Wegner estimate for Gaussian random magnetic fields

Naomasa Ueki

Abstract. – For the Schrödinger operator on L^{2}(ℝ^{2}) with the magnetic field which is a sample path of a stationary Gaussian random field, a Wegner type estimate applicable for the proof of the Anderson localization is proven by referring a recent method by Erdös and Hasler, and the theory of the Malliavin calculus.

1. Introduction

For any L ≥ 1 and ω in a probability space, we consider the self-adjoint operator

(1.1)  H_{L}^{ω} := ∑_{ι=1}^{2}(i∂_{ι} + A_{L,ι}^{ω}(x))^{2}

with the Dirichlet boundary condition on the open square Λ_{L} = (−L/2, L/2)^{2} with the side length L and 0 as its center, where i = \sqrt{−1} and A_{L}^{ω} is a C^{1}-map from Λ_{L} to ℝ^{2} satisfying ∇×A_{L}^{ω} := ∂_{1}A_{L,2}^{ω}−∂_{2}A_{L,1}^{ω} = B^{ω}. Its spectrum depends only on B^{ω} and is independent of the choice of the vector potential A_{L}^{ω}. This is the Schrödinger operator with the magnetic field B^{ω}.

As the magnetic field B^{ω}, we take a Gaussian random field on ℝ^{2}. We assume B^{ω}(x) is stationary with respect to the shift in the space variable x ∈ ℝ^{2}: the random fields B^{ω}(·) and B^{ω}(x + ·) have a same law. Moreover we assume its covariance function V (x − y) = Cov(B^{ω}(x), B^{ω}(y)) is

(1.2)  V (x) = ∫ σ̃(x − y)σ̃(y)dy,

where

(1.3)  σ̃(x) = P(Δ)σ(x),
(1.4)  σ(x) = (\overline{σ}^{2} − |x|^{2})_{+}^{ν},

a_{+} = max\{a, 0\} is the positive part, \overline{σ} ∈ (0, ∞), ν ∈ (3/2, ∞), Δ = ∂_{1}^{2} + ∂_{2}^{2} and P is a non-zero polynomial of the degree less than (ν − 3/2)/2. This special form of the covariance makes possible to apply the theory on the Bessel functions. The condition ν > 3/2 guarantees that the sample path of B^{ω} belongs to the local Sobolev space W_{loc}^{2,p}(ℝ^{2}) of the functions whose derivatives of order ≤ 2 are locally p-th power integrable for any p ∈ [1, ∞). Thus B^{ω}(x) is C^{1} in x by the Sobolev imbedding theorem (see e.g. [1] Theorem 4.12 Part I Case C).

In this paper, we prove the following:

Theorem 1. Under the above assumptions, there exist positive finite constants C_{0}, C_{1} and C_{2} such that

(1.5)  𝔼[Tr[χ_{[E−η,E+η]}(H_{L}^{ω})]] ≤ C_{0}R^{2}ηL^{C_{1}}

for any R ∈ [1, ∞), L ≥ \sqrt{R} ∨ C_{2} and E, η > 0 satisfying E + η ≤ R.

By this theorem and the Lifschitz behavior shown by Theorem 4.3 in [21], the multi scale analysis works well and we obtain the following (cf. [8, 20]):

Corollary . Let F^{ω}(x) be a stationary Gaussian random field with the covariance

Cov(F^{ω}(x), F^{ω}(x′)) = ∫ σ(x − y)σ(x′ − y)dy,

where σ is the function defined in (1.4) with ν > 7/2. Then the operator

H^{ω} := ∑_{ι=1}^{2}(i∂_{ι} + A_{ι}^{ω}(x))^{2}

with a C^{1} vector potential A^{ω} on ℝ^{2} such that ∇ × A^{ω} = −ΔF^{ω} exhibits the Anderson localization in the low energies as follows: there exists a positive finite constant ε_{0} such that [0, ε_{0}] is included in the pure point spectrum of H^{ω}, the corresponding eigenfunctions decay exponentially, and

𝔼[ sup_{t} ∥|x|^{p}e^{−itH^{ω}}1_{I}(H^{ω})1_{K} ∥_{L^{2}(ℝ^{2})→L^{2}(ℝ^{2})}] < ∞

for any p ∈ (0, ∞), I ⊂ [0, ε_{0}] and any compact set K in ℝ^{2}, where ∥ · ∥_{L^{2}(ℝ^{2})→L^{2}(ℝ^{2})} is the operator norm of bounded operators on L^{2}(ℝ^{2}).

In the original Wegner’s estimate [23] for the Anderson model, the motivation was to bound the density of states and the idea was to use the monotone dependence of the eigenvalues with respect to the random elements. To obtain the bound of the density of states, we need the linear dependence in the volume: C_{1} in (1.5) should be 2. On the other hand the above weak type estimates are known to be sufficient for the proof of the Anderson localization by the multi scale analysis initiated by Fröhlich and Spencer [7] and these estimates have been extended to many models [3, 17, 20]. However the main property of the model to prove the estimates has been the monotone dependence of the eigenvalues with respect to the random elements. The Schrödinger operator with random magnetic fields never have such a monotonicity. Until recent, the magnetic case had been treated mainly by Klopp’s method [9, 11, 13, 14, 22]. His method uses the homogeneity with respect to the random element of the eigenvalues of the corresponding Birman-Schwinger operator. However to obtain the homogeneity in the magnetic case, the corresponding random vector potential had been assumed to be small. Very recently, Erdös and Hasler [4, 5, 6] gave a new method to obtain the Wegner type estimate by posing conditions only on the magnetic fields. This is more preferable since the gauge invariance implies that the spectral structure of the magnetic Schrödinger operator depends only on the magnetic field. Their method use the non-degeneracy of the gradient of eigenvalues with respect to the random elements. To obtain the non-degeneracy they assume that the random magnetic field has fine alloy type structures and is dominated from above and below by positive finite constants. In this paper we extend their theory to the above simple Gaussian random fields. Now Gaussian random fields are not bounded and not positive. The unboundedness brings no serious problem since Gaussian random fields decay exponentially at infinity. Now the non-positivity brings the essential problem. Instead of the positivity, we show a non-degeneracy estimate in Section 4 below for our special Gaussian random fields. The key point of this estimate is the existence of a bound which is a quadratic forms of the white noise with infinite rank. Since the rank is infinity, we can show the probability of the decay of the random field is small enough. This is a same situation with that where the non-degeneracy of the Malliavin covariance is proven [12, 15, 16, 19]. Another key point for the Wegner estimate is the integration by parts on the probability space. This is also the key point of the Malliavin calculus. Then we use the same notation used in the Malliavin calculus: as in Nualart [16]

B^{ω}(x) = B + B̃^{ω}(x), B ∈ ℝ and B̃^{ω}(x) = ω(σ̃(x − ·)),

where ω is the isonormal Gaussian process (ω(h))_{h∈L^{2}(ℝ^{2})}: for any h ∈ L^{2}(ℝ^{2}), ω(h) is a Gaussian random variable such that

𝔼[ω(h)] = 0 and 𝔼[ω(h)ω(h′)] = (h, h′)_{L^{2}(ℝ^{2})}.

This ω is also called as the white noise and the notation

B̃^{ω}(x) = ∫_{ℝ^{2}} σ̃(x − y)ω(dy)

is also used (cf. [16] p.8). The σ-field of the probability space is given by that generated by {ω(h) : h ∈ L^{2}(ℝ^{2})} (cf. [16] p.5). Then the measurability of the operator H_{L}^{ω} is obtained as in [3] Chapter 5. To reduce the proof of the theorem to the estimates on the non-degeneracy, we apply also the theory on the Bessel functions.

In the following, we mainly consider the case that

(1.6)  σ̃ = σ

for simplicity. The extension to the case of (1.3) is explained in Remarks 2.1 and 4.1 below. All the estimates in this paper are given as systematic simpler estimates rather than as sharp estimates. As the vector potential, we take as

(1.7)  A_{L,1}^{ω}(x) = (∂_{2}F_{L}^{ω})(x) and A_{L,2}^{ω}(x) = −(∂_{1}F_{L}^{ω})(x)

on Λ_{L}, where

(1.8)  F_{L}^{ω}(x) = ∑_{n∈ℕ^{2}} \frac{Φ_{n,L}(x)}{E_{n,L}} ∫_{Λ_{L}} Φ_{n,L}(y)B^{ω}(y)dy,

and

E_{n,L} = (\frac{π|n|}{L})^{2} and Φ_{n,L}(x) = \frac{2}{L} ∏_{ι=1}^{2} sin (n_{ι}π(\frac{x_{ι}}{L} + \frac{1}{2}))

for n = (n_{1}, n_{2}) ∈ ℕ^{2}. {E_{n,L}, Φ_{n,L}}_{n∈ℕ^{2}} is the eigenvalues and a complete orthonormal system consisting of the eigenfunctions of the negative Dirichlet Laplacian −Δ_{Λ_{L}}^{D} (cf. [18], p-266).

The organization of this paper is as follows. In Section 2, we dominate the norm of the gradient of the eigenvalue from below in terms of the magnetic field. In Section 3, we modify the theory by Erdös and Hasler to prove the estimates of the current used in Section 2. In Section 4, we prove the necessary estimate on the non-degeneracy of the Gaussian random field. In Section 5, we modify the theory by Erdös and Hasler to prove Theorem 1.

2. A Lower bound of the norm of the gradient of the eigenvalue

Let λ_{ℓ}(H_{L}^{ω}) be the ℓ-th eigenvalue of the operator H_{L}^{ω}, which is a functional of the isonormal Gaussian process ω = (ω(·))_{·∈L^{2}(ℝ^{2})}. In Definition 2.6 in [19], the notion of the H-differentiability of a functional F(ω) at a sample path ω_{0} is defined as the existence of DF(ω_{0}) ∈ L^{2}(ℝ^{2}) such that

lim_{ε→0}{F((ω_{0}(·) + ε(Φ, ·))_{·∈L^{2}(ℝ^{2})}) − F(ω_{0})}/ε = (DF(ω_{0}), Φ)_{L^{2}(ℝ^{2})}

for any Φ ∈ L^{2}(ℝ^{2}). λ_{ℓ}(H_{L}^{ω}) is H-differentiable everywhere in this sense since ε ↦ λ_{ℓ}(H_{L}^{ω+εΦ}) is complex analytic for any Φ ∈ L^{2}(ℝ^{2}) by the regular perturbation theory (cf. [18]§XII.2).

In this section, we dominate the norm of the derivative restricted to a finite dimensional subspace from below in terms of the magnetic field. The object is (2.8) below. The derivative is represented as

(Dλ_{ℓ}(H_{L}^{ω}), Φ)_{L^{2}(ℝ^{2})} = ∫_{Λ_{L}} j^{ω}(x) · (DA_{L}^{ω}(x), Φ)_{L^{2}(ℝ^{2})}dx,

(DA_{L,1}^{ω}(x), Φ)_{L^{2}(ℝ^{2})} = ∑_{n∈ℕ^{2}} \frac{(∂_{2}Φ_{n,L})(x)}{E_{n,L}} ∫_{Λ_{L}} dyΦ_{n,L}(y) ∫_{ℝ^{2}} σ(y − z)Φ(z)dz

and

(DA_{L,2}^{ω}(x), Φ)_{L^{2}(ℝ^{2})} = − ∑_{n∈ℕ^{2}} \frac{∂_{1}Φ_{n,L}(x)}{E_{n,L}} ∫_{Λ_{L}} dyΦ_{n,L}(y) ∫_{ℝ^{2}} σ(y − z)Φ(z)dz,

where j^{ω}(x) = (j_{1}^{ω}(x), j_{2}^{ω}(x)) is the current of the eigenfunction ψ_{ℓ} of the eigenvalue λ_{ℓ}(H_{L}^{ω}) defined by

(2.1)  j_{ι}^{ω}(x) = 2 Re \overline{ψ_{ℓ}}(i∂_{ι} + A_{L,ι}^{ω}(x))ψ_{ℓ}.

As the direction Φ of the derivative, we take Φ̃_{ξ,L}(x) defined by 0 on Λ_{3L}^{c} and

\frac{2}{L} ∏_{ι=1}^{2} sin (ξ_{ι}π(\frac{x_{ι}}{L} + \frac{1}{2}))

on Λ_{3L} for ξ = (ξ_{1}, ξ_{2}) ∈ ℝ^{2}. Since supp σ(y − ·) ⊂ Λ_{3L} for any y ∈ Λ_{L} if L ≥ \overline{σ}, we have

∫_{Λ_{3L}} σ(y − z)Φ̃_{ξ,L}(z)dz = σ̂(\frac{ξ}{2L})Φ̃_{ξ,L}(y),

where

σ̂(ξ) = ∫_{ℝ^{2}} exp(−2πiξ · x)σ(x)dx

is the Fourier transform. In a special form of (1.4), the transform is written as

σ̂(ξ) = \frac{\overline{σ}^{ν+1}Γ(ν + 1)}{π^{ν}|ξ|^{ν+1}}J_{ν+1}(2π\overline{σ}|ξ|),

where

J_{ν+1}(t) = ∑_{m=0}^{∞} \frac{(−1)^{m}(t/2)^{2m+ν+1}}{m!Γ(m + ν + 2)}

is the Bessel function of the order ν+1 (cf. [2] 9.1.18 and 11.4.10). The function t^{−ν−1}J_{ν+1}(t) is even, and is known that the zero points {j_{ν+1,s}}_{s∈ℕ} on the interval (0, ∞) are simple and satisfy the asymptotics

(2.2)  j_{ν+1,s} = (s + \frac{ν}{2} + \frac{1}{4})π + O(\frac{1}{s})

(cf. [2] 9.5.12). We should take ξ so that 2π\overline{σ}|ξ|/(2L) is apart from the zero points \{j_{ν+1,s}\}_{s}. On the other hand, we should take the set of ξ so that this set includes sufficiently various elements to obtain a positive lower bound of the norm of the derivatives. One candidate is ℕ^{2} since {Φ_{n,L}}_{n∈ℕ^{2}} is complete in L^{2}(ℝ^{2}). Now we modify ℕ^{2} as {(n; ε, L)}_{n∈ℕ^{2}} ⊂ (0, ∞)^{2}, where

(2.3)  (n; ε, L) := \begin{cases}\{1 + \frac{ε\overline{j}L}{8π\overline{σ}|n|}\}n if |n| ∈ [L\frac{j_{ν+1,s}}{π\overline{σ}} , L\frac{j_{ν+1,s} + \overline{j}/8}{π\overline{σ}}) \\ for some s ∈ ℕ, \\ \{1 − \frac{ε\overline{j}L}{8π\overline{σ}|n|}\}n if |n| ∈ (L\frac{j_{ν+1,s} − \overline{j}/8}{π\overline{σ}}, L\frac{j_{ν+1,s}}{π\overline{σ}} ) \\ for some s ∈ ℕ, \\ n otherwise, \end{cases}

\overline{j} := inf_{s∈ℕ}(j_{ν+1,s+1} − j_{ν+1,s}) ∧ j_{ν+1,1}, and ε ∈ (0, 1) is specified later. Then we have

π\overline{σ}|(n; ε, L)|/L ∈ [0, ∞) \ ⋃_{s∈ℕ} (j_{ν+1,s} − \frac{ε\overline{j}}{8} , j_{ν+1,s} + \frac{ε\overline{j}}{8} ) =: G_{ε}.

The asymptotics of the Bessel function itself is known as

(2.4)  J_{ν+1}(t) = \sqrt{\frac{2}{πt}}\{ cos (t − \frac{2ν + 3}{4}π) + O(\frac{1}{t})\}

(cf. [2] 9.2.1). Thus we can show that

(2.5)  |\sqrt{t}J_{ν+1}(t)| ≥ c_{1}ε on G_{ε} ∩ [\overline{j}/2, ∞)

for some positive constant c_{1}. Indeed we can take T_{ε} ∈ (0, ∞) such that |j_{ν+1,s}−(s+ν/2+1/4)π| < ε\overline{j}/(16) and |\sqrt{πt/2}J_{ν+1}(t) − cos(t − (ν + 1)π/2 − π/4)| ≤ ε\overline{j}/(16π) if (s + ν/2 + 1/4)π, t ≥ T_{ε} by (2.2) and (2.4). Thus we have | cos(t−(ν +1)π/2−π/4)| ≥ ε\overline{j}/(8π) and |\sqrt{πt/2}J_{ν+1}(t)| ≥ ε\overline{j}/(16π) on G_{ε} ∩[T_{ε} +π/2, ∞). By the compactness of [\overline{j}/2, T_{ε} + π/2] and the simplicity of j_{ν+1,s}, we obtain (2.5) (cf. [2] 9.5). By noting also inf_{[0,\overline{j}/2]} J_{ν+1}(t)/t^{ν+1} > 0, we have

|σ̂(\frac{(n; ε, L)}{2L})| ≥ c_{2}((ε(\frac{L}{∥n∥})^{ν+1/2}) ∧ 1).

Thus, as the direction of the derivatives, we take {Φ̃_{(n;ε,L),L}(x)}_{n∈ℕ^{2}:|n|≤R}, where the restricting positive number R is for the estimates in Section 5 below. Then we have

\begin{matrix}|(Dλ_{ℓ}(H_{L}^{ω}), Φ̃_{(n;ε,L),L})_{L^{2}(ℝ^{2})}| ≥ & c_{2}((ε(\frac{L}{R})^{ν+1/2}) ∧ 1)| ∫_{Λ_{L}} dx(∇ × j^{ω})(x) \\ × ∑_{m∈ℕ^{2}} \frac{Φ_{m,L}(x)}{E_{m,L}} ∫_{Λ_{L}} dyΦ_{m,L}(y)Φ̃_{(n;ε,L),L}(y)|. \end{matrix}

By ∥Φ̃_{(n;ε,L),L} − Φ̃_{n,L}∥ ≤ ε\overline{j}L/(4\overline{σ}), E_{m,L} ≥ (π/L)^{2} and E_{n,L} ≤ (πR/L)^{2}, we have

∑_{n∈ℕ^{2}:|n|≤R}(Dλ_{ℓ}(H_{L}^{ω}), Φ̃_{(n;ε,L),L})_{L^{2}(ℝ^{2})}^{2}

≥c_{2}^{2}((ε^{2}(\frac{L}{R})^{2ν+1}) ∧ 1)\{\frac{1}{2π^{4}} (\frac{L}{R})^{4} ∑_{n∈ℕ^{2}:|n|≤R} ( ∫_{Λ_{L}} dx(∇ × j^{ω})(x)Φ_{n,L}(x))^{2}

− \frac{\overline{j}^{2}ε^{2}L^{6}R^{2}}{16π^{4}\overline{σ}^{2}} ∥∇ × j^{ω}∥_{L^{2}(Λ_{L})}^{2}\}.

By (3.2) below, we have c_{3}L^{22}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{6} ≥ ∥∇(∇ × j^{ω})∥_{L^{2}(Λ_{L})}^{2}

= ∑_{ι=1}^{2} ∑_{n∈ℕ^{2}}(∂_{ι}(∇ × j^{ω}), Φ_{n,L}^{(ι)})_{L^{2}(Λ_{L})}^{2} = ∑_{n∈ℕ^{2}} (\frac{π|n|}{L})^{2}(∇ × j^{ω}, Φ_{n,L})_{L^{2}(Λ_{L})}^{2}

≥ (\frac{πR}{L})^{2} ∑_{n∈ℕ^{2}:|n|>R}(∇ × j^{ω}, Φ_{n,L})_{L^{2}(Λ_{L})}^{2},

where, for each ι ∈ {1, 2},

\{Φ_{n,L}^{(ι)} = \frac{2}{L} cos (n_{ι}π(\frac{x_{ι}}{L} + \frac{1}{2})) sin (nι̂π(\frac{xι̂}{L} + \frac{1}{2}))\}_{n∈ℕ^{2}}

is a complete orthonormal system of the orthogonal complement of {ϕ ∈ L^{2}(Λ_{L}) : ϕ is independent of x_{ι}} in L^{2}(Λ_{L}), and ˆ1 = 2 and ˆ2 = 1. By using also (3.1) below, we have

∑_{n∈ℕ^{2}:|n|≤R}(Dλ_{ℓ}(H_{L}^{ω}), Φ̃_{(n;ε,L),L})_{L^{2}(ℝ^{2})}^{2}

≥c_{4}((ε^{2}(\frac{L}{R})^{2ν+1}) ∧ 1)\{(\frac{L}{R})^{4}(∥∇ × j^{ω}∥_{L^{2}(Λ_{L})}^{2} − c_{3} \frac{L^{24}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{6}}{π^{2}R^{2}} )

− c_{5}ε^{2}L^{24}R^{2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{4}}.

Moreover we use ∇ · j^{ω} = ∂_{1}j_{1}^{ω} + ∂_{2}j_{2}^{ω} = 0 and Lemma 3.2 below to change the bound so that its derivatives in ω have simpler representations:

∥∇ × j^{ω}∥_{L^{2}(Λ_{L})}^{2} = ∥∇j^{ω}∥_{L^{2}(Λ_{L})}^{2} ≥ (\frac{π}{L}∥j^{ω}∥_{L^{2}(Λ_{L})})^{2} ≥ \frac{c_{6}}{L^{6}} B(x_{∗}, ω),

where

\begin{aligned}B(x_{∗}, ω) := & ∫_{0}^{c_{7}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2}+R)^{−2}} \frac{dr}{2πr} \\ & × | ∫_{B(c_{7}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2}+R)^{−2}x_{∗},r)} B^{ω}(x)dx|^{2} \end{aligned} ; (2.6)

and x_{∗} ∈ ℤ^{2} such that c_{7}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−2}x_{∗} ∈ Λ_{L}. We now take R and ε as

R(x_{∗}, ω) = (\frac{2c_{3}}{π^{2}c_{6}} )^{1/2}L^{15}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{3}B(x_{∗}, ω)^{−1/2}

and

\begin{aligned}ε(x_{∗}, ω) & = (\frac{c_{6}}{4c_{5}} )^{1/2}B(x_{∗}, ω)^{1/2}L^{−13}R(x_{∗}, ω)^{−3}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−2} \\ & = (\frac{c_{6}}{4c_{5}} )^{1/2}(\frac{π^{2}c_{6}}{2c_{3}} )^{3/2}B(x_{∗}, ω)^{2}L^{−58}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−11}, \end{aligned}

respectively. This ε(x_{∗}, ω) is small enough for the definition (2.3) since we deduce

ε(x_{∗}, ω) ≤ c_{8}L^{−146}R^{−25}

from

(2.7)  B(x_{∗}, ω) ≤ ∫_{0}^{c_{7}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2}+R)^{−2}} \frac{πr^{3}}{2} dr∥B^{ω}∥_{L^{∞}(Λ_{L})}^{2} ≤ c_{9}L^{−44}R^{−7}.

Then we obtain

\begin{matrix}∑_{n∈ℕ^{2}:|n|≤R(x_{∗},ω)}(Dλ_{ℓ}(H_{L}^{ω}), Φ̃_{(n;ε(x_{∗},ω),L),L})_{L^{2}(ℝ^{2})}^{2} \\ ≥ & c_{10}B(x_{∗}, ω)^{ν+15/2}L^{−28ν−192}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−6ν−37}. \end{matrix} ; (2.8)

The integral B(x_{∗}, ω) of the magnetic field in the right hand side is dominated from below in Section 4 below.

Remark 2.1. To extend the results of this section to the case where σ is replaced by σ̃ defined in (1.3), we have only to avoid not only the set \{ξ ∈ ℝ^{2} : J_{ν+1}(π\overline{σ}|ξ|/L)\} but also the set \{ξ ∈ ℝ^{2} : P(−(π|ξ|/L)^{2})\} in the definition (2.3) of (n; ε, L).

3. The estimates of the current

In this section we modify the proof of Lemmas 6.1 and 6.2 in Erdös and Hasler [4] to prove the following:

Lemma 3.1. There exist finite positive constants c_{1} and c_{2} such that

(3.1)  ∥∇ × j^{ω}∥_{L^{2}(Λ_{L})} ≤ c_{1}L^{9}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{2}

and

(3.2)  ∥∇(∇ × j^{ω})∥_{L^{2}(Λ_{L})} ≤ c_{2}L^{11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{3}

for any L ≥ 2, R ≥ 1 and the current j^{ω} of the normalized eigenfunction of the operator H_{L}^{ω} with the eigenvalue less than R.

Lemma 3.2. There exist finite positive constants c_{1} and c_{2}, and x_{0} ∈ Λ_{L} such that

\begin{aligned}& ∫_{Λ_{L}} |j^{ω}(x)|^{2}dx ≥ ∫_{B(x_{00},c_{2}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2}+R)^{−2})} |j^{ω}(x)|^{2}dx \\ ≥ & \frac{c_{1}}{L^{4}} ∫_{0}^{c_{2}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2}+R)^{−2}} | ∫_{B(x_{00},r)} B^{ω}(x)dx|^{2} \frac{dr}{2πr} \end{aligned}

for any x_{00} ∈ B(x_{0}, c_{2}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−2}), L ≥ 2, R ≥ 1 and the current j^{ω} of the normalized eigenfunction of the operator H_{L}^{ω} with the eigenvalue less than R, where B(a, r) = {x ∈ ℝ^{2} : |x − a| ≤ r} for any a ∈ ℝ^{2} and r ≥ 0.

Before proving these, we first prepare the following: Lemma 3.3. There exist finite constants c_{1}, c_{2} and c_{3} such that

(3.3)  ∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})} ≤ c_{1}L∥B^{ω}∥_{W^{2,1}(Λ_{L})} ≤ c_{2}L^{2}∥B^{ω}∥_{W^{2,2}(Λ_{L})}

and

(3.4)  ∥∇A_{L}^{ω}∥_{L^{2}(Λ_{L})} ≤ c_{3}∥B^{ω}∥_{L^{2}(Λ_{L})}

for any L ≥ 2. Proof. By the integration by parts, we have

∫_{Λ_{L}} dyΦ_{n,L}(y)B^{ω}(y)

=\frac{2L}{n_{1}n_{2}π^{2}} \{ ∑_{τ_{1},τ_{2}∈\{1,−1\}}(−τ_{1})^{n_{1}}(−τ_{2})^{n_{2}}B^{ω}(τ_{1} \frac{L}{2}, τ_{2} \frac{L}{2})

+ ∫_{−L/2}^{L/2} dy_{1} ∑_{τ∈\{1,−1\}}(−τ)^{n_{2}}∂_{1}B^{ω}(y_{1}, τ \frac{L}{2}) cos n_{1}π(\frac{y_{1}}{L} + \frac{1}{2})

+ ∫_{−L/2}^{L/2} dy_{2} ∑_{τ∈\{1,−1\}}(−τ)^{n_{1}}∂_{2}B^{ω}(τ \frac{L}{2}, y_{2}) cos n_{2}π(\frac{y_{2}}{L} + \frac{1}{2})

+ ∫_{Λ_{L}} dy∂_{1}∂_{2}B^{ω}(y) ∏_{ι=1}^{2} cos n_{ι}π(\frac{y_{ι}}{L} + \frac{1}{2})\}.

(3.5)

Thus we have

| ∫_{Λ_{L}} dyΦ_{n,L}(y)B^{ω}(y)| ≤ \frac{c_{1}L}{n_{1}n_{2}} ∥B^{ω}∥_{W^{2,1}(Λ_{L})} ≤ \frac{c_{2}L^{2}}{n_{1}n_{2}} ∥B^{ω}∥_{W^{2,2}(Λ_{L})}.

Since

∥∂_{1}Φ_{n,L}∥_{L^{∞}(Λ_{L})} = 2n_{1}πL^{−2} and ∥∂_{2}Φ_{n,L}∥_{L^{∞}(Λ_{L})} = 2n_{2}πL^{−2},

we obtain (3.3). For (3.4), we use the property that {∂^{α}Φ_{n,L}}_{n∈ℕ^{2}} constitutes an orthogonal system in L^{2}(Λ_{L}) for any α ∈ ℤ_{+}^{2}:

∥∇A_{L}^{ω}∥_{L^{2}(Λ_{L})}^{2} = ∥∇^{⊗2}F_{L}^{ω}∥_{L^{2}(Λ_{L})}^{2}

= ∑_{n∈ℕ^{2}} \frac{∥∇^{⊗2}Φ_{n,L}∥_{L^{2}(Λ_{L})}^{2}}{E_{n,L}^{2}} | ∫_{Λ_{L}} Φ_{n,L}(y)B^{ω}(y)dy|^{2}

≤ c_{3} ∑_{n∈ℕ^{2}} | ∫_{Λ_{L}} Φ_{n,L}(y)B^{ω}(y)dy|^{2} = c_{3}∥B^{ω}∥_{L^{2}(Λ_{L})}^{2}.

□

Proof of Lemma 3.1. By (2.1), we have

|∇ × j^{ω}| ≤ 2|∇ × (\overline{ψ_{ℓ}}(i∇ + A_{L}^{ω})ψ_{ℓ}| ≤ 2|∇ψ_{ℓ}|^{2} + 4|A_{L}^{ω}||∇ψ_{ℓ}||ψ_{ℓ}| + 2|B^{ω}||ψ_{ℓ}|^{2}

and

\begin{matrix}∥∇ × j^{ω}∥_{L^{2}(Λ_{L})} ≤ & c_{1}(∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})}^{2} + ∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})}∥ψ_{ℓ}∥_{L^{4}(Λ_{L})} \\ + ∥B^{ω}∥_{L^{∞}(Λ_{L})}∥ψ_{ℓ}∥_{L^{4}(Λ_{L})}^{2}). \end{matrix} ; (3.6)

For the derivative, we have

\begin{matrix}|∇(∇ × j^{ω})| ≤ & 4|∇^{⊗2}ψ_{ℓ}||ψ_{ℓ}| + 4|∇A_{L}^{ω}||∇ψ_{ℓ}||ψ_{ℓ}| + 4|A_{L}^{ω}||∇^{⊗2}ψ_{ℓ}||ψ_{ℓ}| \\ + 4|A_{L}^{ω}||∇ψ_{ℓ}|^{2} + 2|∇B^{ω}||ψ_{ℓ}|^{2} + 4|B^{ω}||∇ψ_{ℓ}||ψ_{ℓ}| \end{matrix}

and

∥∇(∇ × j^{ω})∥_{L^{2}(Λ_{L})}
≤c_{2}(∥∇^{⊗2}ψ_{ℓ}∥_{L^{4}(Λ_{L})}∥ψ_{ℓ}∥_{L^{4}(Λ_{L})} + ∥∇A_{L}^{ω}∥_{L^{2}(Λ_{L})}∥∇ψ_{ℓ}∥_{L^{∞}(Λ_{L})}∥ψ_{ℓ}∥_{L^{∞}(Λ_{L})}
+ ∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}∥∇^{⊗2}ψ_{ℓ}∥_{L^{4}(Λ_{L})}∥ψ_{ℓ}∥_{L^{4}(Λ_{L})} + ∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})}^{2}
+ ∥∇B^{ω}∥_{L^{2}(Λ_{L})}∥ψ_{ℓ}∥_{L^{∞}(Λ_{L})}^{2} + ∥B^{ω}∥_{L^{∞}(Λ_{L})}∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})}∥ψ_{ℓ}∥_{L^{4}(Λ_{L})}).

(3.7)

By the Sobolev inequality, we have ∥ψ_{ℓ}∥_{L^{∞}(Λ_{L})} ≤ c_{3}(∥ψ_{ℓ}∥_{L^{4}(Λ_{L})} + ∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})}), (3.8) ∥∇ψ_{ℓ}∥_{L^{∞}(Λ_{L})} ≤ c_{3}(∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})} + ∥∇^{⊗2}ψ_{ℓ}∥_{L^{4}(Λ_{L})}) (3.9) and

(3.10)  ∥B^{ω}∥_{L^{∞}(Λ_{L})} ≤ c_{4}∥B^{ω}∥_{W^{2,2}(Λ_{L})}

(cf.[1] Theorem 4.12 Part I Case C). In (3.8) and (3.9), we may choose arbitrary L^{p} norms with p > 2. However p = 4 is enough for the present purpose. For any p ∈ [1, ∞), we have

(3.11)  ∥∇^{⊗2}ψ_{ℓ}∥_{L^{p}(Λ_{L})} ≤ c_{p}∥Δψ_{ℓ}∥_{L^{p}(Λ_{L})}

by the Calderon-Zygmund inequality in the form of Corollary 9.10 in [10]. Since we can derive

(3.12)  Δψ_{ℓ} = 2iA_{L}^{ω} · ∇ψ_{ℓ} + (|A_{L}^{ω}|^{2} − λ_{ℓ}(H_{L}^{ω}))ψ_{ℓ}

from the eigenequation by ∇ · A_{L}^{ω} = 0, we have

∥Δψ_{ℓ}∥_{L^{4}(Λ_{L})} ≤ 2∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})} + (∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}^{2} + R)∥ψ_{ℓ}∥_{L^{4}(Λ_{L})}.

By the Gagliardo-Nirenberg inequality (cf.[1] Theorem 4.31 with m = 1) and the Calderon-Zygmund inequality, we have

∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})} ≤ c_{5}∥Δψ_{ℓ}∥_{L^{4/3}(Λ_{L})}.

Since (3.12) is rewritten as

(3.13)  Δψ_{ℓ} = 2A_{L}^{ω} · (i∇ + A_{L}^{ω})ψ_{ℓ} − (|A_{L}^{ω}|^{2} + λ_{ℓ}(H_{L}^{ω}))ψ_{ℓ},

we have

\begin{matrix}∥Δψ_{ℓ}∥_{L^{4/3}(Λ_{L})} \\ ≤ 2∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}L^{1/2}∥(i∇ + A_{L}^{ω})ψ_{ℓ}∥_{L^{2}(Λ_{L})} + (∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}^{2} + R)L^{1/2} \\ ≤ c_{6}L^{4+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R) \end{matrix} ; (3.14)

by using also Lemma 3.3. Thus we have

(3.15)  ∥∇ψ_{ℓ}∥_{L^{4}(Λ_{L})} ≤ c_{7}L^{4+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R).

Similarly we have

\begin{matrix}∥∇ψ_{ℓ}∥_{L^{4/3}Λ_{L})} ≤ ∥(i∇ + A_{L}^{ω})ψ_{ℓ}∥_{L^{4/3}(Λ_{L})} + ∥A_{L}^{ω}ψ_{ℓ}∥_{L^{4/3}(Λ_{L})} \\ ≤ & L^{1/2}(∥(i∇ + A_{L}^{ω})ψ_{ℓ}∥_{L^{2}(Λ_{L})} + ∥A_{L}^{ω}∥_{L^{∞}(Λ_{L})}) \\ ≤ & c_{8}L^{2+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{1/2} \end{matrix} ; (3.16)

and

(3.17)  ∥ψ_{ℓ}∥_{L^{4}(Λ_{L})} ≤ c_{9}∥∇ψ_{ℓ}∥_{L^{4/3}(Λ_{L})} ≤ c_{10}L^{2+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{1/2}.

Thus we have

(3.18)  ∥Δψ_{ℓ}∥_{L^{4}(Λ_{L})} ≤ c_{11}L^{6+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{3/2}.

By applying (3.8)–(3.11), (3.15), (3.17) and (3.18) to each factor in the right hand side of (3.6) and (3.7), we can complete the proof. □ Proof of Lemma 3.2. As in [4], we take x_{0} ∈ Λ_{L} so that |ψ(x_{0})| = max_{Λ_{L}} |ψ_{ℓ}|, and we set

⟨ψ_{ℓ}⟩ := ∫_{B(x_{0},l)} ψ_{ℓ}(x)\frac{dx}{πl^{2}} , ⟨∇ψ_{ℓ}⟩ := ∫_{B(x_{0},l)} ∇ψ_{ℓ}(x)\frac{dx}{πl^{2}}

and

f(x) := ψ_{ℓ}(x) − ⟨ψ_{ℓ}⟩ − ⟨∇ψ_{ℓ}⟩ · (x − x_{0})

for x ∈ B(x_{0}, l). Then we have

sup_{B(x_{0},l)} |f(x)| ≤ c_{1}l∥Δψ_{ℓ}∥_{L^{2}(B(x_{0},l))}

as in [4]. By (3.18) and (3.15), we have

\begin{matrix}∥Δψ_{ℓ}∥_{L^{2}(B(x_{0},l))} ≤ c_{2}\sqrt{l}∥Δψ_{ℓ}∥_{L^{4}((B(x_{0},l))} ≤ c_{2}\sqrt{l}∥Δψ_{ℓ}∥_{L^{4}(Λ_{L})} \\ ≤ & c_{3}\sqrt{l}L^{6+1/2}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{3/2} \end{matrix}

and

|⟨∇ψ_{ℓ}⟩| ≤ c_{4}∥∇ψ_{ℓ}∥_{L^{2}(Λ_{L})}/l ≤ c_{5}L^{5}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)/l.

Thus by putting l = L^{−3/5}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−1/5}|x − x_{0}|^{2/5}, we have

|ψ_{ℓ}(x) − ⟨ψ_{ℓ}⟩| ≤ c_{6}L^{5+3/5}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{6/5}|x − x_{0}|^{3/5}

on B(x_{0}, L^{−1}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−1/3}). Then we see that

|ψ_{ℓ}(x)| ≥ \frac{1}{2L} on B(x_{0}, R_{∗}),

where R_{∗} := c_{7}L^{−11}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)^{−2}. As in (7.18) in [4], we have

∫_{B(x_{00},R_{∗}/2)} |j^{ω}(x)|^{2}dx ≥ \frac{1}{4L^{4}} ∫_{B(x_{00},R_{∗}/2)} |A_{L}^{ω}(x) − ∇θ_{ℓ}(x)|^{2}dx

for any x_{00} ∈ B(x_{0}, R_{∗}/2). By the same proof of Lemma 7.2 in [4], we obtain

∫_{B(x_{00},R_{∗}/2)} |A_{L}^{ω}(x) − ∇θ_{ℓ}(x)|^{2}dx ≥ ∫_{0}^{R_{∗}/2} \frac{dr}{2πr} | ∫_{B(x_{00},r)} B^{ω}(x)dx|^{2}.

□

4. Non-degeneracy of the Gaussian random field

Let

X(R) = ∫_{0}^{R} | ∫_{B(r)} B^{ω}(x)dx|^{2} \frac{dr}{2πr} ,

where B(r) = B(0, r) = {x ∈ ℝ^{2} : |x| ≤ r} for any r ≥ 0.

In this section, we prove the following: Lemma 4.1. For any \overline{R} ∈ (0, ∞), there exist c, c′ ∈ (0, ∞) such that

ℙ(X(R) ≤ t) ≤ exp(−cR^{(2ν+5)/(2ν+4)}/t^{1/(2ν+4)})

for any R ∈ (0, \overline{R}] and t ∈ (0, c′R^{2ν+5}].

As its corollary we have the following: Corollary . For any p, \overline{R} ∈ (0, ∞), there exists c ∈ (0, ∞) such that

𝔼(X(R)^{−p}) ≤ cR^{−p(2ν+5)}

for any R ∈ (0, \overline{R}].

To obtain Lemma 4.1, it is enough to show the following: Lemma 4.2. For any \overline{R} ∈ (0, ∞), there exists c ∈ (0, ∞) such that

𝔼(exp(−sX(R))) ≤ exp(−cRs^{1/(2ν+5)})

for any s ∈ [1, ∞) and R ∈ (0, \overline{R}] satisfying Rs^{1/(2ν+5)} ≥ 1.

In the rest of this section, we prove this lemma. The condition ν > 3/2 can be extended to ν > 1 in the following proof. For any 0 < R_{1} < R, we have

X(R) − X(R_{1})

= ∫_{R_{1}}^{R} \frac{dr}{2πr} | ∫_{B(R_{1})} dxB^{ω}(x) + ∫_{B(r)\B(R_{1})} dx(B

+ ∫_{B(R_{1}+\overline{σ})} σ(x − y)ω(dy) + ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy))|^{2}.

The key point is that ω on B(R_{1} + \overline{σ})^{c} is independent of X(R_{1}), B^{ω}(x) on B(R_{1}) and ω on B(R_{1} + \overline{σ}). To use this property, we proceed as follows:

𝔼(exp(−sX(R)))

=𝔼[ exp \{ − sX(R_{1}) + i ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}} ( ∫_{B(R_{1})} dxB^{ω}(x)

+ ∫_{B(r)\B(R_{1})} dx(B + ∫_{B(R_{1}+\overline{σ})} σ(x − y)ω(dy)))

+ i ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}} ∫_{B(r)\B(R_{1})} dx ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy)\}],

where w(·) is a 1-dimensional Wiener process independent of ω. By taking the expectation with respect to ω on B(R_{1} + \overline{σ})^{c}, we have

𝔼(exp(−sX(R)))

=𝔼[ exp \{ − sX(R_{1}) + i ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}} ( ∫_{B(R_{1})} dxB^{ω}(x)

+ ∫_{B(r)\B(R_{1})} dx(B + ∫_{B(R_{1}+\overline{σ})} σ(x − y)ω(dy)))

− \frac{1}{2}𝔼^{ω}[( ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}} ∫_{B(r)\B(R_{1})} dx ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy))^{2}]\}],

where 𝔼^{ω} is the expectation with respect to ω. By taking the absolute value, we have

𝔼(exp(−sX(R))) ≤ 𝔼(exp(−sX(R_{1})))F (R, R_{1}; s),

where

F (R, R_{1}; s) := 𝔼[ exp \{ − \frac{1}{2}𝔼^{ω}[( ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}}

× ∫_{B(r)\B(R_{1})} dx ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy))^{2}]\}]
=𝔼[ exp (i ∫_{R_{1}}^{R} dw(r)\sqrt{\frac{s}{πr}} ∫_{B(r)\B(R_{1})} dx ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy))]
=𝔼[ exp \{ − s ∫_{R_{1}}^{R} \frac{dr}{2πr} ( ∫_{B(r)\B(R_{1})} dx ∫_{B(R_{1}+\overline{σ})^{c}} σ(x − y)ω(dy))^{2}\}].

For any sequence R = R_{0} > R_{1} > R_{2} > ⋯ > R_{n} ↓ 0, we have

(4.1)  𝔼(exp(−sX(R))) ≤ ∏_{j=1}^{∞} F (R_{j−1}, R_{j}; s).

We next estimate each F (R_{j−1}, R_{j}; s): if we set

X(R_{j−1}, R_{j}) := ∫_{R_{j}}^{R_{j−1}} \frac{dr}{2πr} ( ∫_{B(r)\B(R_{j})} dx ∫_{B(R_{j}+\overline{σ})^{c}} σ(x − y)ω(dy))^{2},

then

1 − F (R_{j−1}, R_{j}; s)
=𝔼[sX(R_{j−1}, R_{j})]
− 𝔼[s^{2}X(R_{j−1}, R_{j})^{2} ∫_{0}^{1} dt_{1} ∫_{0}^{t_{1}} dt_{2} exp(−t_{2}sX(R_{j−1}, R_{j}))]
≥s𝔼[X(R_{j−1}, R_{j})] − \frac{s^{2}}{2} 𝔼[X(R_{j−1}, R_{j})^{2}].

Since the 4-th moment of a centered Gaussian random variable is proportional to the square of the variance of the Gaussian variable, we have

𝔼[X(R_{j−1}, R_{j})^{2}]

≤( ∫_{R_{j}}^{R_{j−1}} \frac{dr}{2πr} 𝔼[( ∫_{B(r)\B(R_{j})} dx ∫_{B(R_{j}+\overline{σ})^{c}} σ(x − y)ω(dy))^{4}]^{1/2})^{2}
=3𝔼[X(R_{j−1}, R_{j})]^{2}.

By the polar coordinate, we have

𝔼[X(R_{j−1}, R_{j})] = ∫_{R_{j}}^{R_{j−1}} \frac{dr}{2πr} ∫_{B(R_{j}+\overline{σ})^{c}} dy( ∫_{B(r)\B(R_{j})} dxσ(x − y))^{2}

= ∫_{R_{j}}^{R_{j−1}} \frac{dr}{r} ∫_{R_{j}+\overline{σ}}^{∞} dr_{0}r_{0}( ∫_{R_{j}}^{r} dr_{1}r_{1} ∫_{0}^{2π} dθ_{1}((\overline{σ} − r_{1} + r_{0})(\overline{σ} + r_{1} − r_{0})

− r_{1}r_{0}(2 sin \frac{θ_{1}}{2} )^{2})_{+}^{ν})^{2}

= ∫_{R_{j}}^{R_{j−1}} \frac{dr}{r} ( ∏_{ι=1}^{2} ∫_{R_{j}}^{r} dr_{ι}r_{ι}) ∫_{\overline{σ}+R_{j}}^{\overline{σ}+(r_{1}∧r_{2})} dr_{0}r_{0}

× ( ∏_{ι=1}^{2} ∫_{0}^{2π} dθ_{ι}((\overline{σ} − r_{ι} + r_{0})(\overline{σ} + r_{ι} − r_{0}) − r_{ι}r_{0}(2 sin \frac{θ_{ι}}{2} )^{2})_{+}^{ν}).

By changing the variables, we have

\begin{matrix}𝔼[X(R_{j−1}, R_{j})] \\ = & ∫_{0}^{1} \frac{dr(R_{j−1} − R_{j})}{R_{j} + (R_{j−1} − R_{j})r} ( ∏_{ι=1}^{2} ∫_{0}^{r} dr_{ι}(R_{j−1} − R_{j})(R_{j} + (R_{j−1} − R_{j})r_{ι})) \\ × ∫_{0}^{r_{1}∧r_{2}} dr_{0}(R_{j−1} − R_{j})(\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0}) \\ × ( ∏_{ι=1}^{2} \frac{\{(2\overline{σ} − (R_{j−1} − R_{j})(r_{ι} − r_{0}))(R_{j−1} − R_{j})(r_{ι} − r_{0})\}^{ν+1/2}}{\{(R_{j} + (R_{j−1} − R_{j})r_{ι})(\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0})\}^{1/2}} \\ × 2 ∫_{0}^{πR_{ι,j}} dθ_{ι}(1 − (2R_{ι,j} sin \frac{θ_{ι}}{2R_{ι,j}} )^{2})_{+}^{ν}), \end{matrix} ; (4.2)

where

R_{ι,j} := \{\frac{(R_{j} + (R_{j−1} − R_{j})r_{ι})(\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0})}{(2\overline{σ} − (R_{j−1} − R_{j})(r_{ι} − r_{0}))(R_{j−1} − R_{j})(r_{ι} − r_{0})}\}^{1/2}.

Since 2θ/π ≤ sin θ ≤ θ for 0 ≤ θ ≤ π/2, we have

(\frac{3}{4})^{ν}π(\frac{1}{2π} ∧ R_{ι,j}) ≤ ∫_{0}^{πR_{ι,j}} dθ_{ι}(1 − (2R_{ι,j} sin \frac{θ_{ι}}{2R_{ι,j}} )^{2})_{+}^{ν} ≤ \frac{π}{2} ∫_{0}^{1} dθ(1 − θ^{2})^{ν}.

Moreover by R_{j} ≤ R_{j} + (R_{j−1} − R_{j})r_{ι} ≤ R_{j−1} for any ι ∈ {0, 1, 2}, we obtain

c_{1}(R_{j−1} − R_{j})^{2ν+5}(\frac{R_{j}}{R_{j−1}} )^{3} ≤ 𝔼[X(R_{j−1}, R_{j})] ≤ c_{2}(R_{j−1} − R_{j})^{2ν+5}(\frac{R_{j−1}}{R_{j}} )^{2}

and

\begin{matrix}F (R_{j−1}, R_{j}; s) \\ ≤ & 1 − c_{1}s(R_{j−1} − R_{j})^{2ν+5}(\frac{R_{j}}{R_{j−1}} )^{3} + \frac{3c_{2}^{2}s^{2}}{2} (R_{j−1} − R_{j})^{2(2ν+5)}(\frac{R_{j−1}}{R_{j}} )^{4} \end{matrix}

if R_{j−1} − R_{j} ≤ \overline{σ}.

We now take \{R_{j}\}_{j} as follows: taking ε ∈ (0, (\overline{σ} ∧ 1)/2) and preparing the sequence

b_{k} := \begin{cases}\frac{1 − 2ε}{(1 − ε)⌈1/ε⌉} for k ∈ [0, ⌈1/ε⌉] ∩ ℕ, \\ ε^{k−⌈1/ε⌉} for k ∈ (⌈1/ε⌉, ∞) ∩ ℕ, \end{cases}

whose elements are in (0, ε) and whose sum is 1, we set R_{0} = R and

R_{j−1} − R_{j} = \frac{b_{k}R}{⌊Rs^{1/(2ν+5)}⌋} for j ∈ ((k − 1)⌊Rs^{1/(2ν+5)}⌋, k⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ,

where ⌊a⌋ = max\{(−∞, a] ∩ ℤ\} and ⌈a⌉ = min\{[a, ∞) ∩ ℤ\} for any a ∈ ℝ. Then we have R_{j−1} − R_{j} ≤ \overline{σ} and

(4.3)  R_{j−1}/R_{j} ≤ 1/ε.

Indeed, we have

R_{j} = \begin{cases}R(1 − \frac{j}{⌊Rs^{1/(2ν+5)}⌋} \frac{1 − 2ε}{(1 − ε)⌈1/ε⌉}) for j ∈ [0, ⌈1/ε⌉⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ, \\ Rε^{k−⌈1/ε⌉}(\frac{ε}{1 − ε} + k − \frac{j}{⌊Rs^{1/(2ν+5)}⌋}) \\ for j ∈ ((k − 1)⌊Rs^{1/(2ν+5)}⌋, k⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ \\ with k ∈ (⌈1/ε⌉, ∞) ∩ ℕ. \end{cases}

If j ∈ [0, ⌈1/ε⌉⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ, then

\frac{R_{j−1}}{R_{j}} = 1 + \frac{1 − 2ε}{⌊Rs^{1/(2ν+5)}⌋(1 − ε)⌈1/ε⌉}/(1 − \frac{j}{⌊Rs^{1/(2ν+5)}⌋} \frac{1 − 2ε}{(1 − ε)⌈1/ε⌉})

takes its maximum at j = ⌈1/ε⌉⌊Rs^{1/(2ν+5)}⌋ and the maximum is

1 + \frac{1 − 2ε}{⌊Rs^{1/(2ν+5)}⌋ε⌈1/ε⌉} ≤ 2 ≤ \frac{1}{ε}.

If j ∈ [2 + (k − 1)⌊Rs^{1/(2ν+5)}⌋, k⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ with k ∈ (⌈1/ε⌉, ∞) ∩ ℕ, then

\frac{R_{j−1}}{R_{j}} = 1 + \frac{1}{⌊Rs^{1/(2ν+5)}⌋}/(\frac{ε}{1 − ε} + k − \frac{j}{⌊Rs^{1/(2ν+5)}⌋})

takes its maximum at j = k⌊Rs^{1/(2ν+5)}⌋ and the maximum is

1 + \frac{1 − ε}{ε⌊Rs^{1/(2ν+5)}⌋} ≤ \frac{1}{ε}.

Finally if j = 1 + (k − 1)⌊Rs^{1/(2ν+5)}⌋ with k ∈ (⌈1/ε⌉, ∞) ∩ ℕ, then

\begin{matrix}R_{j−1} = \frac{Rε^{k−⌈1/ε⌉}}{1 − ε} , \\ R_{j} = R_{j−1} − (R_{j−1} − R_{j}) = Rε^{k−⌈1/ε⌉}(\frac{1}{1 − ε} − \frac{1}{⌊Rs^{1/(2ν+5)}⌋}) \end{matrix}

and

\frac{R_{j−1}}{R_{j}} = \frac{1}{1 − ε}/(\frac{1}{1 − ε} − \frac{1}{⌊Rs^{1/(2ν+5)}⌋}) ≤ \frac{1}{ε}.

Therefore we obtain (4.3).

Since ν > 1, by taking ε sufficiently small, we have

log F (R_{j−1}, R_{j}; s) ≤ −c_{3}b_{k}^{2ν+5}

for j ∈ ((k − 1)⌊Rs^{1/(2ν+5)}⌋, k⌊Rs^{1/(2ν+5)}⌋] ∩ ℕ. By applying this to the right hand side of (4.1), we obtain

log 𝔼(exp(−sX(R))) ≤ −c_{3} ∑_{k=1}^{∞} b_{k}^{2ν+5}⌊Rs^{1/(2ν+5)}⌋ ≤ −c_{4}Rs^{1/(2ν+5)}.

Remark 4.1. The results of this section is extended to the case where σ is replaced by

σ̃(x) = P̃(x)σ(x)

and P̃ is a bounded function on ℝ^{2} such that P̃(x_{0}) ≠ 0 for some x_{0} ∈ ℝ^{2} satisfying |x_{0}| = \overline{σ} and that P̃ is continuous at any points of a neighborhood of x_{0}. In this case, the equation (4.2) is changed to

𝔼[X(R_{j−1}, R_{j})]

= ∫_{0}^{1} \frac{dr(R_{j−1} − R_{j})}{R_{j} + (R_{j−1} − R_{j})r} ( ∏_{ι=1}^{2} ∫_{0}^{r} dr_{ι}(R_{j−1} − R_{j})(R_{j} + (R_{j−1} − R_{j})r_{ι}))

× ∫_{0}^{r_{1}∧r_{2}} dr_{0}(R_{j−1} − R_{j})(\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0}) ∫_{0}^{2π} \frac{dθ_{0}}{2π}

× ( ∏_{ι=1}^{2} \frac{\{(2\overline{σ} − (R_{j−1} − R_{j})(r_{ι} − r_{0}))(R_{j−1} − R_{j})(r_{ι} − r_{0})\}^{ν+1/2}}{\{(R_{j} + (R_{j−1} − R_{j})r_{ι})(\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0})\}^{1/2}}

× ∫_{0}^{πR_{ι,j}} dθ_{ι}(1 − (2R_{ι,j} sin \frac{θ_{ι}}{2R_{ι,j}} )^{2})_{+}^{ν}

× ∑_{τ∈\{+,−\}} P̃((R_{j} + (R_{j−1} − R_{j})r_{ι})(cos (θ_{0} + \frac{τθ_{ι}}{R_{ι,j}} ), sin (θ_{0} + \frac{τθ_{ι}}{R_{ι,j}} ))

− (\overline{σ} + R_{j} + (R_{j−1} − R_{j})r_{0})(cos θ_{0}, sin θ_{0}))).

The upper estimate of this is obtained by the same method. To obtain the lower estimate, we restrict the integral with respect to θ_{0} to an interval I such that inf\{τP(x) : |x + \overline{σ}(cos θ, sin θ)| ≤ δ for some θ ∈ I} > 0 for some δ > 0 and τ ∈ {+, −}.

This extension is applicable to the case of (1.3). Indeed if the degree of P is m, then ν > 2m + 3/2 and σ̃(x) = Q(|x|)(\overline{σ}^{2} − |x|^{2})_{+}^{ν−2m}, where Q is a polynomial of the degree 2m. By the factor theorem, we can write Q(r) = (\overline{σ} − r)^{h}Q̃(r), where h ∈ ℤ ∩ [0, 2m] and Q̃ is a polynomial of the degree 2m − h such that Q̃(\overline{σ}) ≠ 0. Then, since σ̃(x) = (\overline{σ}^{2} − |x|^{2})_{+}^{ν−2m+h}Q̃(|x|)(\overline{σ} + |x|)^{−h}, Q̃(|x|)(\overline{σ} + |x|)^{−h} is bounded on B(\overline{σ}) and Q̃(\overline{σ})(2\overline{σ})^{−h} ≠ 0, the results in this section hold for this case if ν is replaced by ν − 2m + h.

5. Proof of Theorem 1

In this section we modify Erdös and Hasler [4] to prove Theorem 1 by applying the results proved in the preceding sections.

We first cut off high energies:

(5.1)  Tr[χ_{[E−η,E+η]}(H_{L}^{ω})] ≤ Tr[χ_{[(t(E)−η)∨t(0),t(E)+η]}(t(H_{L}^{ω}))]

for any E, η > 0 such that E + η ≤ R, where t(u) := (u + 1)(5R)^{3}/(5R + u + 1)^{3}. By (2.8) and inf_{u∈(0,R)} t′(u) > 0, the right hand side of (5.1) is less than or equal to

c_{1} ∑_{ℓ} χ_{[(t(E)−η)∨t(0),t(E)+η]}(t(λ_{ℓ}(H_{L}^{ω})))L^{28ν+192}

× ∑_{m∈ℕ} χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)m^{6ν+37}

× ∑_{x_{∗}∈(c_{2}L^{−11}m^{−2}ℤ^{2})∩Λ_{L}} B(m, x_{∗}, ω)^{−ν−15/2}

× ∑_{n∈ℕ^{2}} χ̃_{[0,∞)}(R(m, x_{∗}, ω) − |n|)(Dt(λ_{ℓ}(H_{L}^{ω})), Φ̃_{(n;ε(m,x_{∗},ω),L),L})^{2},

where

\begin{matrix}B(m, x_{∗}, ω) = ∫_{0}^{c_{2}L^{−11}m^{−2}} | ∫_{B(x_{∗},r)} B^{ω}(x)dx|^{2} \frac{dr}{2πr} , \\ R(m, x_{∗}, ω) = c_{3}L^{15}m^{3}B(m, x_{∗}, ω)^{−1/2}, \\ ε(m, x_{∗}, ω) = c_{4}B(m, x_{∗}, ω)^{1/2}L^{−13}m^{−2}R(m, x_{∗}, ω)^{−3} \\ = c_{4}c_{3}^{−3}B(m, x_{∗}, ω)^{2}L^{−58}m^{−11}, \end{matrix} ; (5.2)

and, for each interval I, χ̃_{I} is a [0, 1]-valued smooth function on ℝ such that χ̃_{I} = 1 on I and χ̃_{I}(x) = 0 if dist(x, I) ≥ 1. Let F and G be functions on ℝ such that F′ = χ_{[(t(E)−η)∨t(0),t(E)+η]}, G′ = F, and F = G = 0 on (−∞, (t(E) − η) ∨ t(0)]. Then we have

\begin{matrix}χ_{[(t(E)−η)∨t(0),t(E)+η]}(t(λ_{ℓ}(H_{L}^{ω})))(D̃_{n}t(λ_{ℓ}(H_{L}^{ω})))^{2} \\ = & D̃_{n}^{2}G(t(λ_{ℓ}(H_{L}^{ω}))) − F (t(λ_{ℓ}(H_{L}^{ω})))D̃_{n}^{2}t(λ_{ℓ}(H_{L}^{ω})), \end{matrix}

where D̃_{n}(·) := (D(·), Φ̃_{(n;ε(m,x_{∗},ω),L),L})_{L^{2}(ℝ^{2})}. As shown in Lemma 5.2 of [4], we have

(5.3)  ∑_{ℓ} F (t(λ_{ℓ}(H_{L}^{ω})))D̃_{n}^{2}t(λ_{ℓ}(H_{L}^{ω})) ≥ Tr[F (t(H_{L}^{ω}))D̃_{n}^{2}t(H_{L}^{ω})]

and

(5.4)  ∑_{ℓ} D̃_{n}^{2}G(t(λ_{ℓ}(H_{L}^{ω}))) = D̃_{n}^{2} Tr[G(t(H_{L}^{ω}))].

We proceed to estimate the terms including the right hand side of (5.3) under the condition that

(5.5)  ∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R ∈ [m − 2, m + 1].

As in [4], we first show

∥D̃_{n}^{2}t(H_{L}^{ω})∥_{L^{2}(Λ_{L})→L^{2}(Λ_{L})} ≤ c_{5}L^{4}

by decomposing as

D̃_{n}^{2}t(H_{L}^{ω})

=(5R)^{3}(D̃_{n}^{2}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−3}

− 2(5R)^{3} ∑_{k=1}^{3}(D̃_{n}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−k}(D̃_{n}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−4+k}

− (5R)^{3} ∑_{k=1}^{3}(H_{L}^{ω} + 1)(5R + 1 + H_{L}^{ω})^{−k}(D̃_{n}^{2}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−4+k}

+ 2(5R)^{3} ∑_{k+k′≤4^{1≤k,k′≤4}}(H_{L}^{ω} + 1)(5R + 1 + H_{L}^{ω})^{−k}(D̃_{n}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−k′}

× (D̃_{n}H_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−5+k+k′}.

For this, we use

D̃_{n}H_{L}^{ω} = 2(i∇ + A_{L}^{ω}) · (D̃_{n}A_{L}^{ω}) = 2(D̃_{n}A_{L}^{ω}) · (i∇ + A_{L}^{ω}),
D̃_{n}^{2}H_{L}^{ω} = 2|(D̃_{n}A_{L}^{ω})|^{2} + 2(i∇ + A_{L}^{ω}) · (D̃_{n}^{2}A_{L}^{ω})
= 2|D̃_{n}A_{L}^{ω})|^{2} + 2(D̃_{n}^{2}A_{L}^{ω}) · (i∇ + A_{L}^{ω}),
∥(i∇ + A_{L}^{ω})(5R + 1 + H_{L}^{ω})^{−1/2}∥_{L^{2}(Λ_{L})→L^{2}(Λ_{L})⊗ℂ^{2}} ≤ 1

and the following estimates obtained from Lemma 3.3:

(5.6)  ∥D̃_{n}A_{L}^{ω}∥_{L^{∞}(Λ_{L})} ≤ c_{6}L^{2}

and

(5.7)  ∥D̃_{n}^{2}A_{L}^{ω}∥_{L^{∞}(Λ_{L})} ≤ \frac{c_{7}}{L^{144}m^{53/2}} .

To prove (5.7), we also use

∥D̃_{n}^{2}B^{ω}∥_{W^{2,1}(Λ_{L})} ≤ c_{8}L^{2}∥D̃_{n}Φ̃_{(n;ε(m,x_{∗},ω),L),L}∥_{L^{∞}(Λ_{3L})} ≤ \frac{c_{9}}{L^{145}m^{53/2}} ,

which is proven by the following: since we have

(5.8)  B(m, x_{∗}, ω) ≤ \frac{c_{10}}{L^{44}m^{7}}

and

\begin{matrix}|D̃_{n}B(m, x_{∗}, ω)| \\ = & | ∫_{0}^{c_{2}L^{−11}m^{−2}} \frac{dr}{πr} ∫_{B(x_{∗},r)} dx′B^{ω}(x′) \\ × ∫_{B(x_{∗},r)} dx ∫_{ℝ^{2}} dyσ(x − y)Φ̃_{(n;ε(m,x_{∗},ω),L),L}(y)| \\ ≤ & \frac{c_{11}}{L^{45}m^{15/2}} \end{matrix} ; (5.9)

under the condition (5.5) by the same method for (2.7), we have

|D̃_{n}ε(m, x_{∗}, ω)| = 2c_{4}c_{3}^{−3}B(m, x_{∗}, ω)L^{−58}m^{−11}|D̃_{n}B(m, x_{∗}, ω)| ≤ \frac{c_{12}}{L^{147}m^{53/2}} ,

|D̃_{n}(n; ε(m, x_{∗}, ω), L)|

=| ∑_{τ∈\{+,−\},s∈ℕ} χ_{[0,\overline{j}/8)}(τ(π\overline{σ}|n|/L − j_{ν+1,s}))\frac{τ\overline{j}Ln}{8π\overline{σ}|n|}D̃_{n}ε(m, x_{∗}, ω)|
≤\frac{c_{13}}{L^{146}m^{53/2}} ,

and |D̃_{n}Φ̃_{(n;ε(m,x_{∗},ω),L),L}|

=|\frac{2}{L} ∑_{ι=1}^{2} cos ((n; ε(m, x_{∗}, ω), L)_{ι}π(\frac{x_{ι}}{L} + \frac{1}{2}))

× sin ((n; ε(m, x_{∗}, ω), L)ι̂π(\frac{xι̂}{L} + \frac{1}{2}))π(\frac{x_{ι}}{L} + \frac{1}{2})D̃_{n}(n; ε(m, x_{∗}, ω), L)|

≤\frac{c_{14}}{L^{147}m^{53/2}} .

Thus we obtain

| Tr[F (t(H_{L}^{ω}))D̃_{n}^{2}t(H_{L}^{ω})]| ≤ c_{5}L^{4} Tr[F (t(H_{L}^{ω}))].

Let E^{∗}(E) be the root of t(E^{∗}) = t(E) in (5R/2 − 1, ∞) for E ∈ (0, 5R/2 − 1). This is solved as E^{∗}(E)

= \frac{2(5R)^{3}}{\sqrt{(E + 1)^{2}(E + 15R + 1)^{2} + 4(5R)^{3}(E + 1)} + (E + 1)(E + 15R + 1)} − 1

≤ (5R)^{3/2}.

Since t′(u) ≥ t′(R) > 5^{3}/7^{4} > 1/(20) for u ∈ (0, R), F (t(λ_{ℓ}(H_{L}^{ω}))) ≠ 0 implies (E − 20η)_{+} ≤ λ_{ℓ}(H_{L}^{ω})) ≤ E^{∗}((E − 20η)_{+}).

By F ≤ 2η and applying the Weyl bound as in [4], we have

\begin{matrix}Tr[F (t(H_{L}^{ω}))] ≤ 2η#\{spec(H_{L}^{ω}) ∩ [0, E^{∗}((E − 20η)_{+})]\} \\ ≤ c_{15}ηL^{2}E^{∗}((E − 20η)_{+}) ≤ c_{16}ηL^{2}R^{3/2}. \end{matrix}

We next estimate the terms including the right hand side of (5.4). For this we apply the theory of the Malliavin calculus. For any separable Hilbert space H, p ∈ (1, ∞), k ∈ ℤ_{+} and any element F of polynomial functionals

P(H) ={F (ω) = ∑_{m=1}^{M} p_{m}(ω(φ_{1}), ω(φ_{2}), …, ω(φ_{N}))h_{m} : M, N ∈ ℕ,
p_{1}, p_{2}, …: polynomials, φ_{1}, φ_{2}, … ∈ L^{2}(ℝ^{2}), h_{1}, h_{2}, … ∈ H},

we define a norm by

∥F ∥_{𝔻^{k,p}(H)} := ∥∥F ∥_{H}∥_{L^{p}(ℙ)} + ∥∥D^{k}F ∥_{(L^{2}(ℝ^{2}))^{⊗k}⊗H}∥_{L^{p}(ℙ)},

and a Banach space 𝔻^{k,p}(H) by the completion of P(H) with respect to this norm. Let (𝔻^{−k,q}(H), ∥ · ∥_{𝔻^{−k,q}(H)}) be its dual space, where q ∈ (1, ∞) such that 1/p + 1/q = 1 (cf. [12] Chapter V §8, [16] §1.2, §1.3, [19] §4.2). We abbreviate as 𝔻^{k,p}(ℝ) = 𝔻^{k,p}. The derivative operator D can be extended to a continuous operator from 𝔻^{k,p}(H) to 𝔻^{k−1,p}(L^{2}(ℝ^{2}) ⊗ H) for any k ∈ ℤ (cf. [12] Chapter V Theorem 8.5, [19] Proposition 4.13). Let δ be its dual operator: 𝔼[(δ(G), F)_{H}] = 𝔼[(G, DF)_{L^{2}(ℝ^{2})⊗H}] for any G ∈ 𝔻^{−k+1,q}(L^{2}(ℝ^{2})⊗H) and F ∈ 𝔻^{k,p}(H). This is also a continuous operator from 𝔻^{k+1,q}(L^{2}(ℝ^{2})⊗H) to 𝔻^{k,q}(H) for any k ∈ ℤ. Now we apply this fact as follows:

𝔼[(D(D Tr[G(t(H_{L}^{ω}))], Φ_{ñ,L})_{L^{2}(ℝ^{2})}, Φ̃_{ñ,L})_{L^{2}(ℝ^{2})}Ψ(ω)]

=𝔼[(D Tr[G(t(H_{L}^{ω}))], Φ̃_{ñ,L})_{L^{2}(ℝ^{2})}δ(Ψ(ω)Φ̃_{ñ,L})]
=𝔼[Tr[G(t(H_{L}^{ω}))]δ(δ(Ψ(ω)Φ̃_{ñ,L})Φ̃_{ñ,L})]

and

|𝔼[ Tr[G(t(H_{L}^{ω}))]δ(δ(Ψ(ω)Φ̃_{ñ,L})Φ̃_{ñ,L})]|
≤∥ Tr[G(t(H_{L}^{ω}))]∥_{L^{p_{1}} (ℙ)}∥δ(δ(Ψ(ω)Φ̃_{ñ,L})Φ̃_{ñ,L})∥_{L^{p_{2}} (ℙ)}
≤c_{17}∥ Tr[G(t(H_{L}^{ω}))]∥_{L^{p_{1}} (ℙ)}∥δ(Ψ(ω)Φ̃_{ñ,L})Φ̃_{ñ,L}∥_{𝔻^{1,p_{2}} (L^{2}(ℝ^{2}))}
≤c_{18}∥ Tr[G(t(H_{L}^{ω}))]∥_{L^{p_{1}} (ℙ)}∥Ψ(ω)Φ̃_{ñ,L}∥_{𝔻^{2,p_{3}} (L^{2}(ℝ^{2}))}∥Φ̃_{ñ,L}∥_{𝔻^{1,p_{4}} (L^{2}(ℝ^{2}))}
≤c_{19}∥ Tr[G(t(H_{L}^{ω}))]∥_{L^{p_{1}} (ℙ)}∥Ψ(ω)∥_{𝔻^{2,p_{5}}} ∥Φ̃_{ñ,L}∥_{𝔻^{2,p_{4}} (L^{2}(ℝ^{2}))}^{2},

(5.10) where ñ := (n; ε(m, x_{∗}, ω), L), p_{1}, p_{2}, p_{3}, p_{4}, p_{5} ∈ (1, ∞) satisfying 1/p_{1} + 1/p_{2} = 1, 1/p_{3} + 1/p_{4} = 1/p_{2}, 1/p_{4} + 1/p_{5} = 1/p_{3}, and

Ψ(ω) := χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)χ̃_{[0,∞)}(R(m, x_{∗}, ω) − |n|)B(m, x_{∗}, ω)^{−ν−15/2}.

To justify this estimate, we should show Tr[G(t(H_{L}^{ω}))] ∈ L^{p_{1}}(ℙ), Ψ(ω) ∈ 𝔻^{2,p_{5}} and Φ̃_{ñ,L} ∈ 𝔻^{2,p_{4}}(L^{2}(ℝ^{2})). This is reduced to show the finiteness of the norms by Proposition 4.21 in [19].

We next proceed to estimates of each norm in the right hand side of (5.10). For the first norm, a sufficient uniform estimate

Tr[G(t(H_{L}^{ω}))] ≤ c_{20}R^{2}ηL^{2}.

is obtained by applying the Feynman-Kac-Itô formula and the diamagnetic inequality as in [4]. By the Hölder inequality, we have

\begin{matrix}∥Ψ(ω)∥_{𝔻^{2,p_{5}}} \\ ≤ & ∥χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)∥_{𝔻^{2,p_{6}}} \\ × ∥χ̃_{[0,∞)}(R(m, x_{∗}, ω) − |n|)∥_{𝔻^{2,p_{7}}} ∥B(m, x_{∗}, ω)^{−ν−15/2}∥_{𝔻^{2,p_{8}}} , \end{matrix} ; (5.11)

where p_{6}, p_{7}, p_{8} ∈ (1, ∞) satisfying 1/p_{6} + 1/p_{7} + 1/p_{8} = 1/p_{5}. Since D^{2}χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)

=χ′_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)(D∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2})^{⊗2}
+ χ′′_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)D^{2}∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2},
∥D∥∂^{α}B^{ω}∥_{L^{2}(Λ_{L})}^{2}∥_{L^{2}(ℝ^{2})}^{2}
= ∫_{ℝ^{2}} dy|2 ∫_{Λ_{L}} dx(∂^{α}B^{ω})(x)(∂^{α}σ)(x − y)|^{2} ≤ c_{α,1}mL^{2}

and

\begin{matrix}∥D^{2}∥∂^{α}B^{ω}∥_{L^{2}(Λ_{L})}^{2}∥_{L^{2}(ℝ^{2})^{⊗2}}^{2} \\ = & ∫_{ℝ^{2}} dy ∫_{ℝ^{2}} dy′|2 ∫_{Λ_{L}} dx(∂^{α}σ)(x − y)(∂^{α}σ)(x − y′)|^{2} ≤ c_{α,2}L^{4} \end{matrix}

under (5.5) for any α = (α_{1}, α_{2}) ∈ ℤ_{+}^{2} satisfying α_{1} + α_{2} ≤ 2, we have

\begin{matrix}∥χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)∥_{𝔻^{2,p_{6}}} \\ ≤ & c_{21}mL^{2}ℙ(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R ∈ [m − 2, m + 1])^{1/p_{6}}. \end{matrix}

The probability is estimated as ℙ(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R ∈ [m − 2, m + 1])

≤L^{2}ℙ((m − R − 2)/L^{2} ≤ ∥B^{ω}∥_{W^{2,2}(Λ_{1})}^{2})
≤L^{2}𝔼[exp(c_{22}(∥B^{ω}∥_{W^{2,2}(Λ_{1})}^{2} − (m − R − 2)/L^{2}))]
≤c_{23}L^{2} exp(−(m − R − 2)/L^{2}))

by Chebyshev’s inequality and Fernique’s theorem (cf. [12] p.402). Thus we have

∥χ̃_{[m−1,m]}(∥B^{ω}∥_{W^{2,2}(Λ_{L})}^{2} + R)∥_{𝔻^{2,p_{6}}} ≤ c_{24}mL^{2+2/p_{6}} exp(−(m − R − 1)/(L^{2}p_{6}))).

Similarly we have ∥χ̃_{[0,∞)}(R(m, x_{∗}, ω) − |n|)∥_{𝔻^{2,p_{7}}}

≤ c_{25}ℙ(R(m, x_{∗}, ω) − |n| ∈ [−1, ∞))^{1/p_{9}}(1 + ∥∥DR(m, x_{∗}, ω)∥_{L^{2}(ℝ)}^{2}∥_{L^{p_{10}} (ℙ)}
+ ∥∥D^{2}R(m, x_{∗}, ω)∥_{L^{2}(ℝ)^{⊗2}}∥_{L^{p_{10}} (ℙ)})

and

\begin{matrix}ℙ(R(m, x_{∗}, ω) − |n| ∈ [−1, ∞))^{1/p_{9}} ≤ & 𝔼[(\frac{R(m, x_{∗}, ω) + 1}{|n|})^{3p_{9}}]^{1/p_{9}} \\ ≤ & ∥R(m, x_{∗}, ω) + 1∥_{L^{3p_{9}} (ℙ)}^{3}/|n|^{3}, \end{matrix}

where p_{9}, p_{10} ∈ (1, ∞) satisfying 1/p_{9} + 1/p_{10} = 1/p_{7}. Thus the remained estimates for ∥Ψ(ω)∥_{𝔻^{2,p_{5}}} are those of ∥∥D^{k}B(m, x_{∗}, ω)^{−h}∥_{L^{2}(ℝ)^{⊗k}}∥_{L^{p}(ℙ)} with (k, h, p) ∈ {0, 1, 2} × (0, ∞) × (1, ∞). Since

\begin{matrix}∥DB(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})}^{2} \\ = & ∫_{ℝ^{2}} dy| ∫_{0}^{c_{2}L^{−11}m^{−2}} \frac{dr}{πr} ∫_{B(x_{∗},r)} dxσ(x − y) ∫_{B(x_{∗},r)} dx′B^{ω}(x′)|^{2} \\ ≤ & \frac{c_{26}∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{2}}{L^{88}m^{16}} \end{matrix} ; (5.12)

and

\begin{matrix}∥D^{2}B(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})^{⊗2}}^{2} \\ = & ∫_{ℝ^{2}} dy ∫_{ℝ^{2}} dy′| ∫_{0}^{c_{2}L^{−11}m^{−2}} \frac{dr}{πr} \\ × ∫_{B(x_{∗},r)} dxσ(x − y) ∫_{B(x_{∗},r)} dx′σ(x′ − y′)|^{2} \\ ≤ & \frac{c_{27}}{L^{88}m^{16}} , \end{matrix} ; (5.13)

are deduced as in (2.7), we have

\begin{matrix}∥∥DB(m, x_{∗}, ω)^{−h}∥_{L^{2}(ℝ)^{⊗k}}∥_{L^{p}(ℙ)} \\ ≤ & \frac{hc_{26}^{1/2}}{L^{44}m^{8}} ∥B(m, x_{∗}, ω)^{−h−1}∥_{L^{p′} (ℙ)}∥∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}∥_{L^{p′′} (ℙ)} \end{matrix}

and ∥∥D^{2}B(m, x_{∗}, ω)^{−h}∥_{L^{2}(ℝ)^{⊗k}}∥_{L^{p}(ℙ)}

≤\frac{h(h + 1)c_{26}}{L^{88}m^{16}} ∥B(m, x_{∗}, ω)^{−h−2}∥_{L^{p′} (ℙ)}∥∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}∥_{L^{2p′′} (ℙ)}^{2}

+ \frac{hc_{27}^{1/2}}{L^{44}m^{8}} ∥B(m, x_{∗}, ω)^{−h−1}∥_{L^{p}(ℙ)},

where p′, p′′ ∈ (1, ∞) such that 1/p′ + 1/p′′ = 1/p. We here do not use the condition (5.5) as in (5.8) and (5.9), since we have a simpler estimate

(5.14)  ∥∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}∥_{L^{p′′} (ℙ)} ≤ c_{p′′}.

We can now apply the corollary of Lemma 4.1 to obtain

(5.15)  ∥B(m, x_{∗}, ω)^{−1}∥_{L^{p}(ℙ)} ≤ c_{p}(L^{11}m^{2})^{2ν+5}

for any p ∈ [1, ∞). Finally, since

\begin{matrix}∥Dε(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})} = 2c_{4}c_{3}^{−3}L^{−58}m^{−11}B(m, x_{∗}, ω)∥DB(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})} \\ ≤ c_{28}∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{3}L^{−146}m^{−27} \end{matrix}

and

∥D^{2}ε(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})^{⊗2}}
=2c_{4}c_{3}^{−3}L^{−58}m^{−11}(∥DB(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})}^{2} + B(m, x_{∗}, ω)∥D^{2}B(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})^{⊗2}})
≤c_{28}∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{2}L^{−146}m^{−27}

are deduced from (5.2), (5.12), (5.13) and B(m, x_{∗}, ω) ≤ c_{29}∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{2}L^{−44}m^{−8}, we have ∥D^{2}Φ̃_{ñ,L}∥_{L^{2}(ℝ^{2})^{⊗3}}

≤c_{30}(L^{2}∥Dε(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})}^{2} + L∥D^{2}ε(m, x_{∗}, ω)∥_{L^{2}(ℝ^{2})^{⊗2}})
≤c_{31}(∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{6}L^{−290}m^{−54} + ∥B^{ω}∥_{L^{∞}(B(x_{∗},1))}^{2}L^{−145}m^{−27})

and

∥Φ̃_{ñ,L}∥_{𝔻^{2,p_{4}} (L^{2}(ℝ^{2}))} ≤ c_{32}

by (5.14).

Thus we obtain

𝔼[Tr[χ_{[E−η,E+η]}(H_{L}^{ω})]]
≤c_{33}ηR^{2}L^{c_{34}} ∑_{m∈ℕ} m^{c_{35}} exp(−(m − R − 1)/(L^{2}p_{6})) ∑_{n∈ℕ^{2}} |n|^{−3}
≤c_{36}ηR^{2}L^{c_{37}}

for L ≥ \sqrt{R} ∨ c_{38}.

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Naomasa Ueki Graduate School of Human and Environmental Studies Kyoto University Kyoto 606-8501 Japan e-mail: ueki@math.h.kyoto-u.ac.jp