). Then

ν_{n}^{+} = Λ − (1 + ϵ)μ_{n−i}, ν_{n}^{−} = Λ − (1 − ϵ)μ_{n−j},  (11)

for some integers i, j and all sufficiently large n.

4. Let us prove that λ_{n} ≤ ν_{n+k}^{−} for some integer k and all sufficiently large n. Denote δ = (λ_{1} −Λ+τ)/2 and let us write R_{+} = R_{+}^{(1)} +R_{+}^{(2)}, where 0 ≤ R_{+}^{(1)} ≤ δI and rank R_{−}^{(2)} < ∞. Denote by N_{S}(α, β) the number of eigenvalues of S in the interval (α, β). Writing S = S_{−} − R_{+}^{(1)} − R_{+}^{(2)}, we get for any λ ∈ (λ_{1}, Λ):

N_{S}(Λ − τ, λ) = N_{S}(λ_{1} − 2δ, λ)

≥ N_{S_{−}−R_{+}^{(1)}}(λ_{1} − 2δ, λ) − rank R_{−}^{(2)} ≥ N_{S_{−}}(λ_{1} − δ, λ) − rank R_{−}^{(2)}.

The second inequality above follows from σ(R_{+}^{(1)}) ⊂ [0, δ] (see [2, Lemma 9.4.3]). These inequalities for the eigenvalue counting functions can be rewritten as λ_{n} ≤ ν_{n+k}^{−} with some integer k. In the same way, one proves that λ_{n} ≥ ν_{n−k}^{+} for large n and some integer k. Taken together with (11), this yields the required result.

3 Preliminaries and reduction to Toeplitz operators

Let K ⊂ ℝ^{2} be a compact set; we return to the discussion of the spectrum of X(K^{c}) and start with some general remarks. First we would like to point out that the spectral asymptotics that we are interested in is independent of the “holes” in the domain K:

δ_{q}(K) = δ_{q}(P c(K)), Δ_{q}(K) = Δ_{q}(P c(K)).  (12)

Indeed, let us write K^{c} = Ω ∪ Σ, where Ω is the unbounded connected component of K^{c} and Ω and Σ are disjoint. With respect to the direct sum decomposition L^{2}(K^{c}) = L^{2}(Ω) ⊕ L^{2}(Σ), we have X(K^{c}) = X(Ω) ⊕ X(Σ). By the compactness of the embedding H_{A}^{1}(Σ) ⊂ L^{2}(Σ), the operator X(Σ) has a compact resolvent. Thus, on any bounded interval of the real line the spectrum of X(K^{c}) differs from the spectrum of X(Ω) by at most finitely many eigenvalues. By (10), this yields (12).

Next, we apply the abstract reasoning of section 2.1 to the quadratic form a[u] = ∥u∥_{H_{A}^{1}(K^{c})}^{2} with domain d[a] = H_{A}^{1}(K^{c}), considering L^{2}(ℝ^{2}) as the main Hilbert space H. We consider the operator R(K^{c}) (see (7)) and write R(K^{c}) = X_{0}^{−1} − W. Proposition 2.2 suggests that in order to find the rate of accumulation of the eigenvalues of R(K^{c}) to Λ_{q}^{−1}, one should study the spectrum of the Toeplitz type operators P_{q}W P_{q}. This is done in the next section. Denote by μ_{1}^{q} ≥ μ_{2}^{q} ≥ … the eigenvalues of P_{q}W P_{q}. We will prove

Proposition 3.1. Let K ⊂ ℝ^{2} be a compact set and q ≥ 0. Then

lim sup_{n→∞}(n!μ_{n}^{q})^{1/n} ≤ \frac{B}{2} (Cap (K))^{2},
lim inf_{n→∞}(n!μ_{n}^{q})^{1/n} ≥ \frac{B}{2} (Cap_{−}(K))^{2}.

If K is a C^{∞} smooth curve, then one has

lim_{n→∞}(n!μ_{n}^{q})^{1/n} = \frac{B}{2} (Cap (K))^{2}.

Now we can prove our main statements.

Proof of Theorem 1.1 and Theorem 1.2. Combining Proposition 3.1, Proposition 2.2 and (12), we get the estimates for the quantities

lim_{n→∞} sup[n!(Λ_{q}^{−1} − (λ_{n}^{q})^{−1})]^{1/n} ≤ \frac{B}{2} (Cap (K))^{2},
lim_{n→∞} inf[n!(Λ_{q}^{−1} − (λ_{n}^{q})^{−1})]^{1/n} ≥ \frac{B}{2} (Cap_{−}(P c(K))^{2}

for any compact K. If K is a C^{∞} smooth curve, we get

lim_{n→∞}[n!(Λ_{q}^{−1} − (λ_{n}^{q})^{−1})]^{1/n} = \frac{B}{2} (Cap (K))^{2}.

An elementary argument shows that

lim_{n→∞} \left\{{sup \atop inf}\right\}[n!(Λ_{q}^{−1} − (λ_{n}^{q})^{−1})]^{1/n} = lim_{n→∞} \left\{{sup \atop inf}\right\}[n!(λ_{n}^{q} − Λ_{q})]^{1/n}.

This yields the required statements.

Proof of (9). Let D be a disc such that K ⊂ D. By Proposition 2.1(i), we get

D^{c} ⊂ K^{c} ⊂ ℝ^{2} ⇒ R(D^{c}) ≤ R(K^{c}) ≤ X_{0}^{−1}

and so

0 ≤ X_{0}^{−1} − R(K^{c}) ≤ X_{0}^{−1} − R(D^{c}).  (13)

Thus, W = X_{0}^{−1} − R(K^{c}) is non-negative; let us address compactness.

It is well known that if 0 ≤ V_{1} ≤ V_{2} are self-adjoint operators and V_{2} is compact, then V_{1} is also compact. Thus, by (13), in order to prove the compactness of W, it suffices to check that X_{0}^{−1} − R(D^{c}) is compact. Let Γ = ∂D. Employing the same arguments as in the proof of (12), we see that X(Γ^{c})^{−1} − R(D^{c}) is the inverse of the magnetic operator on the disc and hence a compact operator. Thus, it suffices to prove that the difference

X_{0}^{−1} − X(Γ^{c})^{−1} = (X_{0}^{−1} − R(D^{c})) − (X(Γ^{c})^{−1} − R(D^{c}))

is compact. Let us compute the quadratic form of this difference. Let f, g ∈ L^{2}(ℝ^{2}), X_{0}^{−1}f = u, X(Γ^{c})^{−1}g = v. We have

((X_{0}^{−1} − X(Γ^{c})^{−1})f, g) = (u, X(Γ^{c})v) − (X_{0}u, v).

Integrating by parts and noting that v ∈ Dom (X(Γ^{c})) vanishes on Γ, we get

(u, X(Γ^{c})v) − (X_{0}u, v) = ∫_{Γ}(n_{A}v(s)^{+} + n_{A}v(s)^{−})u(s)ds  (14)

where n_{A}v(s) = (∇ − iA(s))v · n(s), n(s) is the exterior normal to Γ at the point s and the superscripts + and − indicate that the limits of the functions are taken on the circle Γ by approaching it from the outside or inside. Take a smooth cut-off function ω ∈ C_{0}^{∞}(ℝ^{2}) such that ω(x) = 1 in the neighborhood of D. Then we can replace u, v by u_{1} = ωu, v_{1} = ωv in the r.h.s. of (14). By the local elliptic regularity we have u_{1} ∈ H^{2}(ℝ^{2}), v_{1} ∈ H^{2}(Γ^{c}), and the corresponding Sobolev norms of u_{1}, v_{1} can be estimated via the L^{2}-norms of f, g. Now it remains to notice that the trace mapping u_{1} ↦ u_{1}|_{Γ} is compact as considered from H^{2}(ℝ^{2}) to L^{2}(Γ), and the mappings v_{1} ↦ (n_{A}v_{1})^{±} are compact as considered from H^{2}(Γ^{c}) to L^{2}(Γ). It follows that the difference X_{0}^{−1} − X(Γ^{c})^{−1} is compact, as required.

4 The spectrum of Toeplitz operators

4.1 Restriction operators and the associated Toeplitz operators

Let μ be a finite measure in ℝ^{2} with a compact support. Consider the restriction operator

γ_{0} : C_{0}^{∞}(ℝ^{2}) ∋ u ↦ u |_{supp(μ)}∈ L^{2}(μ).

We are interested in two special cases, namely when μ is the restriction of the Lebesgue measure to a set with Lipschitz boundary and when μ is the arc length measure on a simple smooth curve. In both cases γ_{0} can be extended by continuity to a bounded and compact operator γ : H_{A}^{1}(ℝ^{2}) → L^{2}(μ). Next, let J : H_{A}^{1}(ℝ^{2}) → L^{2}(ℝ^{2}) be the embedding operator, J : u ↦ u. Then the adjoint J^{∗} : L^{2}(ℝ^{2}) → H_{A}^{1}(ℝ^{2}) acts as J^{∗} : u ↦ X_{0}^{−1/2}u.

For q ≥ 0, consider the operators T_{q}(μ) in L^{2}(ℝ^{2}) defined by the quadratic form

(T_{q}(μ)u, u)_{L^{2}(ℝ^{2})} = ∫_{supp μ}|(P_{q}u)(x)|^{2}dμ(x), u ∈ L^{2}(ℝ^{2}).

This operator can be represented as

T_{q}(μ) = (γJ^{∗}X_{0}^{1/2}P_{q})^{∗}(γJ^{∗}X_{0}^{1/2}P_{q}) = Λ_{q}(γJ^{∗}P_{q})^{∗}(γJ^{∗}P_{q}).

Since γ is compact by assumption, the operator T_{q}(μ) is also compact. Fix q ≥ 0; let s_{1}^{q} ≥ s_{2}^{q} ≥ … be the eigenvalues of T_{q}(μ) in L^{2}(ℝ^{2}).

Proposition 4.1. (i) Let μ be the restriction of the Lebesgue measure onto a bounded domain K ⊂ ℝ^{2} with a Lipschitz boundary. Then

lim_{n→∞}(n!s_{n}^{q})^{1/n} = \frac{B}{2} (Cap (K))^{2}.

(ii) Let μ be the arc length measure of a C^{∞} smooth simple curve of a finite length. Then

lim_{n→∞}(n!s_{n}^{q})^{1/n} = \frac{B}{2} (Cap (supp μ))^{2}.

Before proving this proposition, we need to recall the description of the subspaces L_{q}.

4.2 The structure of subspaces L_{q}

Denote Ψ(z) = \frac{1}{4}B|z|^{2}. Let us define the creation and annihilation operators (first introduced in this context by Fock [5])

𝔔 = −2ie^{−Ψ}\overline{∂}e^{Ψ} = −2i\overline{∂} − \frac{B}{2} iz
𝔔^{∗} = −2ie^{Ψ}∂e^{−Ψ} = −2i∂ + \frac{B}{2} i\overline{z}.

The Landau Hamiltonian can be expressed as

X_{0} = 𝔔^{∗}𝔔 + B = 𝔔𝔔^{∗} − B.  (15)

The spectrum and spectral subspaces of X_{0} can be described in the following way. The equation (X_{0} − B)u = 0 is equivalent to

𝔔u = −2ie^{−Ψ} ¯∂(e^{Ψ}u) = 0.

This means that f = e^{Ψ}u is an entire analytic function such that e^{−Ψ}f ∈ L^{2}(ℂ). The space of such functions f is called Fock or Segal-Bargmann space F^{2} (see [6] for an extensive discussion). So L_{0} = e^{−Ψ}F^{2}. Further eigenspaces L_{q}, q = 1, 2, …, are obtained as L_{q} = (𝔔^{∗})^{q}L_{0}. The operators 𝔔^{∗}, 𝔔 act between the subspaces L_{q} as

𝔔^{∗} : L_{q} ↦ L_{q+1}, 𝔔 : L_{q} ↦ L_{q−1}, 𝔔 : L_{0} ↦ {0},  (16)

and are, up to constant factors, isometries on L_{q}. In particular, the substitution

L_{q} ∋ u = C_{q}^{−1}(𝔔^{∗})^{q}e^{−Ψ}f, f ∈ F^{2}, C_{q} = \sqrt{q!(2B)^{q}}  (17)

gives a unitary equivalence of spaces L_{q} and F^{2}.

4.3 Proof of Proposition 4.1

(i) The proof is given in [3, Lemma 3.1] for q = 0 and [3, Lemma 3.2] for q ≥ 0. (ii) For q = 0 the result again follows from Lemma 3.1 in [3]. Although the reasoning there concerns the operators T_{q}(v) = (vP_{q})^{∗}(vP_{q}) where the function v is separated from zero on a compact, it goes through for T_{q}(μ). Only notational changes are required; one simply has to replace the measure v(z)dm(z) by dμ(z). For q ≥ 1 below we apply the reduction to the lowest Landau level similar to the proof of Lemma 3.2 in [3]. Denote dμ̃(z) = e^{−Ψ(z)}dμ(z). Applying the unitary equivalence (17), we get for u ∈ L_{q}

(T_{q}(μ)u, u)_{L^{2}(ℝ^{2})} = C_{q}^{−2}∥(2∂ − B\overline{z})^{q}f∥_{L^{2}(μ̃)}^{2}.  (18)

In particular, for q = 0

(T_{0}(μ)u, u)_{L^{2}(ℝ^{2})} = C_{0}^{−2}∥f∥_{L^{2}(μ̃)}^{2}.  (19)

Below we separately prove the upper and lower bound for the quadratic form (18).

1. Upper bound. Consider the open δ-neighborhood U_{δ} ⊂ ℂ^{1} of the curve Γ. As it follows from the Cauchy integral formula, for some constant C_{1}(q, δ), the inequality

∥∂^{k}f∥_{L^{2}(μ̃)}^{2} ≤ C_{1}(q, δ) ∫_{U_{δ}}|f(z)|^{2}dm(z).

holds for all functions f ∈ F^{2}. Thus, we have the estimate

∥(2∂ − B\overline{z})^{q}f∥_{L^{2}(μ̃)}^{2} ≤ C_{2}(q, δ) ∫_{U_{δ}}|f(z)|^{2}dm(z).

Using (18), (19), we arrive at the estimate

T_{q}(μ) ≤ CT_{0}(χ_{U_{δ}} (x)dx),  (20)

where χ_{U_{δ}} is the characteristic function of the set U_{δ}. Now we can again apply the estimate of [3, Lemma 3.1] to the eigenvalues s_{1}(δ) ≥ s_{2}(δ) ≥ … of T_{0}(χ_{U_{δ}} (x)dx). This estimate together with (20) yields

lim_{n→∞}(n!s_{n})^{1/n} ≤ lim_{n→∞}(n!s_{n}(δ))^{1/n} ≤ \frac{B}{2} (Cap (U_{δ}))^{2}.

Finally, by the continuity of capacity, Cap (U_{δ}) → Cap (Γ) as δ → 0, and this proves the upper bound.

2. Lower bound. The lower bound for the spectrum of T_{q}(μ) requires a little more work. Let σ : [0, s] → ℂ be the parameterization of Γ by the arc length. Since f is analytic, we have

(∂f)(σ(t)) = ρ(t)\frac{d}{dt}f(σ(t))  (21)

with some smooth factor ρ(t), |ρ(t)| = 1. Next, due to the compactness of the embedding H^{1}(0, s) ⊂ L^{2}(0, s), for any β > 0 there exists a subspace of H^{1}(0, s) of a finite codimension such that for any u in this subspace,

∫_{0}^{s}|u(t)|^{2}dt ≤ β^{2} ∫_{0}^{s}|u′(t)|^{2}dt.  (22)

It follows from (21) and (22) that for any β > 0 there exists a subspace of F^{2} of a finite codimension such that for any f in this subspace

∫_{Γ}|f(z)|^{2}dμ̃(z) ≤ β^{2} ∫_{Γ}|∂f(z)|^{2}dμ̃(z).

Arguing by induction, we obtain that for any β > 0 there exists a subspace N = N(β, q) ⊂ F^{2} of finite codimension such that for all f ∈ N(β, q)

∫_{Γ}|∂^{k}f(z)|^{2}dμ̃(z) ≤ β^{2} ∫_{Γ}|∂^{q}f(z)|^{2}dμ̃(z), ∀k = 0, 1, …, q − 1.  (23)

Using (23) and choosing β sufficiently small, we can estimate the form (18) from below as follows:

∥(2∂ − B\overline{z})^{q}f∥_{L^{2}(μ̃)}^{2} ≥ (∥(2∂)^{q}f∥_{L^{2}(μ̃)} − ∑_{k=0}^{q−1} C_{q,k}∥∂^{k}f∥_{L^{2}(μ̃)})^{2}

≥ ∥(2∂)^{q}f∥_{L^{2}(μ̃)}^{2}(1 − ∑_{k=0}^{q−1} C_{q,k}β)^{2} = C_{1}∥∂^{q}f∥_{L^{2}(μ̃)}^{2} ≥ C_{2}∥f∥_{L^{2}(μ̃)}^{2}

for all f ∈ N(β, q). Using (18) and (19), we arrive at the lower bound

T_{q}(μ) ≥ CT_{0}(μ) + F

where F is a finite rank operator. For the eigenvalues of T_{0}(μ) the required lower estimates are already obtained by reference to [3, Lemma 3.1]; this completes the proof of the lower bound.

5 Proof of Proposition 3.1

We will prove separately upper and lower bounds.

5.1 Proof of the upper bound

1. Let U ⊂ ℝ^{2} be an open bounded set with a Lipschitz boundary, K ⊂ U, and let ω ∈ C_{0}^{∞}(ℝ^{2}) be such that ω|_{K} = 1 and ω|_{U^{c}} = 0. Denote ω̃ = 1 − ω. Note that for any ψ ∈ H, the function ω̃P_{q}ψ belongs both to Dom (X_{0}) and to the form domain of X(K^{c}). Thus, by Proposition 2.1(ii) (with A = X_{0} and B = X(K^{c})), we have W X_{0}ω̃P_{q}ψ = 0. Thus, we have

(W P_{q}ψ, P_{q}ψ) = \frac{1}{Λ_{q}^{2}} (W X_{0}P_{q}ψ, X_{0}P_{q}ψ)

= \frac{1}{Λ_{q}^{2}} (W X_{0}(ω + ω̃)P_{q}ψ, X_{0}(ω + ω̃)P_{q}ψ) = \frac{1}{Λ_{q}^{2}} (W X_{0}ωP_{q}ψ, X_{0}ωP_{q}ψ).

Since W = X_{0}^{−1} − R(K^{c}) ≤ X_{0}^{−1}, we have

(W X_{0}ωP_{q}ψ, X_{0}ωP_{q}ψ) ≤ (X_{0}^{−1}X_{0}ωP_{q}ψ, X_{0}ωP_{q}ψ) = ∥ωP_{q}ψ∥_{H_{A}^{1}}^{2}.

Using (15), we get

∥ωP_{q}ψ∥_{H_{A}^{1}}^{2} = ∥𝔔^{∗}ωP_{q}ψ∥^{2} − B∥ωP_{q}ψ∥^{2} ≤ ∥𝔔^{∗}ωP_{q}ψ∥^{2}

= ∥ω𝔔^{∗}P_{q}ψ − 2i(∂ω)P_{q}ψ∥^{2} ≤ 2∥𝔔^{∗}P_{q}ψ∥_{L^{2}(U)}^{2} + C_{1}∥P_{q}ψ∥_{L^{2}(U)}^{2}.

2. Due to the compactness of the embedding H_{0}^{1}(U) ⊂ L^{2}(U), for any β > 0 there exists a subspace of H_{0}^{1}(U) of a finite codimension such that for all elements u of this subspace,

∫_{U}|u(x)|^{2}dx ≤ β^{2} ∫_{U}|∇u(x)|^{2}dx = β^{2} ∫_{U}|2∂u(x)|^{2}dx.

Taking β sufficiently small, we obtain

∥𝔔^{∗}u∥_{L^{2}(U)} ≥ ∥2∂u∥_{L^{2}(U)} − \frac{B}{2} ∥\overline{z}u∥_{L^{2}(U)} ≥ ∥2∂u∥_{L^{2}(U)} − \frac{B}{2} sup_{U}|z|∥u∥_{L^{2}(U)}

≥ (1 − \frac{B}{2} β sup_{U}|z|)∥2∂u∥_{L^{2}(U)} ≥ \frac{1}{2}∥2∂u∥_{L^{2}(U)} ≥ \frac{1}{2β} ∥u∥_{L^{2}(U)}

for all u in our subspace. It follows that on a subspace of ψ ∈ L^{2}(ℝ^{2}) of a finite codimension,

(W P_{q}ψ, P_{q}ψ)_{L^{2}(ℝ^{2})} ≤ 2∥𝔔^{∗}P_{q}ψ∥_{L^{2}(U)}^{2} + 4β^{2}∥𝔔^{∗}P_{q}ψ∥_{L^{2}(U)}^{2} ≤ C∥P_{q+1}ψ∥_{L^{2}(U)}^{2};

the last inequality holds true by (16). Thus, we have

P_{q}W P_{q} ≤ C_{3}P_{q+1}χ_{U}P_{q+1} + F,

where χ_{U} is the characteristic function of U, and F is a finite rank operator.

3. From Proposition 4.1 we get

lim sup_{n→∞}(n!μ_{n})^{1/n} ≤ \frac{1}{2}B(Cap U)^{2}.

Since U can be chosen such that Cap U is arbitrarily close to Cap K, by the continuity property of capacity, we get the required upper bound.

5.2 Proof of the lower bound

1. Let γ, J, μ be as in section 4.1. Consider the quadratic form in L^{2}(ℝ^{2})

∥u∥_{H_{A}^{1}(ℝ^{2})}^{2} + ∫_{supp μ}|u(x)|^{2}dμ(x) = ∥X_{0}^{1/2}u∥_{L^{2}(ℝ^{2})}^{2} + ∥γJ^{∗}X_{0}^{1/2}u∥_{L^{2}(ℝ^{2})}^{2}

defined for u ∈ H_{A}^{1}(ℝ^{2}). This form is closed and positively defined on L^{2}(ℝ^{2}). Denote by X̃ the corresponding self-adjoint operator in L^{2}(ℝ^{2}). We have

X̃ = X_{0} + X_{0}^{1/2}Jγ^{∗}γJ^{∗}X_{0}^{1/2} = X_{0}^{1/2}(I + Jγ^{∗}γJ^{∗})X_{0}^{1/2}

and therefore

X_{0}^{−1} − X̃^{−1} = X_{0}^{−1/2}[Jγ^{∗}γJ^{∗}(I + Jγ^{∗}γJ^{∗})^{−1}]X_{0}^{−1/2}.

Since γ is compact by assumption, we have Jγ^{∗}γJ^{∗} ≤ I on a subspace of a finite codimension. Thus,

X_{0}^{−1} − X̃^{−1} ≥ \frac{1}{2}X_{0}^{−1/2}Jγ^{∗}γJ^{∗}X_{0}^{−1/2}

on a subspace of finite codimension, and so

P_{q}(X_{0}^{−1} − X̃^{−1})P_{q} ≥ \frac{1}{2Λ_{q}} (γJ^{∗}P_{q})^{∗}(γJ^{∗}P_{q}) + F  (24)

where F is a finite rank operator.

2. Now let K ⊂ ℝ^{2} be a compact with a non-empty interior. Let K_{1} ⊂ K be a set with a Lipschitz boundary. Let μ be the restriction of the Lebesgue measure on K_{1}. By Proposition 2.1(i), we have X_{0}^{−1} ≥ X̃^{−1} ≥ R(K^{c}). It follows that

P_{q}(X_{0}^{−1} − R(K^{c}))P_{q} ≥ P_{q}(X_{0}^{−1} − X̃^{−1})P_{q}.  (25)

From here, using (24) and Proposition 4.1(i), we get the required lower bound in the first part of Proposition 3.1. Finally, consider the case of K being a smooth curve. Let μ be the arc measure of the curve. Then, again by (24) and (25), and applying Proposition 4.1(ii), we get the second part of Proposition 3.1.

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Alexander Pushnitski Department of Mathematics King’s College London Strand London WC2R 2LS U.K. alexander.pushnitski@kcl.ac.uk Grigori Rozenblum Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Chalmers Tvärgatan, 3, S-412 96 Gothenburg, Sweden grigori@math.chalmers.se