Wegner estimates and localization for Gaussian random potentials

By

Naomasa Ueki ^{∗}

Abstract

A Wegner estimate is proven for a Schrödinger operator with a bounded random vector potential and a Gaussian random scalar potential. The estimate is used to prove the strong dynamical localization and the exponential decay of the eigenfunctions. For the proof, Klopp’s method using a vector field on a probability space and Germinet and Klein’s bootstrap multi scale analysis are applied. Moreover Germinet and Klein’s characterization of the Anderson metal-insulator transport transition is extended to the above operator.

§1. Introduction

In this paper we give a Wegner estimate for a random Schrödinger operator

∑_{j=1}^{d}(i∂_{j} + A_{j}^{ω})^{2} + V^{ω}

on L^{2}(ℝ^{d}) and apply the estimate to study the spectral structure, where i = \sqrt{−1}, ∂_{j} = ∂/(∂x_{j}) and \{(A_{1}^{ω}(x), …, A_{d}^{ω}(x), V^{ω}(x)), x ∈ ℝ^{d}, ω ∈ Ω\} are random fields on ℝ^{d}. For this subject, we have many works in the case that the vector potential A^{ω}(x) = (A_{1}^{ω}(x), …, A_{d}^{ω}(x)) is not random and the scalar potential V^{ω}(x) is of the alloy-type: ∑_{m∈ℤ^{d}} λ_{m}^{ω}u_{m}(x), where {λ_{m}^{ω}, m ∈ ℤ^{d}, ω ∈ Ω} is a family of independently and identically distributed real random variables and u_{m}(x) are real deterministic functions on ℝ^{d} [1], [5], [6], [7], [20], [22], [24], [25], [36], [39]. For the case that the vector potential A^{ω}(x) is also random, Hislop and Klopp [20] recently give a Wegner estimate in the case that A_{j}^{ω}(x), j = 1, …, d, are alloy-type bounded random fields and V^{ω}(x) = 0. More recently Klopp, Nakamura, Nakano and Nomura [23] showed Anderson localization for the corresponding discrete model. Apart from the alloy-type potential, Fischer, Hupfer, Leschke and Müller [11] and Hupfer, Leschke, Müller and Warzel [21] give a Wegner estimate for the case that the vector potential A^{ω}(x) is not random and the scalar potential V^{ω}(x) is a random field which is stationary in x ∈ ℝ^{d}. Their most fundamental example of the random field is a Gaussian random field. For other type of potentials, see [26]. The random potential (A^{ω}(x), V^{ω}(x)) treated in this paper is a combination of a deterministic function and a Gaussian random field which is stationary in x ∈ ℝ^{d}. For this we extend Klopp’s method [22] used in [20] so that his method can be applied to the Gaussian random potentials. The main idea is to represent the random field as a random Fourier series by referring the spectral representation (cf. [9]) of stationary random fields (see Lemma 2.2 below). This representation has the same form with the alloy-type potential such that the support of each single site potential u_{m}(x) is noncompact. Because of technical reasons, we give only a weak form of the Wegner estimate only on a low energy interval under the condition that the covariance function of the Gaussian random field is compact. Under these restrictions, our estimate is a generalization of the estimates in [11] and [21] to the case that the vector potential and the scalar potential are correlated.

∗

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-

8501, Japan

One difficulty of the random vector potential is same as that of the alloy type scalar potential ∑_{m∈ℤ^{d}} λ_{m}^{ω}u_{m}(x) with nonsign definite single site potentials {u_{m}(x)}_{m} treated in [6], [20], [22] and [39]: the quadratic form associated to the Schrödinger operator is not monotone as a function of random variables. Then a standard method extending a parameter to the complex space used in [5] and [27], which is used also for the Gaussian scalar potential in [11] and [21], seems to be not applicable. For many of such cases, Klopp’s method [22] using a vector field on a probability space is the only effective method at present. This vector field is a number operator, whose eigenfunctions are homogeneous polynomials. Therefore his method is effective when the main part of the Schrödinger operator to give the Wegner estimate is a homogeneous polynomial on the probability space. In [20] Hislop and Klopp applies Klopp’s method [22] to the random vector potential by using only the effect from the linear part ∑_{j}(i∂_{j}A_{j}^{ω}(x) + A_{j}^{ω}(x)i∂_{j}) and neglecting the effect from the square part ∑_{j}(A_{j}^{ω}(x))^{2} under the condition that the vector potential A^{ω} is small enough. The supremum norm of the vector potential A^{ω} affects the energy interval where we can show the Wegner estimate. In this paper we assume that the vector potential is bounded and the scalar potential is the n-th power W_{ω}^{0}(x)^{n} of a Gaussian random field W_{ω}^{0}(x). We use only the effect from the scalar potential to give the result. As n becomes larger, the effect from the scalar potential becomes stronger and the energy interval where we can show the Wegner estimate holds becomes wider (see (2.3) below). To clarify this situation, we use a Birman-Schwinger type operator whose parts of the square root are random (see (2.6) below). To treat this operator we need extra works not needed in [20]: we should consider the differentiation of the square root of the resolvent operator with respect to a parameter in the potential (see Lemma 2.3 below).

Our Wegner estimate can be used to prove the localization by the multi scale analysis. Since the pioneer work by Fröhlich and Spencer [13], this proof of localization has been extended, improved and simplified by many works. For this aspect, see [37] and the references therein. In particular Germinet and Klein give an effective multiscale analysis to deduce the strong dynamical localization from a weak initial estimate for a wide class of operators with short correlated potentials [14], [17]. In this paper we show that their methods are applicable to our case. The only task is to extend their theory to the operators unbounded below (see Corollary of Theorem 3.2 below). For this it is enough to use a control of Gaussian random fields in Lemma 5.3 in Fischer, Leschke and Müller [12] based on Fernique’s theory [10].

Moreover Germinet and Klein [16], [18] recently show that the strong dynamical localization is equivalent with the initial estimate under some conditions. From this result, they show that some exponent β^{−}(E), which is defined as a function of the energy parameter E in (4.8) below, jumps from zero to a positive value as the energy parameter changes from an insulator region to a metallic region. Therefore this exponent gives a characterization of a transport property. In this paper we show that this theory is also extended to our case (see Theorem 4.1 and Proposition 4.3 below). For this we use again the estimate of the scalar potential based on Fernique’s theory [10]. Since our Wegner estimate is weaker than the estimate used in [16], the lower bound of the size of the jump of the exponent β^{−}(E) is different from that in [16]. We give a lower bound of the size in terms of the exponents appeared in the Wegner estimate (cf. Proposition 4.3 below).

The organization of this paper is as follows. In Section 2 we give a Wegner estimate. In Section 3 we modify Germinet and Klein’ s theory on the multiscale analysis [14] to show the strong dynamical localization. In Section 4 we modify Germinet and Klein’ s theory on a characterization of the transport transition [16] so that their theory can be applied to the operators in this paper.

§2. Wegner estimate

§2.1. Main theorem

Let {(W_{ω}^{j}(x))_{j=0,1,··· ,g}, x ∈ ℝ^{d}, ω ∈ Ω} be an ℝ^{g+1}-valued stationary ergodic Gaussian random field with mean zero and the covariance R^{jk}(x) = E[W_{ω}^{j}(x)W_{ω}^{k}(0)] represented as

R^{jk}(x) = ∑_{ι=1}^{r} ∫ σ_{ι}^{j}(x + y)\overline{σ_{ι}^{k}(y)}dy

for some complex valued C^{s} functions σ_{ι}^{j}(x), 0 ≤ j ≤ g, 1 ≤ ι ≤ r, on ℝ^{d} with compact support and s ≥ d + 4. Let A(x, w) be an ℝ^{d} valued C^{1} function of (x, w) ∈ ℝ^{d} × ℝ^{g} such that ∇_{w}A(x, w) is a C^{2} function of (x, w) and w · ∇_{w}A and ∇_{w}A are bounded. For each L > 0, let Λ_{L} := (−L/2, L/2)^{d} be an open box and, for L > 3, let χ̃_{L} be a smooth function on ℝ^{d} such that 0 ≤ χ̃_{L} ≤ 1, χ̃_{L} = 1 on Λ_{L−3}, χ̃_{L} = 0 on Λ_{L−1}^{c} and |∇χ̃_{L}| ≤ 2. We consider a self-adjoint operator on L^{2}(ℝ^{d}) defined by

H̃_{L}^{ω} := ∑_{j=1}^{d}(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x)))^{2} − χ̃_{L}W_{ω}^{0}(x)^{n}, (2.1)

where \overline{W_{ω}}(x) =^{t} (W_{ω}^{1}(x), …, W_{ω}^{g}(x)). If we define a positive operator by

Ĥ_{L}^{ω} := ∑_{j=1}^{d}(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x)))^{2}, (2.2)

then we see that χ̃_{L}W_{ω}^{0}(x)^{n} is Ĥ_{L}^{ω}-compact. Therefore the spectrum in (−∞, 0) of the operator H̃_{L}^{ω} consists only of its discrete spectrum. For the spectrum, we will prove the following Wegner estimate in this section:

Theorem 2.1. For any

E_{0} < −(2∥w · ∇_{w}A∥_{∞}/n)^{2}, (2.3)

there exists a finite constant Q(E_{0}) such that

P (dist(E, σ(H̃_{L}^{ω})) ≤ η) ≤ Q(E_{0})η^{1−d/s}L^{d(2−d/(2s))} (2.4)

for any E ≤ E_{0}, L ≥ 4 and η > 0, where, for any self-adjoint operator A, σ(A) is the spectral set of A.

This estimate can be used to show results on the spectral localization of the self-adjoint operator

H^{ω} := ∑_{j=1}^{d}(i∂_{j} + A_{j}(x, \overline{W_{ω}}(x)))^{2} − W_{ω}^{0}(x)^{n}, (2.5)

on L^{2}(ℝ^{d}). This is the subject of the next section.

Remark 2.1. In the case that A is independent of w and n = 1, Fisher, Hupfer, Leschke and Müller [11] and Hupfer, Leschke, Müller and Warzel [21] give a strong form of the Wegner estimate under more general conditions, where the compactness of the covariance function is not assumed. Their estimate is as follows: let H_{L,X}^{ω} be the restriction of the operator ∑_{j=1}^{d}(i∂_{j} +A_{j}(x))^{2}−W_{ω}^{0}(x) to L^{2}(Λ_{L}) by the Dirichlet or Neumann boundary conditions. Then there exists an explicitly written function C_{W} (E) such that

E[♯(σ(H_{L,X}^{ω}) ∩ [E − η, E + η])] ≤ C_{W} (E + η)ηL^{d}

for all E ∈ ℝ and η ≥ 0. From this estimate, we see that the function C_{W} (E) is an upper bound of the density of states. This estimate is the first motivation of Wegner’s estimate [40].

Remark 2.2. (i) Theorem 2.1 is extended to the operator

{ \atop H_{L,1}^{ω}} := H̃_{L}^{ω} + χ̃_{L}V (x, \overline{W_{ω}}(x)),

where V (x, w) be a real bounded continuous function of (x, w) ∈ ℝ^{d} × ℝ^{g} such that V (x, w) is C^{1} in w and ∇_{w}V and w ·∇_{w}V are bounded. For this operator, (2.4) holds only for E_{0} < 𝔈_{0}, where

𝔈_{0} := \begin{cases}−(2𝔄/n)^{2} if 𝔙 ≤ 0, \\ −\{𝔙/(\sqrt{𝔄^{2} + n𝔙} − 𝔄)\}^{2} if 𝔙 > 0, \end{cases}

𝔄 := ∥w · ∇_{w}A∥_{∞} and 𝔙 := sup(w · ∇_{w}V − nV).

(ii) Theorem 2.1 is extended to the operator

{ \atop H_{L,2}^{ω}} := Ĥ_{L}^{ω} + χ̃_{L}p_{n}(\overline{W_{ω}^{0}}(x))

without changing the statement, where p_{n}(t_{1}, …, t_{h}) is a real homogeneous polynomial of degree n and \{\overline{W_{ω}^{0}} = (W_{ω}^{0,k}(x))_{k=1,2,... ,h} : x ∈ ℝ^{d}, ω ∈ Ω\} is an ℝ^{h} valued random field such that \{(\overline{W_{ω}^{0}}(x), \overline{W_{ω}}(x)) : x ∈ ℝ^{d}, ω ∈ Ω\} is an ℝ^{g+h} valued random field satisfying the same conditions for the random field \{(W_{ω}^{0}(x), \overline{W_{ω}}(x)) : x ∈ ℝ^{d}, ω ∈ Ω\} stated at the front of this section.

(iii) Theorem 2.1 is extended to the operator

{ \atop H_{L,3}^{ω}} := Ĥ_{L}^{ω} − χ̃_{L}q_{n}(W_{ω}^{0}(x))

without changing the statement, where q_{n}(t) is a polynomial such that tq′_{n}(t) ≥ nq_{n}(t). For example, we can take as q_{2m−1}(t) = t^{2m} + at^{2m−1}.

§2.2. Proof of the Wegner estimate

For the proof of Theorem 2.1, we introduce a Birman-Schwinger type operator by

Γ(ω, L, E) = (Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}(Ĥ_{L}^{ω} − E)^{−1/2}. (2.6)

It is a different point from [6], [20] and [22] that the operator (Ĥ_{L}^{ω} − E)^{−1/2} depends on ω. We deduce (2.4) only by the effect from the part χ̃_{L}W_{ω}^{0}(x)^{n}.

For any p > 0, let I_{p} be the trace or super-trace class: a continuous linear operator A on L^{2}(ℝ^{d}) is said to belong to I_{p} if |||A|||_{p} := (Tr[|A|^{p}])^{1/p} < ∞, where |A| := (A^{∗}A)^{1/2} (cf. [6], [7], [20], [33]). We use ∥ · ∥_{p} for the norm on L^{p}(ℝ^{d}) and ||| · ||| for the operator norm.

Then we have the following:

Lemma 2.1. Let q ≥ min{q ∈ 4ℕ : q > d}. Let A ∈ L_{loc}^{4}(ℝ^{d} → ℝ^{d}) such that ∇ · A ∈ L_{loc}^{2}(ℝ^{d}). We set H(A) = ∑_{j=1}^{d}(i∂_{j} + A_{j})^{2}. Then we have the following:

(i) There exists a finite constant c_{1} depending only on q and d such that

|||(H(A) − E)^{−1/2}k|||_{q} ≤ c_{1}∥k∥_{q}/|E|^{(q−d)/(2q)}

for any E < 0 and k ∈ L^{q}(ℝ^{d}).

(ii) There exists a finite constant c_{2} depending only on q and d such that

|||(H(A) − E)^{−1/2}k(H(A) − E)^{−1/2}|||_{q/2} ≤ c_{2}∥k∥_{q/2}/|E|^{(q−d)/q}

for any E < 0 and k ∈ L^{q/2}(ℝ^{d}).

(iii) There exists a finite constant c_{3} depending only on q and d such that

|||(H(A) − E)^{−1/2}h · (i∇ + A)(H(A) − E)^{−1/2}|||_{q} ≤ c_{3}∥h∥_{q}/|E|^{(q−d)/(2q)} for any E < 0 and ℂ^{d} valued L^{q} function h on ℝ^{d}.

(iv) There exists a finite constant c_{4} depending only on q and d such that

|||(H(A) − E)^{−1}k|||_{q/2} ≤ c_{4}∥k∥_{q/2}/|E|^{(q−d)/q}

for any E < 0 and k ∈ L^{q/2}(ℝ^{d}).

Proof. (i) By the diamagnetic inequality, we have

|(H(A) − E)^{−1/2}φ| ≤ (H(0) − E)^{−1/2}|φ| (2.7)

for any φ ∈ C_{0}^{∞}(ℝ^{d}) (see (4.9) in [29] and (A.23) in [21]). We now assume that q ∈ 2ℕ. By using (2.7) successively, we have

||(H(A) − E)^{−1/2}k|^{q}φ| ≤ |(H(0) − E)^{−1/2}|k||^{q}|φ|,

where, for any operator L, |L| := \sqrt{L^{∗}L}. From this, Lemma 15.11 in [34] and Theorem 4.1 in [33], we have

|||(H(A) − E)^{−1/2}k|||_{q} ≤ |||(H(0) − E)^{−1/2}|k||||_{q} ≤ c_{1}∥k∥_{q}/|E|^{(q−d)/(2q)}.

For q ∉ 2ℕ, we use the Stein interpolation theorem (cf. [32] Theorem IX.21). (ii) Since

|||(H(A) − E)^{−1/2}k(H(A) − E)^{−1/2}|||_{q/2}

≤ |||(H(A) − E)^{−1/2}\sqrt{|k|}|||_{q}|||\sqrt{|k|}(H(A) − E)^{−1/2}|||_{q},

(i) implies (ii).

(iii) Since

|||(i∂_{j} + A_{j})(H(A) − E)^{−1/2}|||^{2} ≤ 1

for any j, we have

|||(H(A) − E)^{−1/2}h · (i∇ + A)(H(A) − E)^{−1/2}|||_{q}

≤ ∑_{j=1}^{d} |||(H(A) − E)^{−1/2}h_{j}|||_{q}.

Therefore (i) implies (iii).

(iv) For q ∈ 4ℕ, as in (i), we have

||(H(A) − E)^{−1}k|^{q/2}φ| ≤ |(H(0) − E)^{−1}|k||^{q/2}|φ|

and

|||(H(A) − E)^{−1/2}k|||_{q/2} ≤ |||(H(0) − E)^{−1/2}|k||||_{q/2} ≤ c_{4}∥k∥_{q/2}/|E|^{(q−d)/q}.

For q ∉ 4ℕ, we use the interpolation theorem.

In the following we fix q ∈ ℕ such that q ≥ min{q ∈ 4ℕ : q > d}. Lemma 2.1 (ii) implies Γ(ω, L, E) ∈ I_{q/2}. We take a smooth function ρ having the form ρ = σ^{q}, where σ is a smooth function on ℝ such that 0 ≤ σ ≤ 1 on ℝ, σ = 1 on (−∞, −1/2] and σ = 0 on [1/2, ∞). For any self-adjoint operator A, let E(·; A) be the spectral projection of A. As in [6], [20] and [22], we have

P (dist(E, σ(H̃_{L}^{ω})) ≤ η)
≤ E[Tr[E([1 − κ, 1 + κ] : Γ(ω, L, E))]]
≤ E[Tr[ρ_{κ}(Γ(ω, L, E) − 1 − 3κ/2) − ρ_{κ}(Γ(ω, L, E) − 1 + 3κ/2)]]
= ∫_{−3κ/2}^{3κ/2} dtE[Tr[(−ρ_{κ})′(Γ(ω, L, E) − 1 + t)]],

where κ = η/|E| and ρ_{κ}(μ) = ρ(μ/κ).

To consider an associated vector field used in [6], [20] and [22], we refer the theory of the spectral representation of stationary field (cf. [9]) to give the following representation:

Lemma 2.2. The random field W_{ω}(x) = (W_{ω}^{j}(x))_{j=0}^{g} is represented as

W_{ω}^{j}(x) = ∑_{ι=1}^{2r} ∑_{m∈ℤ^{d}} W_{ω}^{ι,m}e_{ι,m}^{j}(x) (2.8)

in L^{p}(Λ_{L} × Ω) for any 1 ≤ p < ∞, where {W_{ω}^{ι,m}}_{1≤ι≤2r,m∈ℤ^{d}} is a family of independently and identically distributed random variables with the standard normal distribution and e_{ι,m}^{j}, 1 ≤ ι ≤ 2r, m ∈ ℤ^{d}, 0 ≤ j ≤ g, are C^{s} functions such that, for any 0 ≤ ℓ ≤ s,

sup_{x∈Λ_{L}} |∇^{ℓ}e_{ι,m}^{j}(x)| ≤ cL^{s−ℓ−d/2}|m|_{∞}^{ℓ−s}, (2.9)

where c is some finite constant depending only on the covariance R = (R^{jk}(x)).

Proof. Let {W_{ι}(h) : h ∈ L^{2}(ℝ^{d})}_{1≤ι≤2r} be 2r independent copies of the isonormal Gaussian random field: for each h_{1}, h_{2}, …, h_{m} ∈ L^{2}(ℝ^{d}), W_{ι}(h_{1}), W_{ι}(h_{2}), …, W_{ι}(h_{m}) are Gaussian random variables with mean zero and

E[W_{ι}(h_{1})W_{ι}(h_{2})] = ∫_{ℝ^{d}} h_{1}(ξ)h_{2}(ξ)dξ

(cf. Definition 1.1.1 in Nualart [30]). We define the Fourier transform of f ∈ L^{1}(ℝ^{d}) by

f̂(ξ) = ∫_{ℝ^{d}} e^{−2πiξ·x}f(x)dx.

Then the random field W_{ω}(x) has the following representation:

W_{ω}^{j}(x) = ∑_{ι=1}^{r}{W_{ι}(Re(e^{2πiξ·x}σ̂_{ι}^{j}(ξ))) + W_{r+ι}(Im(e^{2πiξ·x}σ̂_{ι}^{j}(ξ)))}.

We here note that

e^{2πiξ·x}σ̂_{ι}^{j}(ξ) = { \atop σ_{ι}^{j}(· + x)}(ξ).

If supp σ_{ι}^{j} ⊂ Λ_{R} and x ∈ Λ_{L}, then supp σ_{ι}^{j}(· + x) ⊂ Λ_{R+L}. Then we take a complete orthonormal basis {h_{m}(y) := ∏_{k=1}^{d} k_{m_{k}}(y_{k}) : m ∈ ℤ^{d}} of L^{2}(Λ_{R+L}), where y = (y_{1}, y_{2}, ⋯, y_{d}), k_{0}(y_{1}) = 1/\sqrt{R + L}, and, for m_{1} > 0,

k_{−m_{1}}(y_{1}) = \sqrt{\frac{2}{R + L}} sin \frac{2m_{1}πy_{1}}{R + L} and k_{m_{1}}(y_{1}) = \sqrt{\frac{2}{R + L}} cos \frac{2m_{1}πy_{1}}{R + L} .

We extend these functions to ℝ^{d} by h_{m}(y) = 0 on Λ_{R+L}^{c}. Then, for each x ∈ Λ_{L}, we have

σ_{ι}^{j}(y + x) = ∑_{m∈ℤ^{d}} h_{m}(y) ∫ h_{m}(z)σ_{ι}^{j}(z + x)dz

and

e^{2πiξ·x}σ̂_{ι}^{j}(ξ) = ∑_{m∈ℤ^{d}} ĥ_{m}(ξ) ∫ h_{m}(y)σ_{ι}^{j}(y + x)dy

in L^{2}(ℝ^{d}). Let (ℤ^{d})_{e} := {m = (m_{1}, m_{2}, …, m_{d}) ∈ ℤ^{d} : ♯{j : m_{j} < 0} is even} and (ℤ^{d})_{o} := {m = (m_{1}, m_{2}, …, m_{d}) ∈ ℤ^{d} : ♯{j : m_{j} < 0} is odd}. Then ĥ_{m}(ξ) is real valued for m ∈ (ℤ^{d})_{e} and is purely imaginary valued for m ∈ (ℤ^{d})_{o}. Thus we have

Re(e^{2πiξ·x}σ̂_{ι}^{j}(ξ)) = ∑_{m∈(ℤ^{d})_{e}} ĥ_{m}(ξ)e_{ι,m}^{j}(x) + ∑_{m∈(ℤ^{d})_{o}} iĥ_{m}(ξ)e_{ι,m}^{j}(x),

where

e_{ι,m}^{j}(x) := \begin{cases}∫ h_{m}(y) Re σ_{ι}^{j}(y + x)dy for m ∈ (ℤ^{d})_{e}, \\ ∫ h_{m}(y) Im σ_{ι}^{j}(y + x)dy for m ∈ (ℤ^{d})_{o}. \end{cases}

Similarly we have

Im(e^{2πiξ·x}σ̂_{ι}^{j}(ξ)) = ∑_{m∈(ℤ^{d})_{e}} ĥ_{m}(ξ)e_{r+ι,m}^{j}(x) − ∑_{m∈(ℤ^{d})_{o}} iĥ_{m}(ξ)e_{r+ι,m}^{j}(x),

where

e_{r+ι,m}^{j}(x) := \begin{cases}∫ h_{m}(y) Im σ_{ι}^{j}(y + x)dy for m ∈ (ℤ^{d})_{e}, \\ ∫ h_{m}(y) Re σ_{ι}^{j}(y + x)dy for m ∈ (ℤ^{d})_{o}. \end{cases}

Then we have (2.8) for each x ∈ Λ_{L} in L^{2}(Ω), where

W_{ω}^{ι,m} := \begin{cases}W_{ι}(ĥ_{m}) for m ∈ (ℤ^{d})_{e}, \\ W_{ι}(iĥ_{m}) for m ∈ (ℤ^{d})_{o} \end{cases}

and

W_{ω}^{r+ι,m} := \begin{cases}W_{r+ι}(ĥ_{m}) for m ∈ (ℤ^{d})_{e}, \\ W_{r+ι}(−iĥ_{m}) for m ∈ (ℤ^{d})_{o}. \end{cases}

By the integration by parts, for each 0 ≠ m ∈ ℤ^{d} and 0 ≤ ℓ < s, there exist m′ ∈ ℤ^{d} and k ∈ {1, 2, …, d}, such that

| ∫ h_{m}(y)∇^{ℓ}σ_{ι}^{j}(y + x)dy|

= (\frac{R + L}{2π|m|_{∞}} )^{s−ℓ} | ∫ h_{m′}(y)(\frac{∂}{∂y^{k}} )^{s−ℓ}∇^{ℓ}σ_{ι}^{j}(y + x)dy|,

where |m|_{∞} is the maximum norm on ℤ^{d}. Since |h_{m′}(y)| ≤ \sqrt{2/(R + L)}^{d}, we obtain (2.9). Then we see that {∑_{ι=1}^{2r} ∑_{|m|_{∞}≤N} W_{ω}^{ι,m}e_{ι,m}^{j}}_{N} is a Cauchy sequence in L^{p}(Λ_{L} × Ω). Therefore (2.8) holds in L^{p}(Λ_{L} × Ω).

Remark 2.3. If 0 ≤ ℓ < s−d, we see that {∑_{ι=1}^{2r} ∑_{|m|_{∞}≤N} W_{ω}^{ι,m}∇^{ℓ}e_{ι,m}^{j}}_{N} is a Cauchy sequence in L^{p}(Λ_{L} × Ω) by (2.9). Therefore, for each k ∈ ℤ_{+}, if s ≥ d + 2 + k, then we see that {∑_{ι=1}^{2r} ∑_{|m|_{∞}≤N} W_{ω}^{ι,m}∇^{k}e_{ι,m}^{j}(x)}_{N} converges to ∇^{k}W_{ω}^{j}(x) uniformly on Λ_{L} and W_{ω}^{j}(x) is a C^{k} function of x almost surely. Since we assume s ≥ d + 4, W_{ω}^{j}(x) is a C^{2} function of x.

We use the following lemma on differentiations in random variables:

Lemma 2.3. For any ε > 0, 1 ≤ ι ≤ 2r and m ∈ ℤ^{d}, we set
W_{ω}^{j}(x, ε; ι, m) := W_{ω}^{j}(x) + εe_{ι,m}^{j}(x),

Ĥ_{L}^{ω}(ε; ι, m) := ∑_{j=1}^{d}(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x, ε; ι, m)))^{2}

and

Γ(ω, L, E, ε; ι, m)
:= (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x, ε; ι, m)^{n}
×(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}.

Then we have the following:

(i) In the weak sense, we have

\{\frac{∂}{∂ε}(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}\}_{ε=0} = Ξ(ω, L, E; ι, m),

where Ξ(ω, L, E; ι, m) is a bounded operator defined by the following integral in the weak sense:

Ξ(ω, L, E; ι, m)

:= \frac{−1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du exp(−utĤ_{L}^{ω})

× ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)), i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}

× exp(−(1 − u)tĤ_{L}^{ω}),

where \overline{e_{ι,m}}(x) :=^{t} (e_{ι,m}^{1}(x), …, e_{ι,m}^{g}(x)) and [A, B]_{+} := AB + BA for any operators A and B.

(ii) The operator Ξ(ω, L, E; ι, m) defined in (i) maps the elements of ⋃_{a>0} Ran E([a, ∞) : Γ(ω, L, E)) to the domain of Ĥ_{L}^{ω}, where, for any operator A, Ran A is the range of A.

(iii) On ⋃_{a>0} Ran E([a, ∞) : Γ(ω, L, E)), it holds that

[(Ĥ_{L}^{ω} − E)^{1/2}, Ξ(ω, L, E; ι, m)]_{+}

= −(Ĥ_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}.

(iv) For any f ∈ C_{0}^{1}((0, ∞)), we have

\{\frac{∂}{∂ε} Tr[f^{q}(Γ(ω, L, E, ε; ι, m))]\}_{ε=0}
= Tr[(f^{q})′(Γ(ω, L, E))Γ(ω, L, E; ι, m)],

where

Γ(ω, L, E; ι, m)

:= (Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}ne_{ι,m}^{0}(x)W_{ω}^{0}(x)^{n−1}(Ĥ_{L}^{ω} − E)^{−1/2}

−Γ(ω, L, E)(Ĥ_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}.

We postpone the proof of this lemma to the next subsection. We use a vector field 𝔸 on the space of sample paths defined by

𝔸F (W_{ω}(·)) = ∑_{ι=1}^{2r} ∑_{m∈ℤ^{d}} W_{ω}^{ι,m} (\frac{∂}{∂ε}F (W_{ω}(·, ε; ι, m)))_{ε=0}

in L^{1}(Ω), for any functional F of the sample path W_{ω}(·), when the limit exists. For this vector field, we have the following:

Lemma 2.4. For any f ∈ C_{0}^{1}((0, ∞)), we have

𝔸 Tr[f^{q}(Γ(ω, L, E))] = Tr[(f^{q})′(Γ(ω, L, E))Γ^{(1)}(ω, L, E)],

where

Γ^{(1)}(ω, L, E)

:= Γ(ω, L, E)\{n − (Ĥ_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}\overline{W_{ω}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}\}.

Wegner estimates 13

Proof. By Lemma 2.3 (iv), we have

| ∑_{ι=1}^{2r} ∑_{|m|_{∞}≤M} W_{ω}^{ι,m} (\frac{∂}{∂ε} Tr[f^{q}(Γ(ω, L, E, ε; ι, m))])_{ε=0}

− Tr[(f^{q})′(Γ(ω, L, E))Γ^{(1)}(ω, L, E)]| ≤ | Tr[(f^{q})′(Γ(ω, L, E))(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}nW_{ω}^{0}(x)^{n−1}

×R_{ω}^{0}(x, M)(Ĥ_{L}^{ω} − E)^{−1/2}]| +| Tr[(f^{q})′(Γ(ω, L, E))Γ(ω, L, E)(Ĥ_{L}^{ω} − E)^{−1/2}

× ∑_{j=1}^{d}[χ̃_{L}\overline{R_{ω}}(x, M) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}]|,

where

R_{ω}^{j}(x, M) := ∑_{ι=1}^{2r} ∑_{|m|_{∞}≤M} W_{ω}^{ι,m}e_{ι,m}^{j}(x) − W_{ω}^{j}(x)

and \overline{R_{ω}}(x, M) :=^{t} (R_{ω}^{1}(x, M), …, R_{ω}^{d}(x, M)). By the Hölder inequality for the norms ||| · |||_{p} (cf. Theorem 2.8 in [33]), this is less than or equal to

|||(f^{q})′(Γ(ω, L, E))|||_{q/(q−2)}|||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}nW_{ω}^{0}(x)^{n−1}

×R_{ω}^{0}(x, M)(Ĥ_{L}^{ω} − E)^{−1/2}|||_{q/2}

+|||(f^{q})′(Γ(ω, L, E))Γ(ω, L, E)|||_{q/(q−1)} (2.10)

×|||(Ĥ_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}\overline{R_{ω}}(x, M) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}|||_{q}.

For any p > 1, we have

|||(f^{q})′(Γ(ω, L, E))|||_{p} ≤ c Tr[E(supp f : Γ(ω, L, E))]^{1/p}

≤ c|||Γ(ω, L, E)|||_{q/2}^{q/(2p)}.

Similarly we have |||(f^{q})′(Γ(ω, L, E))Γ(ω, L, E)|||_{p} ≤ c|||Γ(ω, L, E)|||_{q/2}^{q/(2p)}.

14 Naomasa Ueki

Thus, by Lemma 2.1, the quantity in (2.10) is dominated by ∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{(q−2)/2}∥χ̃_{L}nW_{ω}^{0}(x)^{n−1}R_{ω}^{0}(x, M)∥_{q/2}

+∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{(q−1)/2} ∑_{j=1}^{d} ∥χ̃_{L}\overline{R_{ω}}(x, M) · ∇_{w}A_{j}(x, W_{ω}^{j}(x))∥_{q}.

By the Hölder inequality on the probability space, we have

E[| ∑_{ι=1}^{2r} ∑_{|m|_{∞}≤M} W_{ω}^{ι,m} (\frac{∂}{∂ε} Tr[f^{q}(Γ(ω, L, E, ε; ι, m))])_{ε=0}

− Tr[(f^{q})′(Γ(ω, L, E))Γ^{(1)}(ω, L, E)]|]

≤ c(E[∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{q/2}]^{(q−2)/q}E[∥χ̃_{L}nW_{ω}^{0}(x)^{n−1}R_{ω}^{0}(x, M)∥_{q/2}^{q/2}]^{2/q}

+E[∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{q/2}]^{(q−1)/q} ∑_{j=1}^{d} E[∥χ̃_{L}\overline{R_{ω}}(x, M) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x))∥_{q}^{q}]^{1/q}).

This converges to 0 as M → ∞ since, for any p > 1, W_{ω}^{0}(x) ∈ L^{p}(Λ_{L} × Ω) and R_{ω}(x, M) converges to 0 in L^{p}(Λ_{L} × Ω) by Lemma 2.2.

We now obtain E[Tr[(−ρ_{κ})′(Γ(ω, L, E) − 1 + t)Γ^{(1)}(ω, L, E)]]

= ∑_{ι=1}^{2r} ∑_{m∈ℤ^{d}} I(L, E, t; ι, m),

where I(L, E, t; ι, m) (2.11)

:= E[W_{ω}^{ι,m} Tr[(−ρ_{κ})′(Γ(ω, L, E) − 1 + t)Γ(ω, L, E; ι, m)]].

Since

(φ, (Ĥ_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}W_{ω}^{j}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}φ)

≤ 2∥w · ∇_{w}A∥_{∞}/\sqrt{|E|}

for any φ ∈ L^{2}(ℝ^{d}) such that ∥φ∥_{2} = 1, we obtain

∑_{ι=1}^{2r} ∑_{m∈ℤ^{d}} I(L, E, t; ι, m) ≥ c(n, κ, E)E[Tr[(−ρ_{κ})′(Γ(ω, L, E) − 1 + t)]]

for −3κ/2 ≤ t ≤ 3κ/2, where

c(n, κ, E) = (1 − 2κ)(n − 2∥w · ∇_{w}A∥_{∞}/\sqrt{|E|})_{+}.

If E is less than −(2∥w · ∇_{w}A∥_{∞}/n)^{2} defined in (2.4), then c(n, κ, E) does not vanish for small enough κ and we have the following bound:

P (dist(E, σ(H̃_{L}^{ω})) ≤ η) (2.12)

≤ \frac{1}{c(n, κ, E)} ∫_{−3κ/2}^{3κ/2} dt ∑_{ι=1}^{2r} ∑_{m∈ℤ^{d}} I(L, E, t; ι, m).

We estimate the each term in the summation:

Lemma 2.5. There exists a finite constant c independent of L, E, m and κ such that

sup_{−3κ/2≤t≤3κ/2} |I(L, E, t; ι, m)| ≤ cL^{d}/|E|^{(q−d)/2} (2.13)

for any 1 ≤ ι ≤ 2r, m ∈ ℤ^{d} and 0 < κ < 1/8.

Proof. Since Γ(ω, L, E) should be greater than 1 − 2κ for ρ′_{κ}(Γ(ω, L, E) − 1 + t) to be nonzero, we have

I(L, E, t; ι, m)
= E[W_{ω}^{ι,m} Tr[(−ρ_{κ})′(Γ(ω, L, E) − 1 + t)
×τ(Γ(ω, L, E) + 2κ)Γ(ω, L, E; ι, m)]],

where τ(μ) = υ(μ)^{q} and υ is a smooth function on ℝ such that 0 ≤ υ ≤ 1, 0 ≤ υ′ ≤ 3 on ℝ, υ = 0 on (−∞, 1/2] and υ = 1 on [1, ∞). This is rewritten as

I(L, E, t; ι, m) = ∑_{j=1}^{2} I_{j}(L, E, t; ι, m),

where

I_{1}(L, E, t; ι, m)
= E[W_{ω}^{ι,m} Tr[{−ρ_{κ}(· − 1 + t)τ(· + 2κ)}′(Γ(ω, L, E))
×Γ(ω, L, E; ι, m)]]

16 Naomasa Ueki

and

I_{2}(L, E, t; ι, m)
= E[W_{ω}^{ι,m} Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t)
×τ′(Γ(ω, L, E) + 2κ)Γ(ω, L, E; ι, m)]].

By Lemma 2.3 (iv), we have I_{1}(L, E, t; ι, m) (2.14)

= E[(1 − (W_{ω}^{ι,m})^{2}) Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t)τ(Γ(ω, L, E) + 2κ)]]. In fact, if we denote Γ_{ι,m,λ}(ω, L, E) the operator obtained by replacing W_{ω}^{ι,m} by λ in Γ(ω, L, E), then (2.14) is rewritten as

E[ ∫_{ℝ} \frac{dλ}{\sqrt{2π}} e^{−λ^{2}/2}λ\frac{∂}{∂λ} Tr[−ρ_{κ}(Γ_{ι,m,λ}(ω, L, E) − 1 + t)

×τ(Γ_{ι,m,λ}(ω, L, E) + 2κ)]]

= E[ ∫_{ℝ} \frac{dλ}{\sqrt{2π}} e^{−λ^{2}/2}(1 − λ^{2}) Tr[ρ_{κ}(Γ_{ι,m,λ}(ω, L, E) − 1 + t)

×τ(Γ_{ι,m,λ}(ω, L, E) + 2κ)]].

We here simply estimate as | Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t)τ(Γ(ω, L, E) + 2κ)]|

≤ Tr[E((1/2 − 2κ, ∞) : Γ(ω, L, E))]

≤ (1/2 − 2κ)^{−q/2} Tr[E((0, ∞) : Γ(ω, L, E))Γ(ω, L, E)^{q/2}] (2.15)

≤ (1/2 − 2κ)^{−q/2}|||Γ(ω, L, E)|||_{q/2}^{q/2}
≤ (1/2 − 2κ)^{−q/2}c_{d,q}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{q/2}/|E|^{(q−d)/2}

by Lemma 2.1 (ii). Then we obtain |I_{1}(L, E, t; ι, m)|

≤ (1/2 − 2κ)^{−q/2}c_{d,q} ∫_{ℝ^{d}} dx(χ̃_{L}(x))^{q/2}

×E[|1 − (W_{ω}^{ι,m})^{2}||W_{ω}^{0}(x)|^{nq/2}]/|E|^{(q−d)/2} (2.16)

≤ (1/2 − 2κ)^{−q/2}c_{d,q}L^{d}E[|1 − (W_{ω}^{ι,m})^{2}|^{2}]^{1/2}
×E[|W_{ω}^{0}(0)|^{nq}]^{1/2}/|E|^{(q−d)/2}.

For I_{2}(L, E, t; ι, m), we use Lemma 2.1 (ii) and (iii) to estimate parts of Γ(ω, L, E; ι, m) as

|||(H̃_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}(H̃_{L}^{ω} − E)^{−1/2}|||_{q/2}
≤ c_{d,q}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}/|E|^{(q−d)/q}

and

|||(H̃_{L}^{ω} − E)^{−1/2} ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(H̃_{L}^{ω} − E)^{−1/2}|||_{q}

≤ c_{d,q}∥χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x))∥_{q}/|E|^{(q−d)/(2q)}.

On the other hand, as in (2.15), we have

|||ρ_{κ}(Γ(ω, L, E) − 1 + t)τ′(Γ(ω, L, E) + 2κ)|||_{q/(q−2)}
≤ 3(1/2 − 2κ)^{−(q−2)/2}|||Γ(ω, L, E)|||_{q/2}^{(q−2)/2}

and

|||ρ_{κ}(Γ(ω, L, E) − 1 + t)τ′(Γ(ω, L, E) + 2κ)Γ(ω, L, E)|||_{q/(q−1)}
≤ 3(1/2 − 2κ)^{−(q−3)/2}|||Γ(ω, L, E)|||_{q/2}^{(q−1)/2}.

Then, as in (2.16), we obtain

|I_{2}(L, E, t; ι, m)| ≤ (1/2 − 2κ)^{−(q−2)/2}c_{d,q}c_{R}L^{d}/|E|^{(q−d)/2},

where c_{R} is a constant depending only on the covariance of W_{ω}^{0}(0).

This bound is simple and the growth as L → ∞ is not big. However this does not decay as |m|_{∞} → ∞. To obtain the decay, we give another bound by using the spectral shift function as in [6], [7] and [20]:

Lemma 2.6. For any ℓ ∈ ℕ such that ℓ > q/2, there exists a finite constant c independent of L, m and κ such that

sup_{−3κ/2≤t≤3κ/2} |I(L, E, t; ι, m)| (2.17)

≤ c|E|^{−(q−d)/2}(|E|^{−q/(4ℓ)} + 1) (\frac{L^{s−d/2}}{κ|m|_{∞}^{s}} )^{q/(2ℓ)} L^{d}

for any 1 ≤ ι ≤ 2r, m ∈ ℤ^{d} and 0 < κ < 1/2.

Proof. (2.14) is rewritten as

I(L, E, t; ι, m)
= E[(1 − (W_{ω}^{ι,m})^{2}) Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t)
−ρ_{κ}(Γ_{ι,m}(ω, L, E) − 1 + t)]],

where Γ_{ι,m}(ω, L, E) is the operator obtained by replacing W_{ω}^{ι,m} by 0 in Γ(ω, L, E). In fact, E[1 − (W_{ω}^{ι,m})^{2}] = 0 and Γ_{ι,m}(ω, L, E) is statistically independent of W_{ω}^{ι,m}. By Lemma 2.7 below and Theorem 2.1 in [6], [7] and [20], we have the spectral shift function ξ(λ : Γ(ω, L, E)^{ℓ}, Γ_{ι,m}(ω, L, E)^{ℓ}) for the pair (Γ(ω, L, E)^{ℓ}, Γ_{ι,m}(ω, L, E)^{ℓ}) such that

∥ξ(· : Γ(ω, L, E)^{ℓ}, Γ_{ι,m}(ω, L, E)^{ℓ})∥_{2ℓ/q}
≤ |||Γ(ω, L, E)^{ℓ} − Γ_{ι,m}(ω, L, E)^{ℓ}|||_{q/(2ℓ)}^{q/(2ℓ)}
≤ {c_{1}|E|^{−ℓ(q−d)/q}(|E|^{−1/2} + 1)Q(ω, L, ι, m)}^{q/(2ℓ)}.

As in [6], [7] and [20], we apply the Birman-Krein identity [2], [41] as follows:

Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t) − ρ_{κ}(Γ_{ι,m}(ω, L, E) − 1 + t)]

= ∫_{ℝ} \{\frac{∂}{∂λ}ρ_{κ}(λ^{1/ℓ} − 1 + t)\}ξ(λ : Γ(ω, L, E)^{ℓ}, Γ_{ι,m}(ω, L, E)^{ℓ})dλ.

This is estimated as

| Tr[ρ_{κ}(Γ(ω, L, E) − 1 + t) − ρ_{κ}(Γ_{ι,m}(ω, L, E) − 1 + t)]| ≤ {c_{1}|E|^{−ℓ(q−d)/q}(|E|^{−1/2} + 1)Q(ω, L, ι, m)}^{q/(2ℓ)}

× (∫_{ℝ} |\frac{∂}{∂λ}ρ_{κ}(λ^{1/ℓ} − 1 + t)|^{2ℓ/(2ℓ−q)}dλ)^{(2ℓ−q)/(2ℓ)}

Since

(∫_{ℝ} |\frac{∂}{∂λ}ρ_{κ}(λ^{1/ℓ} − 1 + t)|^{2ℓ/(2ℓ−q)}dλ)^{(2ℓ−q)/(2ℓ)}

≤ c_{2} (∫_{ℝ} |\frac{∂}{∂λ}ρ_{κ}(λ − 1 + t)|^{2ℓ/(2ℓ−q)}dλ)^{(2ℓ−q)/(2ℓ)}

≤ \frac{c_{3}}{κ^{q/(2ℓ)}} (∫_{ℝ} ( − \frac{∂}{∂λ}ρ_{κ})(λ − 1 + t)dλ)^{(2ℓ−q)/(2ℓ)}

= c_{3}/κ^{q/(2ℓ)}

and

E[|1 − (W_{ω}^{ι,m})^{2}|Q(ω, L, ι, m)^{q/(2ℓ)}] ≤ c_{4} (\frac{L^{s−d/2}}{κ|m|_{∞}^{s}} )^{q/(2ℓ)} ,

Wegner estimates 19

we obtain (2.17).

We next show the estimate of the difference of the operators used in the proof of the preceding lemma:

Lemma 2.7. For any ℓ ∈ ℕ, the operator Γ(ω, L, E)^{ℓ} − Γ_{ι,m}(ω, L, E)^{ℓ} belongs to the super trace class I_{q/(2ℓ)} and satisfies

|||Γ(ω, L, E)^{ℓ} − Γ_{ι,m}(ω, L, E)^{ℓ}|||_{q/(2ℓ)}

≤ c|E|^{−ℓ(q−d)/q}(|E|^{−1/2} + 1)Q(ω, L, ι, m),

where c is a finite constant, Q(ω, L, ι, m)

:= \frac{L^{s−d/2}}{|m|_{∞}^{s}} |W_{ω}^{ι,m}|L^{d/q}(∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q} + ∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q} + L^{d/q})

×(∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2} + ∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2}

+∥χ̃_{L} ∑_{k=0}^{n−1} W_{ω}^{0}(x)^{n−1+k}W_{ω,ι,m}^{0}(x)^{k}∥_{q/2} + L^{2d/q})

×(∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2} + ∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2} + L^{2d/q})^{ℓ−2}

and W_{ω,ι,m}^{0}(x) is the random field obtained by replacing W_{ω}^{ι,m} by 0 in W_{ω}^{0}(x).

Proof. For simplicity, we treat only the case that V (x, w) = 0. We divide the operator as

Γ(ω, L, E)^{ℓ} − Γ_{ι,m}(ω, L, E)^{ℓ} = ∑_{j=0}^{ℓ} I_{j} + ∑_{j=1}^{ℓ} J_{j},

where

I_{0} := (Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}{(Ĥ_{L}^{ω} − E)^{−1}χ̃_{L}W_{ω}^{0}(x)^{n}}^{ℓ−1}

×((Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}),

I_{j} := (Ĥ_{L}^{ω} − E)^{−1/2}{χ̃_{L}W_{ω}^{0}(x)^{n}(Ĥ_{L}^{ω} − E)^{−1}}^{ℓ−1−j}χ̃_{L}W_{ω}^{0}(x)^{n}

×((Ĥ_{L}^{ω} − E)^{−1} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1})χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}

×\{({ \atop H_{L,ι,m}^{ω}} − E)^{−1}χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}\}^{j−1}({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}

for 1 ≤ j ≤ ℓ − 1,

I_{ℓ} := ((Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2})\{χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}

×({ \atop H_{L,ι,m}^{ω}} − E)^{−1}\}^{ℓ−1}χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}

J_{j} := (Ĥ_{L}^{ω} − E)^{−1/2}{χ̃_{L}W_{ω}^{0}(x)^{n}(Ĥ_{L}^{ω} − E)^{−1}}^{ℓ−j}

×χ̃_{L}(W_{ω}^{0}(x)^{n} − W_{ω,ι,m}^{0}(x)^{n})\{({ \atop H_{L,ι,m}^{ω}} − E)^{−1}χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}\}^{j−1}

×({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2} for 1 ≤ j ≤ ℓ

and { \atop H_{L,ι,m}^{ω}} is the operator obtained by replacing W_{ω}^{ι,m} by 0 in Ĥ_{L}^{ω}. By the Hölder inequality for the norms ||| · |||_{p} (cf. Theorem 2.8 in [33]), we have

|||I_{0}|||_{q/(2ℓ)}

≤ |||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q} × |||(Ĥ_{L}^{ω} − E)^{−1}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q/2}^{ℓ−1}

×|||(Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}|||_{q}.

By Lemma 2.1 (i) and (iv), we have

|||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q} ≤ c_{1}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q}/|E|^{(q−d)/(2q)}

and

|||(Ĥ_{L}^{ω} − E)^{−1}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q/2} ≤ c_{2}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}/|E|^{(q−d)/q}.

By the expression

(Ĥ_{L}^{ω} − E)^{−1/2} = \frac{1}{\sqrt{π}} ∫_{0}^{∞} \frac{dt}{\sqrt{t}}e^{tE} exp(−tĤ_{L}^{ω})

((A.22) in [21]) and the Du Hamel formula (cf. [4]), we have

(Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}

= \frac{1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du exp(−utĤ_{L}^{ω})

× ∑_{k=1}^{d}\{(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x))) (2.18)

×χ̃_{L}(A_{k}(x, \overline{W_{ω,ι,m}}(x)) − A_{k}(x, \overline{W_{ω}}(x))) +χ̃_{L}(A_{k}(x, \overline{W_{ω,ι,m}}(x)) − A_{k}(x, \overline{W_{ω}}(x)))

×(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω,ι,m}}(x)))\} exp(−(1 − u)t{ \atop H_{L}^{ω,ι,m}}).

Wegner estimates 21

Since ∥∇_{w}A(x, w)∥_{∞} < ∞, this is estimated as

|||(Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}|||_{q}

≤ c_{3}|W_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du

× ∑_{k=1}^{d}(||| exp(−utĤ_{L}^{ω})(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x)))|||

×|||χ̃_{L} exp(−(1 − u)t{ \atop H_{L,ι,m}^{ω}})|||_{q}

+|||χ̃_{L} exp(−utĤ_{L}^{ω})|||_{q}|||(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x)))

× exp(−(1 − u)t{ \atop H_{L,ι,m}^{ω}})|||).

Since

||| exp(−utĤ_{L}^{ω})(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x)))||| ≤ 1/\sqrt{ut}

and

|||χ̃_{L} exp(−utĤ_{L}^{ω})|||_{q}
≤ |||χ̃_{L}(Ĥ_{L}^{ω} − E)^{−1/2}|||_{q}|||(Ĥ_{L}^{ω} − E)^{1/2} exp(−utĤ_{L}^{ω})|||
≤ c_{4}L^{d/q}\sqrt{−E + 1/(ut)},

we have

|||(Ĥ_{L}^{ω} − E)^{−1/2} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1/2}|||_{q}

≤ c_{5}|W_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} L^{d/q}/|E|^{1−d/(2q)}

and |||I_{0}|||_{q/(2ℓ)}

≤ c_{6}|W_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} L^{d/q}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q}

×∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{ℓ−1}/|E|^{ℓ(q−d)/q+1/2}.

|||I_{ℓ}|||_{q/(2ℓ)} is estimated similarly. By

(Ĥ_{L}^{ω} − E)^{−1} − ({ \atop H_{L,ι,m}^{ω}} − E)^{−1}

= (Ĥ_{L}^{ω} − E)^{−1} ∑_{k=1}^{d}\{(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x)))

×χ̃_{L}(A_{k}(x, \overline{W_{ω,ι,m}}(x)) − A_{k}(x, \overline{W_{ω}}(x))) +χ̃_{L}(A_{k}(x, \overline{W_{ω,ι,m}}(x)) − A_{k}(x, \overline{W_{ω}}(x)))

×(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω,ι,m}}(x)))\}({ \atop H_{L,ι,m}^{ω}} − E)^{−1}

22 Naomasa Ueki

and

|||(Ĥ_{L}^{ω} − E)^{−1/2}(i∂_{k} + χ̃_{L}A_{k}(x, \overline{W_{ω}}(x)))||| ≤ 1,

we have |||I_{j}|||_{q/(2ℓ)}

≤ c_{7}|W_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} L^{d/q}\{∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{ℓ−1−j}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q}∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2}^{j}

+∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{ℓ−j}∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q}∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2}^{j−1}}/|E|^{ℓ(q−d)/q+1/2}. It is also easy to show

|||J_{j}|||_{q/(2ℓ)}

≤ c_{8}|X_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} L^{d/q}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q}∥χ̃_{L}W_{ω}^{0}(x)^{n}∥_{q/2}^{ℓ−1−j}

×∥ ∑_{k=1}^{n} χ̃_{L}W_{ω}^{0}(x)^{n−k}W_{ω,ι,m}^{0}(x)^{k−1}∥_{q/2}∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2}^{j−1}/|E|^{ℓ(q−d)/q}

for 1 ≤ j ≤ ℓ − 1 and |||J_{ℓ}|||_{q/(2ℓ)}

≤ c_{9}|W_{ω}^{ι,m}|\frac{L^{s−d/2}}{|m|_{∞}^{s}} L^{d/q}∥ ∑_{k=1}^{n} χ̃_{L}W_{ω}^{0}(x)^{n−k}W_{ω,ι,m}^{0}(x)^{k−1}∥_{q/2}

×∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q/2}^{ℓ−2}∥χ̃_{L}W_{ω,ι,m}^{0}(x)^{n}∥_{q}/|E|^{ℓ(q−d)/q},

where a_{3}(ℓ, q, d) is a number determined by ℓ, q and d.

To complete the proof of Theorem 2.1, we apply Lemma 2.5 to the terms with |m|_{∞} ≤ L^{1−d/(2s)}/κ^{1/s} and Lemma 2.6 to other terms in (2.12). Then we obtain (2.4).

§2.3. Proof of Lemma 2.3 (i) As in (2.18), we have

(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2} − (Ĥ_{L}^{ω} − E)^{−1/2}

= \frac{1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du exp(−utĤ_{L}^{ω}(ε; ι, m))

× ∑_{j=1}^{d}\{(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x, ε; ι, m))) (2.19)

×χ̃_{L}(A_{j}(x, \overline{W_{ω}}(x)) − A_{j}(x, \overline{W_{ω}}(x, ε; ι, m)))
+χ̃_{L}(A_{j}(x, \overline{W_{ω}}(x)) − A_{j}(x, \overline{W_{ω}}(x, ε; ι, m)))
×(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x)))\} exp(−(1 − u)tĤ_{L}^{ω}).

Using

A_{j}(x, \overline{W_{ω}}(x, ε; ι, m)) − A_{j}(x, \overline{W_{ω}}(x)) = ε\overline{e_{ι,m}}(x) · ∫_{0}^{1} dv∇_{w}A_{j}(x, \overline{W_{ω}}(x, εv; ι, m)) (2.20)

= ε\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x))

+ε\overline{e_{ι,m}}(x) · ∫_{0}^{1} dv\{∇_{w}A_{j}(x, \overline{W_{ω}}(x, εv; ι, m)) − ∇_{w}A_{j}(x, \overline{W_{ω}}(x))\},

we expand as

(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2} − (Ĥ_{L}^{ω} − E)^{−1/2}

= εΞ(ω, L, E; ι, m) + ∑_{j=1}^{3} R_{j}(ε, ω, L, E; ι, m),

24 Naomasa Ueki

where R_{1}(ε, ω, L, E; ι, m)

:= \frac{−ε}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du exp(−utĤ_{L}^{ω}(ε; ι, m))

× ∑_{j=1}^{d} \{(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x, ε; ι, m)))

×χ̃_{L}\overline{e_{ι,m}}(x) · ∫_{0}^{1} dv\{∇_{w}A_{j}(x, \overline{W_{ω}}(x, εv; ι, m)) − ∇_{w}A_{j}(x, \overline{W_{ω}}(x))\} +χ̃_{L}\overline{e_{ι,m}}(x) · ∫_{0}^{1} dv\{∇_{w}A_{j}(x, \overline{W_{ω}}(x, εv; ι, m)) − ∇_{w}A_{j}(x, \overline{W_{ω}}(x))\}

×(i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x)))\}

× exp(−(1 − u)tĤ_{L}^{ω}), R_{2}(ε, ω, L, E; ι, m)

:= \frac{−ε^{2}}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du exp(−utĤ_{L}^{ω}(ε; ι, m))

× ∑_{j=1}^{d} χ̃_{L}^{2}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x))

× ∫_{0}^{1} dv\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x, εv; ι, m)) exp(−(1 − u)tĤ_{L}^{ω}) and

R_{3}(ε, ω, L, E; ι, m)

:= \frac{−ε}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du(exp(−utĤ_{L}^{ω}(ε; ι, m)) − exp(−utĤ_{L}^{ω}))

× ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)), i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}

× exp(−(1 − u)tĤ_{L}^{ω}).

Since supp χ̃_{L} is compact, for any φ and ψ ∈ L^{2}(ℝ^{d}), we have |(φ, R_{1}(ε, ω, L, E; ι, m)ψ)| ≤ o(ε) ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1} du\{∥(i∇ + χ̃_{L}A(x, \overline{W_{ω}}(x, ε; ι, m)))

× exp(−utĤ_{L}^{ω}(ε; ι, m))φ∥_{2}∥ψ∥_{2}

+∥φ∥_{2}∥(i∇ + χ̃_{L}A(x, \overline{W_{ω}}(x))) exp(−(1 − u)tĤ_{L}^{ω})ψ∥_{2}\}.

Since

∥(i∇ + χ̃_{L}A(x, \overline{W_{ω}}(x))) exp(−tĤ_{L}^{ω})ψ∥_{2}
= (exp(−tĤ_{L}^{ω})ψ, Ĥ_{L}^{ω} exp(−tĤ_{L}^{ω})ψ)^{1/2}
≤ (2te)^{−1/2}∥ψ∥_{2},

we have

(φ, R_{j}(ε, ω, L, E; ι, m)ψ) = o(ε) (2.21)

for j = 1. It is also easy to show (2.21) for j = 2. For R_{3}(ε, ω, L, E; ι, m), we use the Du Hamel formula again and apply similar estimates. Then we have (2.21) for j = 3. By the same method, we can show that Ξ(ω, L, E; ι, m) is a bounded operator.

(ii) It is enough to show that Ĥ_{L}^{ω}Ξ(ω, L, E; ι, m)E((a, ∞) : Γ(ω, L, E)) is a bounded operator for any a > 0. In this proof, we write E^{j} := χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)) and A_{j} := χ̃_{L}A_{j}(x, \overline{W_{ω}}(x)), for simplicity. Since ∇_{w}A(x, w) is a C^{2} function of (x, w) and \overline{W_{ω}}(x) is a C^{2} function of x by Remark 2.3, E^{j} and A^{j} are C^{2} functions of x. Therefore, if we write [A, B]_{−} = AB − BA for any operators A and B, then we have

[Ĥ_{L}^{ω}, ∑_{j=1}^{d}[E^{j}, i∂_{j} + A_{j}]_{+}]_{−}

= ∑_{j,k=1}^{d}(i∂_{j} + A_{j})E_{(j,k)}(i∂_{k} + A_{k}) + ∑_{k=1}^{d} E_{(k)}(i∂_{k} + A_{k}) + E_{(0)},

where E_{(j,k)}, E_{(k)} and E_{(0)} are continuous functions with compact support. By using this, we divide as follows:

Ĥ_{L}^{ω}Ξ(ω, L, E; ι, m) = ∑_{j=1}^{7} H_{j},

where

H_{1} := \frac{−2}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1/2} du exp(−utĤ_{L}^{ω}) ∑_{j=1}^{d}(i∂_{j} + A_{j})E^{j}Ĥ_{L}^{ω}

× exp(−(1 − u)tĤ_{L}^{ω}),

H_{2} := \frac{1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1/2} du exp(−utĤ_{L}^{ω}) ∑_{j=1}^{d}(i∂_{j}E^{j})Ĥ_{L}^{ω}

× exp(−(1 − u)tĤ_{L}^{ω}),

H_{3} := \frac{−1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1/2} du exp(−utĤ_{L}^{ω})

× ∑_{j,k=1}^{d}(i∂_{j} + A_{j})E_{(j,k)}(i∂_{k} + A_{k}) exp(−(1 − u)tĤ_{L}^{ω}),

H_{4} := \frac{−1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1/2} du exp(−utĤ_{L}^{ω}) ∑_{k=1}^{d} E_{(k)}(i∂_{k} + A_{k})

× exp(−(1 − u)tĤ_{L}^{ω}),

H_{5} := \frac{−1}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{0}^{1/2} du exp(−utĤ_{L}^{ω})E_{(0)} exp(−(1 − u)tĤ_{L}^{ω}),

H_{6} := \frac{−2}{\sqrt{π}} ∫_{0}^{∞} dt\sqrt{t}e^{tE} ∫_{1/2}^{1} duĤ_{L}^{ω} exp(−utĤ_{L}^{ω}) ∑_{j=1}^{d} E^{j}(i∂_{j} + A_{j})

× exp(−(1 − u)tĤ_{L}^{ω})

and

H_{7} := \frac{−1}{\sqrt{π}} ∫_{1/2}^{∞} dt\sqrt{t}e^{tE} ∫_{1/2}^{1} duĤ_{L}^{ω} exp(−utĤ_{L}^{ω})

×( ∑_{j=1}^{d} i∂_{j}E^{j}) exp(−(1 − u)tĤ_{L}^{ω}).

As in the proof of (i), it is easy to show that H_{2}, H_{3}, H_{4}, H_{5} and H_{7} are bounded operators.

For H_{1}, we estimate as

|(φ, exp(−utĤ_{L}^{ω}) ∑_{j=1}^{d}(i∂_{j} + A_{j})E^{j}Ĥ_{L}^{ω} exp(−(1 − u)tĤ_{L}^{ω})ψ)|

≤ ∥(i∇ + A) exp(−utĤ_{L}^{ω})φ∥_{2}∥E^{·}∥_{∞}∥Ĥ_{L}^{ω} exp(−(1 − u)tĤ_{L}^{ω})ψ∥_{2}, and

∥(i∇ + A) exp(−utĤ_{L}^{ω})φ∥_{2} ≤ c_{1}(ut)^{−1/2}∥φ∥_{2}.

It is also easy to see that

∥Ĥ_{L}^{ω} exp(−(1 − u)tĤ_{L}^{ω})ψ∥_{2} ≤ c_{2}((1 − u)t)^{−1}∥ψ∥_{2}.

However this bound is too big to integrate with respect to the variable t. We now take φ from the range of E((a, ∞) : Γ(ω, L, E)). For this we have

∥(Ĥ_{L}^{ω} − E)^{1/2}ψ∥_{2} = ∥χ̃_{L}X_{ω}^{0}(x)^{n}(Ĥ_{L}^{ω} − E)^{1/2}Γ(ω, L, E)^{−1}ψ∥_{2}
≤ c_{3}∥Γ(ω, L, E)^{−1}ψ∥_{2} ≤ c_{3}∥ψ∥_{2}/a.

Then we have

∥(Ĥ_{L}^{ω} − E) exp(−(1 − u)tĤ_{L}^{ω})ψ∥_{2} ≤ c_{4}(((1 − u)t)^{−1/2} + 1)∥ψ∥_{2}/a

and

∥Ĥ_{L}^{ω} exp(−(1 − u)tĤ_{L}^{ω})ψ∥_{2} ≤ c_{4}(((1 − u)t)^{−1/2} + 1)∥ψ∥_{2}/a + |E|∥ψ∥_{2}.

Therefore H_{1}E((a, ∞) : Γ(ω, L, E)) is bounded. Similarly H_{6}E((a, ∞) : Γ(ω, L, E)) is bounded.

(iii) We take ψ ∈ L^{2}(ℝ^{d}) and an element ϕ of the domain of Ĥ_{L}^{ω}. We first show that

(ψ, ((Ĥ_{L}^{ω} − E)^{1/2} − (Ĥ_{L}^{ω}(ε; ι, m) − E)^{1/2})ϕ) = O(ε) (2.22)

as ε → 0. For this, we divide the left hand side as

(ψ, ((Ĥ_{L}^{ω} − E)^{−1/2} − (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2})(Ĥ_{L}^{ω} − E)ϕ)
+(ψ, ((Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}((Ĥ_{L}^{ω} − E) − (Ĥ_{L}^{ω}(ε; ι, m) − E))ϕ).

By (i), the first term is O(ε). The second term is dominated by

ε∥ψ∥_{2}(∥(i∇ + χ̃_{L}A(x, \overline{W_{ω}}(x)))ϕ∥_{2} + ∥ϕ∥_{2}).

Therefore we obtain (2.22).

We take φ ∈ ⋃_{a>0} Ran E([a, ∞] : Γ(ω, L, E)). By (i), (ii) and (2.22), we have

(ψ, [(Ĥ_{L}^{ω} − E)^{1/2}, Ξ(ω, L, E; ι, m)]_{+}φ)

= \frac{1}{ε}(ψ, ((Ĥ_{L}^{ω}(ε; ι, m) − E)^{1/2}((Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2} − (Ĥ_{L}^{ω} − E)^{−1/2})

+((Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2} − (Ĥ_{L}^{ω} − E)^{−1/2})(Ĥ_{L}^{ω} − E)^{1/2})φ) + O(ε).

Since (Ĥ_{L}^{ω} − E)^{1/2}(Ĥ_{L}^{ω} − E)^{−1/2} = (Ĥ_{L}^{ω} − E)^{−1/2}(Ĥ_{L}^{ω} − E)^{1/2} = 1, the right hand side is equal to

\frac{−1}{ε}(ψ, ((Ĥ_{L}^{ω}(ε; ι, m) − E)^{1/2} − (Ĥ_{L}^{ω} − E)^{1/2})(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}

+(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}((Ĥ_{L}^{ω}(ε; ι, m) − E)^{1/2} − (Ĥ_{L}^{ω} − E)^{1/2})φ) +O(ε)

= \frac{−1}{ε}(ψ, (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}(Ĥ_{L}^{ω}(ε; ι, m) − Ĥ_{L}^{ω})

×(Ĥ_{L}^{ω} − E)^{−1/2}φ) + O(ε).

By (i) and (ii), we see that this is equal to

−(ψ, (Ĥ_{L}^{ω} − E)^{−1} ∑_{j=1}^{d}[χ̃_{L}\overline{e_{ι,m}}(x) · ∇_{w}A_{j}(x, \overline{W_{ω}}(x)),

i∂_{j} + χ̃_{L}A_{j}(x, \overline{W_{ω}}(x))]_{+}(Ĥ_{L}^{ω} − E)^{−1/2}φ) + o(1).

(iv) As in [35], by the Fourier transform and the Du Hamel formula, we write as

f(Γ(ω, L, E, ε; ι, m)) − f(Γ(ω, L, E))

= ∫_{ℝ} dζf̂′(ζ) ∫_{0}^{1} du exp(u2πiζΓ(ω, L, E, ε; ι, m)) (2.23)

×(Γ(ω, L, E, ε; ι, m) − (Γ(ω, L, E, ε)) exp((1 − u)2πiζΓ(ω, L, E)).

By (i), we have

Γ(ω, L, E, ε; ι, m) − Γ(ω, L, E, ε) = εΓ′(ω, L, E; ι, m) + o(ε),

where

Γ′(ω, L, E; ι, m)
:= (Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}Ξ(ω, L, E; ι, m)
+(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}ne_{ι,m}^{0}(x)W_{ω}^{0}(x)^{n−1}(Ĥ_{L}^{ω} − E)^{−1/2}
+Ξ(ω, L, E; ι, m)χ̃_{L}W_{ω}^{0}(x)^{n}(Ĥ_{L}^{ω} − E)^{−1/2}.

Then we obtain

f(Γ(ω, L, E, ε; ι, m)) − f(Γ(ω, L, E))

= ε ∫_{ℝ} dζf̂′(ζ) ∫_{0}^{1} du exp(u2πiζ(Γ(ω, L, E)) (2.24)

×Γ′(ω, L, E; ι, m) exp((1 − u)2πiζ(Γ(ω, L, E)) + o(ε).

Wegner estimates 29

We expand the trace as

Tr[f^{q}(Γ(ω, L, E, ε; ι, m))] − Tr[f^{q}(Γ(ω, L, E))] = ∑_{k=1}^{q} I_{k},

where

I_{k} := ∑_{1≤ℓ_{1}<···<ℓ_{k}≤q} Tr[f(Γ(ω, L, E)) ⋯

×(f(Γ(ω, L, E, ε; ι, m)) −^{⌣^{ℓ_{1}}} f(Γ(ω, L, E)))

× ⋯ (f(Γ(ω, L, E, ε; ι, m)) −^{⌣^{ℓ_{k}}} f(Γ(ω, L, E))) ⋯ f(Γ(ω, L, E))]

Since supp f ⊂ (0, ∞), the rank of f(Γ(ω, L, E)) is finite. Thus, using (2.24), we have

I_{1} = q Tr[{f(Γ(ω, L, E; ι, m)) − f(Γ(ω, L, E))}f^{q−1}(Γ(ω, L, E))]
= ε Tr[(f^{q})′(Γ(ω, L, E))Γ′(ω, L, E; ι, m)] + o(ε).

By changing the order of the operators, we have Tr[(f^{q})′(Γ(ω, L, E))Γ′(ω, L, E; ι, m)] = Tr[(f^{q})′(Γ(ω, L, E))(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}ne_{ι,m}^{0}(x)W_{ω}^{0}(x)^{n−1}(Ĥ_{L}^{ω} − E)^{−1/2}]

+ Tr[(f^{q})′(Γ(ω, L, E))Γ(ω, L, E)[(Ĥ_{L}^{ω} − E)^{1/2}, Ξ(ω, L, E; ι, m)]_{+}]. By using also (iii), we obtain

I_{1} = ε Tr[(f^{q})′(Γ(ω, L, E))Γ(ω, L, E; ι, m)] + o(ε).

Similarly we obtain

I_{k} = O(ε^{k})

for 2 ≤ k ≤ q − 1. By (2.23), we have

I_{q} ≤ ∥f̂′∥_{1}^{q}|||Γ(ω, L, E, ε; ι, m) − Γ(ω, L, E)|||_{q}^{q}.

We divide the second factor as |||Γ(ω, L, E, ε; ι, m) − Γ(ω, L, E)|||_{q}

≤ |||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q}
×|||(Ĥ_{L}^{ω} − E)^{−1/2} − (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}|||
+|||(Ĥ_{L}^{ω} − E)^{−1/2} − (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}|||
×|||χ̃_{L}W_{ω}^{0}(x, ε; ι, m)^{n}(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}|||_{q}.

By Lemma 2.1 (i), we have

|||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}W_{ω}^{0}(x)^{n}|||_{q} = O(1),
|||(Ĥ_{L}^{ω} − E)^{−1/2}χ̃_{L}(W_{ω}^{0}(x)^{n} − W_{ω}^{0}(x, ε; ι, m)^{n})|||_{q} = O(ε)

and

|||χ̃_{L}W_{ω}^{0}(x, ε; ι, m)^{n}(Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}|||_{q} = O(1).

By (2.19) and (2.20), we have

|||(Ĥ_{L}^{ω} − E)^{−1/2} − (Ĥ_{L}^{ω}(ε; ι, m) − E)^{−1/2}||| = O(ε).

Therefore we obtain

|||Γ(ω, L, E, ε; ι, m) − Γ(ω, L, E)|||_{q} = O(ε)

and

I_{q} = O(ε^{q}).

Lemma 2.3 is now proven.

§3. Bootstrap multiscale analysis and localization

In this section we modify the theory in Germinet and Klein [14] so that the operator (2.5) can be treated. For this we weaken the assumptions on the Wegner estimate and the semiboundedness of the operator in [14]. In this section we treat a random operator

H^{ω} := ∑_{j=1}^{d}(i∂_{j} + A_{j}^{ω}(x))^{2} + V^{ω}(x) (3.1)

with a general random field (A^{ω}(x), V^{ω}(x)). We do not use the specific form of (A^{ω}(x), V^{ω}(x)) in the last section. The application to the operator (2.5) is discussed in Subsection 3.4 below. As in [14], we write C = C(a, b, …) if C is a positive finite constant depending only on the parameters a, b, . . ..

§3.1. Main theorems

We first assume the following regularity and stationarity:

(R) For a. e. ω, V^{ω} ∈ L_{loc}^{2}(ℝ^{d}) and A^{ω} = A^{(1),ω} + A^{(2),ω}, where A^{(1),ω} and A^{(2),ω} are satisfying the following:

(i) A^{(1),ω} ∈ L_{loc}^{4}(ℝ^{d} → ℝ^{d}) and ∇ · A^{(1),ω} ∈ L_{loc}^{2}(ℝ^{d});

(ii) A^{(2),ω} ∈ C(ℝ^{d} → ℝ^{d}), ∇ · A^{(2),ω} ∈ L_{loc}^{2}(ℝ^{d}) and dA^{(2),ω} ∈ C(ℝ^{d} → Λ^{2}(ℝ^{d})), where d is the exterior derivative of A^{(2),ω} regarded as the 1-form ∑_{j=1}^{d} A_{j}^{(2),ω}(x)dx_{j} and Λ^{2}(ℝ^{d}) is the 2-fold exterior power of ℝ^{d}.

(S) The random field (A^{(1),ω}(x), dA^{(2),ω}(x), V^{ω}(x)) is stationary with respect to the shift in the variable x ∈ ℤ^{d}.

As in [14], we assume the independence at distance:

(IAD) There exists ρ > 0 such that for any finite number of bounded subsets B_{1}, B_{2}, …, B_{m} of ℝ^{d} with dist(B_{i}, B_{j}) > ρ for i ≠ j, the random fields {(A^{(1),ω}(x), dA^{(2),ω}(x), V^{ω}(x)) : x ∈ B_{i}}, i = 1, 2, …, m, are independent.

Referring Lemma 3.5 below, we assume that a Fernique type estimate holds: (F) There exist finite positive constants β, δ_{1} and δ_{2} such that

(3.2) P ( ess sup_{x∈Λ_{L}} V_{−}^{ω}(x) > η) ≤ δ_{1} exp(−δ_{2}η^{β}/ log L)

for any L ≥ 2 and η ≥ 0, where V_{−}^{ω}(x) := max{−V^{ω}(x), 0}.

Under these assumptions the operator H^{ω} in (3.1) is essentially self-adjoint on C_{0}^{∞}(ℝ^{d}) by Lemma 3.1 below (cf. Leinfelder and Simader [29]).

Moreover, since the bound in Theorem 2.1 is not linear in the variable of energy, we assume that the Wegner estimate holds in a weaker form than that of [14]. For any L > 0, let H_{L}^{ω} be the self-adjoint operator on L^{2}(Λ_{L}) defined by the restriction of H^{ω} with the Dirichlet boundary condition. In the following we fix an open interval Ĩ_{0} and assume the following:

(W) There exist 1 ≤ b < ∞, 0 < h ≤ 1 and 0 < QĨ_{0} < ∞ such that

P (dist(σ(H_{L}^{ω}), E) ≤ η) ≤ QĨ_{0}η^{h}L^{bd}

for any E ∈ Ĩ_{0}, η > 0 and L ∈ 2ℕ.

For any x ∈ ℝ^{d} and L > 0, let Λ_{L}(x) be the open box of side L centered at x. Let H_{x,L}^{ω} be the self-adjoint operator on L^{2}(Λ_{L}(x)) defined by the restriction of H^{ω} with the Dirichlet boundary condition. Let χ_{x,L} and Γ_{x,L} be the characteristic functions of Λ_{L}(x) and \overline{Λ_{L−1}(x)} − Λ_{L−3}(x), respectively. We use the following definitions in [14]:

Definition 3.1. Let θ > 0, E ∈ ℝ, x ∈ ℤ^{d} and L ∈ 6ℕ. A box Λ_{L}(x) is called (θ, E)-suitable for ω if E ∉ σ(H_{x,L}^{ω}) and

|||Γ_{x,L}(H_{x,L}^{ω} − E)^{−1}χ_{x,L/3}||| ≤ L^{−θ}.

Definition 3.2. Let m > 0, E ∈ ℝ, x ∈ ℤ^{d} and L ∈ 6ℕ. A box Λ_{L}(x) is called (m, E)-regular for ω if E ∉ σ(H_{x,L}^{ω}) and

|||Γ_{x,L}(H_{x,L}^{ω} − E)^{−1}χ_{x,L/3}||| ≤ exp(−mL/2).

We fix a compact interval I_{0} included in Ĩ_{0} and we set M_{I_{0}} := max I_{0}. Then the bootstrap multiscale analysis in [14] is modified as follows:

Theorem 3.1 (Bootstrap Multiscale Analysis, cf. Theorem 3.4 in [14]). For any θ > bd/h, there exists \overline{L_{θ}} = \overline{L_{θ}}(d, ρ, M_{I_{0}}, QĨ_{0}, β, δ_{1}, δ_{2}, b, h, θ) ∈ 6ℕ satisfying the following: if there exist E_{0} ∈ I_{0} and \overline{L_{θ}} ≤ L ∈ 6ℕ such that

(3.3) P (Λ_{L} is (θ, E_{0}) − suitable) > 1 − 841^{−d},

then there exists δ_{0} = δ_{0}(d, ρ, M_{I_{0}}, QĨ_{0}, β, δ_{1}, δ_{2}, θ, L) > 0 such that, for any 0 < ζ < 1 and 1 < α < ζ^{−1}, there are L_{0} = L_{0}(d, ρ, M_{I_{0}}, QĨ_{0}, β, δ_{1}, δ_{2}, θ, L, ζ, α) ∈ 6ℕ and m_{ζ} = m_{ζ}(ζ, L_{0}) > 0 satisfying

P (R(m_{ζ}, L_{k}, I_{0}(E_{0}, δ_{0}), x, y)) > 1 − exp(−L_{k}^{ζ})

for any k ∈ ℤ_{+} and x, y ∈ ℤ^{d} with |x−y|_{∞} > L_{k} +2ρ, where L_{k+1} = [L_{k}^{α}]_{6ℕ} := max{N ∈ 6ℕ : N ≤ L_{k}^{α}}, I_{0}(E_{0}, δ_{0}) := [E_{0} − δ_{0}, E_{0} + δ_{0}] ∩ I_{0} and

R(m, L, I, x, y)

:= {ω : for every E ∈ I, either Λ_{L}(x) or Λ_{L}(y) is (m, E)-regular}.

As in [14], the following is obtained from Theorem 3.1 (cf. Theorem 3.8 in [14]):

Theorem 3.2 (Decay of the kernel, cf. Theorem 3.8 in [14]). Let \overline{L_{θ}} and I_{0}(E_{0}, δ_{0}) be the number and the interval given in Theorem 3.1. If there exists E_{0} ∈ I_{0} such that (3.3) holds for some L ≥ \overline{L_{θ}} and θ > bd/h, then, for any 0 < ζ < 1, there exists a finite constant C_{ζ} = C_{ζ}(ζ, d, ρ, M_{I_{0}}, QĨ_{0}, β, δ_{1}, δ_{2}, b, h, θ) such that

E[ sup_{f∈B_{1}(ℝ)} |||χ_{x}f(H^{ω})E(I_{0}(E_{0}, δ_{0}) : H^{ω})χ_{y}|||_{2}^{2}]

≤ C_{ζ} exp(−|x − y|_{∞}^{ζ})

for any x, y ∈ ℤ^{d}, where | · |_{∞} is the maximal norm, χ_{x} is the operator of the multiplication with the characteristic function of the open box Λ_{1}(x) with side 1 centered at x and B_{1}(ℝ) is the set of all real valued Borel functions f on ℝ with sup |f| ≤ 1.

From this theorem, we obtain the strong dynamical localization in the Hilbert-Schmidt norm

E[ sup_{t∈ℝ} ||||X|^{n/2}E(I_{0}(E_{0}, δ_{0}) : H^{ω}) exp(−itH^{ω})χ_{0}|||_{2}^{2}] < ∞ (3.4)

for any n ≥ 0 (cf. Corollary 3.10 in [14]) and the strong dynamical localization in the operator norm

E[ sup_{t∈ℝ} ||||X|^{n/2}E(I_{0}(E_{0}, δ_{0}) : H^{ω}) exp(−itH^{ω})χ_{0}|||] < ∞ (3.5)

for any n ≥ 0 (cf. (1.8) in [14]), where |X|^{n/2} is the operators of multiplication with |x|^{n/2}. The equivalence of these two notions shown in [16] holds also in our case as we discuss in Subsection 4.1 below. From these, we obtain the weak dynamical localization,

sup_{t∈ℝ} ||||X|^{n/2}E(I_{0}(E_{0}, δ_{0}) : H^{ω}) exp(−itH^{ω})χ_{0}||| < ∞ for a. e. ω,

and the absence of the continuous spectrum

σ_{c}(H^{ω}) ∩ I_{0}(E_{0}, δ_{0}) = ∅ for a. e. ω

as is explained in [14] (cf. [37]).

On the other hand, we obtain the following from Theorem 3.1 as in [14]:

Theorem 3.3 (Semi Uniformly Localized Eigenfunction, cf. Theorem 3. 11 in [14]). Let \overline{L_{θ}} and I_{0}(E_{0}, δ_{0}) be the number and the interval given in Theorem 3.1. If there exists E_{0} ∈ I_{0} such that (3.3) holds for some L ≥ \overline{L_{θ}} and θ > bd/h, then, for any ε > 0, there exists m_{ε} > 0 satisfying the following for a. e. ω : σ_{c}(H^{ω})∩I_{0}(E_{0}, δ_{0}) = ∅ and, if {ϕ_{j,ω}}_{j∈ℕ} is the normalized eigenfunctions of H^{ω} with energy E_{j,ω} in I_{0}(E_{0}, δ_{0}), then, for any ν > d/4, there exist C_{ε,ω}, C̃_{ω} ∈ (0, ∞) and {x_{j,ω}}_{j∈ℕ} ⊂ ℤ^{d} such that

∥χ_{x}ϕ_{j,ω}∥_{2} ≤ C_{ε,ω} exp(m_{ε}(log |x_{j,ω}|_{∞})_{+}^{1+ε} − m_{ε}|x − x_{j,ω}|_{∞}) (3.6)

and

|x_{j,ω}|_{∞} ≥ C̃_{ω}j^{1/(4ν)}

for any j ∈ ℕ and x ∈ ℤ^{d}.

Moreover we assume that the coefficients belongs to the local Kato class K_{loc}(ℝ^{d}) defined as follows: a real function f on ℝ^{d} is said to be in the Kato class K(ℝ^{d}) if

lim_{α↓0} sup_{x∈ℝ} ∫_{|x−y|≤α} g_{d}(x − y)|f(y)|dy = 0

and is said to be in the local Kato class K_{loc}(ℝ^{d}) if χ_{K}f ∈ K_{loc}(ℝ^{d}) for all compact K ⊂ ℝ^{d}, where

g_{d}(x) := \begin{cases}− log |x| if d = 2, \\ |x|^{2−d} if d ≥ 3. \end{cases}

Then we obtain the following as in Lemma 2.5 in [22]:

Proposition 3.1. We assume

(K)  ∑_{j=1}^{d}(A_{j}^{(1),ω})^{2}, ∇ · A_{j}^{ω}, |V^{ω}| ∈ K_{loc}(ℝ^{d}).

Then, in the situation of Theorem 3.3, the eigenfunction ϕ_{j,ω} is a continuous function satisfying

|ϕ_{j,ω}(x)| ≤ C_{d,ε,ω} exp(m′_{ε}(log |x_{j,ω}|_{∞})_{+}^{1+ε} − m′_{ε}|x − x_{j,ω}|_{∞}),

where C_{d,ε,ω} and m′_{ε} are finite constants independent of j.

§3.2. Verification of Assumptions in [14]

The theory in Germinet and Klein [14] is discussed under several assumptions in an abstract setting. We have already introduced some of these conditions. In this subsection we show that the other assumptions hold in their forms or in weakened forms.

We first prepare the following proposition, which we often use in the following:

Lemma 3.1. Under the assumption (F), we have the following:

(i) For any 1 ≤ p < ∞, there exists a finite constant c = c(δ_{1}, δ_{2}, β, p) such that

E[ ess sup_{x∈Λ_{L}} V_{−}^{ω}(x)^{p}] ≤ c(log L)^{p/β}

for any L ≥ 2.

(ii) For almost all ω, there exists a finite constant c_{ω} such that

V_{−}^{ω}(x) ≤ c_{ω}{log(|x|_{∞} + 2)}^{2/β}

for almost every x ∈ ℝ^{d}.

Proof. (i) By Fubini’s theorem, we have

E[ ess sup_{x∈Λ_{L}} V_{−}^{ω}(x)^{p}] = ∫_{0}^{∞} λd_{λ}P ( ess sup_{x∈Λ_{L}} V_{−}^{ω}(x)^{p} ≤ λ)

= ∫_{0}^{∞} P ( ess sup_{x∈Λ_{L}} V_{−}^{ω}(x)^{p} > η)dη.

By applying (3.2) to the right hand side we obtain the result.

(ii) By taking δ_{3} > 1/\sqrt{δ_{2}}, we have

∑_{L∈ℕ} P ( ess sup_{x∈Λ_{L}} V_{−}^{ω}(x) > (δ_{3} log L)^{2/β}) < ∞.

By the Borel-Cantelli lemma, we see that, for almost all ω, there exists L_{ω} ∈ ℕ such that

ess sup_{x∈Λ_{L}} V_{−}^{ω}(x) ≤ (δ_{3} log L)^{2/β}

for any L ≥ L_{ω} Then for any L ≥ L_{ω} and almost every x ∈ Λ_{L} − Λ_{L−1}, we have

V_{−}^{ω}(x) ≤ (δ_{3} log L)^{2/β} ≤ {δ_{3} log(2|x|_{∞} + 1)}^{2/β}.

Therefore we obtain the result.

We write Λ_{L}(x) ⊏ Λ_{L′}(x′) if Λ_{L}(x) ⊂ Λ_{L′−3}(x′). Then the assumption (SLI) in [14] relating the Simon-Lieb inequality is verified in the following weakened form:

Proposition 3.2. There exists a finite constant c = c(d, M_{I_{0}}) such that, for any L, ℓ′, ℓ′′ ≥ 2, x, y, y′ ∈ ℝ^{d} with Λ_{ℓ′′}(y) ⊏ Λ_{ℓ′}(y′) ⊏ Λ_{L}(x) and any E ∈ I_{0} − σ(H_{x,L}) − σ(H_{y′,ℓ′}),

|||Γ_{x,L}(H_{x,L} − E)^{−1}χ_{y,ℓ′′}|||

(3.7) ≤ (( ess sup_{Γ_{y′,ℓ′}} V_{−}^{ω})^{1/2} + c)|||Γ_{y′,ℓ′}(H_{y′,ℓ′} − E)^{−1}χ_{y,ℓ′′}|||

×|||Γ_{x,L}(H_{x,L} − E)^{−1}Γ_{y′,ℓ′}|||.

The assumption (SLI) in [14] requires that the first factor in the right hand side of (3.7) is bounded in ω. Instead of the boundedness, we use the assumption (F).

To prove this proposition, we modify Lemma 2.5.3 in [37] as follows:

Lemma 3.2. There exists a finite constant c = c(d) satisfying the following: let B̃, B be bounded open regions in ℝ^{d} such that B̃ ⊂ B and δ := dist(∂B̃, ∂B) > 0. Let g ∈ L^{2}(B) and u ∈ L^{2}(B) such that (i∇+A^{ω})u ∈ L^{2}(B) and ((i∇ + A^{ω})u, (i∇ + A^{ω})φ) + (u, V^{ω}φ) = (g, φ) for any φ ∈ C_{0}^{∞}(B). Then it holds that

∥(i∇ + A^{ω})u∥_{L^{2}(B̃)} ≤ (( ess sup_{B} V_{−}^{ω})^{1/2} + \frac{c}{δ} )∥u∥_{L^{2}(B)} + ∥g∥_{L^{2}(B)}.

Then Lemma 2.5.2 in [37] is modified as follows:

Lemma 3.3. There exists a finite constant c = c(d, M_{I_{0}}) satisfying the following: let Λ_{L′}(x′) ⊂ Λ_{L}(x) and A′ ⊂ Λ_{L′}(x′), A ⊂ Λ_{L}(x). Let ϕ ∈ C_{0}^{1}(Λ_{L}(x)) and B be an open neighborhood of supp ∇ϕ such that δ_{1} := dist(∂B, supp ∇ϕ) > 0, B ∩ A = ∅ and B ⊂ Λ_{L}(x). Then it holds that

|||χ_{A′}(ϕ(H_{x,L} − E)^{−1} − (H_{x′,L′} − E)^{−1}ϕ)χ_{A}|||

≤ (( ess sup_{B} V_{−}^{ω})^{1/2} + \frac{c}{δ_{1}} )|||χ_{A′}(H_{x′,L′} − E)^{−1}χ_{B}|||

×|||χ_{B}(H_{x,L} − E)^{−1}χ_{A}|||

for any E ∈ I_{0} − σ(H_{x,L}) − σ(H_{x′,L′}).

Proof of Proposition 3.2. We have only to take Λ_{L′}(x′) = Λ_{ℓ′}(y′), A′ = Λ_{ℓ′′}(y), A = \overline{Λ_{L−1}(x)}−Λ_{L−3}(x), B = Λ_{ℓ′−1}(y′)−\overline{Λ_{ℓ′−3}(y′)} and ϕ so that ϕ = 1 on Λ_{ℓ′−5/2}(y′) and ϕ = 0 on Λ_{ℓ′−3/2}(y′)^{c} in Lemma 3.3.

Let T be the operator of multiplication with the function (1+|x|^{2})^{ν}, where ν is a fixed number greater than d/4. Let H_{±} be the weighted spaces defined by

H_{±} := L^{2}(ℝ^{d}, (1 + |x|^{2})^{±2ν}dx).

The set D_{+}^{ω} := {ϕ ∈ D(H^{ω}) ∩ H_{+}, H^{ω}ϕ ∈ H_{+}} is dense in H_{+} and an operator core for H^{ω}, for a. e. ω, where D(H^{ω}) is the domain of H^{ω}.

ψ ∈ H_{−} is called a generalized eigenfunction of H^{ω} with generalized eigenvalue E if

∫ \overline{H^{ω}φ}ψdx = E ∫ \overline{φ}ψdx

for any φ ∈ D_{+}^{ω}. Then the assumption (EDI) in [14] on an eigenfunction decay inequality is verified in the following weakened form:

Proposition 3.3. There exists a finite constant c = c(d, M_{I_{0}}) satisfying the following: for any generalized eigenfunction ψ of H^{ω} with generalized eigenvalue E ∈ I_{0}, x ∈ ℝ^{d} and L ≥ 4 such that E ∉ σ(H_{x,L}^{ω}), it holds that

∥χ_{x}ψ∥_{2} ≤ c(( ess sup_{Λ_{L}(x)} V_{−}^{ω})^{1/2} + 1)|||Γ_{x,L}(H_{x,L}^{ω} − E)^{−1}χ_{x}||| × ∥Γ_{x,L}ψ∥_{2}. (3.8)

As in the assumption (SLI), the assumption (EDI) in [14] requires that the second factor in the right hand side of (3.8) is bounded in ω. This proposition is proven by Lemma 3.2.

The assumption (NE) in [14] on the average number of eigenvalues is verified by the following:

Proposition 3.4. For any a ∈ ℝ, there exists a finite constant C_{a} = C_{a}(a, d, β, δ_{1}, δ_{2}) such that

E[♯{(−∞, a) ∩ σ(H_{L}^{ω})}] ≤ C_{a}L^{d} (3.9)

for any L ≥ 1.

Proof. Let H_{L}^{ω,a} be the operator obtained by replacing V^{ω} by −(V^{ω} −a− 1)_{−} in the definition of H_{L}^{ω}. Then, since H_{L}^{ω} − a − 1 ≥ H_{L}^{ω,a}, we have

♯{(−∞, a) ∩ σ(H_{L}^{ω})} ≤ ♯{(−∞, −1) ∩ σ(H_{L}^{ω,a})}.

Let { \atop H_{L+3}^{ω,a}} be the operator obtained by replacing A(x, \overline{W_{ω}}(x)) and W_{0}^{ω}(x)^{n} by A^{ω} and (V^{ω} − a − 1)_{−}, respectively, in the definition of { \atop H_{L+3}^{ω}} in (2.1). Then, since σ(H_{L}^{ω,a}) ⊂ σ({ \atop H_{L+3}^{ω,a}}), we have

♯\{(−∞, −1) ∩ σ(H_{L}^{ω,a})\} ≤ ♯\{(−∞, −1) ∩ σ({ \atop H_{L+3}^{ω,a}})\}.

Let { \atop H_{L+3}^{ω,−}} be the operator obtained by replacing A(x, \overline{W_{ω}}(x)) by A^{ω} in the definition of { \atop H_{L+3}^{ω}} in (2.2), and Γ^{a}(ω, L + 3) be the operator obtained by replacing { \atop H_{L+3}^{ω}}, E and W_{0}^{ω}(x)^{n} by { \atop H_{L+3}^{ω,−}}, −1 and (V^{ω} − a − 1)_{−}, respectively, in the definition of Γ(ω, L + 3, E) in (2.6). Then, by the Birman-Schwinger principle (cf. Theorem 8.1 in [34]), we have

♯\{(−∞, −1) ∩ σ({ \atop H_{L+3}^{ω,a}})\} ≤ ♯\{(1, ∞) ∩ σ(Γ^{a}(ω, L + 3))\}.

This is less than or equal to

|||Γ^{a}(ω, L + 3)|||_{q/2}^{q/2}.

By Lemma 2.1 (ii), we see this is dominated by

∥{ \atop χ_{L+3}}(V^{ω} − a − 1)_{−}∥_{q/2}^{q/2}.

By Lemma 3.1 (i), we have

E[∥{ \atop χ_{L+3}}(V^{ω} − a − 1)_{−}∥_{q/2}^{q/2}] ≤ ∑_{b∈Λ_{L+3}∩ℤ^{d}} E[ ess sup_{x∈Λ_{2}(b)} |(V^{ω} − a − 1)_{−}|^{q/2}]

≤ C(δ_{1}, δ_{2}, β, a, q)♯{Λ_{L+3} ∩ ℤ^{d}}.

Therefore we obtain (3.9).

The following ensures that the assumption (SGEE) in [14] relating a generalized eigenfunction expansion holds:

38 Naomasa Ueki

Proposition 3.5. For any a < b and p > 0, we have E[Tr[T^{−1}E([a, b] : H^{ω})T^{−1}]^{p}] < ∞. (3.10)

The assumption (SGEE) in [14] requires (3.10) only for p = 2. However we need (3.10) for some p > 2 since our scalar potential V^{ω} is not bounded below.

Proof of Proposition 3.5. By taking a complete orthonormal system {φ_{m}} of L^{2}(ℝ^{d}), we write the trace as

Tr[T^{−1}E([a, b] : H^{ω})T^{−1}] = ∑_{m} ∥E([a, b] : H^{ω})T^{−1}φ_{m}∥_{2}^{2}. (3.11)

Since (H^{ω} − a + 1)^{−1} ≥ (b − a + 1)^{−1} on the range of E([a, b] : H^{ω}), we have ∥E([a, b] : H^{ω})T^{−1}φ_{m}∥_{2} (3.12)

≤ (b − a + 1)∥(H^{ω} − a + 1)^{−1}E([a, b] : H^{ω})T^{−1}φ_{m}∥_{2}.

Since (H^{ω} − a + 1)^{−1}E([a, b] : H^{ω})

= E([a, b] : H^{ω})({ \atop H^{ω,+}} + 1)^{−1}

+(H^{ω} − a + 1)^{−1}E([a, b] : H^{ω})(a + V_{−}^{ω})({ \atop H^{ω,+}} + 1)^{−1},

we have

∥(H^{ω} − a + 1)^{−1}E([a, b] : H^{ω})T^{−1}φ_{m}∥_{2}

≤ ∥E([a, b] : H^{ω})({ \atop H^{ω,+}} + 1)^{−1}T^{−1}φ_{m}∥_{2} (3.13)

+∥(H^{ω} − a + 1)^{−1}E([a, b] : H^{ω})(a + V_{−}^{ω})({ \atop H^{ω,+}} + 1)^{−1}T^{−1}φ_{m}∥_{2},

where { \atop H^{ω,+}} = (i∇ + A^{ω})^{2} + V_{+}^{ω}. The second term is less than of equal to

∥(a + V_{−}^{ω})({ \atop H^{ω,+}} + 1)^{−1}T^{−1}φ_{m}∥_{2},

since (H^{ω} − a + 1)^{−1} ≤ 1 on the range of E([a, b] : H^{ω}). Therefore the quantity in (3.11) is dominated by

Tr[T^{−1}({ \atop H^{ω,+}} + 1)^{−2}T^{−1}] + Tr[T^{−1}({ \atop H^{ω,+}} + 1)^{−1}(a + V_{−}^{ω})^{2}({ \atop H^{ω,+}} + 1)^{−1}T^{−1}].

By using the diamagnetic inequality as in the proof of Lemma 2.1, we see that this is less than or equal to

Tr[T^{−1}(−Δ + 1)^{−2}T^{−1}] + Tr[T^{−1}(−Δ + 1)^{−1}(a + V_{−}^{ω})^{2}(−Δ + 1)^{−1}T^{−1}].

By the explicit representation of the resolvent kernel we see that the first term is finite if d < 4. By using also the boundedness of E[∏_{ι=1}^{2N}(a + V_{−}^{ω}(x_{ι}))^{2}], we see that

E[Tr[T^{−1}(−Δ + 1)^{−1}(a + V_{−}^{ω})^{2}(−Δ + 1)^{−1}T^{−1}]^{2N}]

is finite for any N ∈ ℕ if d < 4. For the higher dimensional case, we take ℓ ∈ ℕ so that ℓ > d/4. By repeating the estimates as in (3.12) and (3.13), we have

∥E([a, b] : H^{ω})T^{−1}φ_{m}∥_{2}

≤ C ∑_{j_{1},j_{2},... ,j_{ℓ}∈\{1,2\}} ∥(a + V_{−}^{ω})^{j_{1}}({ \atop H^{ω,+}} + 1)^{−1}(a + V_{−}^{ω})^{j_{2}}({ \atop H^{ω,+}} + 1)^{−1} ⋯

×(a + V_{−}^{ω})^{j_{ℓ}}({ \atop H^{ω,+}} + 1)^{−1}T^{−1}φ_{m}∥_{2}.

The rest of the proof is similar.

§3.3. Proof of the main theorems

Theorem 3.1 is proven by extending the four theorems in Section 5 in [14] and combining them as in [14]. The first one of them is extended as follows:

Proposition 3.6 (cf. Theorem 5.1 in [14]). For any E_{0} ∈ I_{0}, θ > bd/h, odd integer Y with Y ≥ 11 and 0 < p < hθ − bd, there exists Z_{θ} = Z_{θ}(d, ρ, M_{I_{0}}, QĨ_{0}, β, δ_{1}, δ_{2}, b, h, θ, p, Y) ∈ 6ℕ satisfying the following: if there exists Z_{θ} ≤ L_{0} ∈ 6ℕ such that

P (Λ_{L_{0}} is (θ, E_{0})-suitable) > 1 − (3Y − 4)^{−2d},

then there exists κ = κ(p, Y, L_{0}) ∈ ℤ_{+} such that

P (Λ_{L_{k}} is (θ, E_{0})-suitable) > 1 − L_{k}^{−p}

for any k ≥ κ, where L_{k+1} = Y L_{k} for any k ∈ ℤ_{+}.

Proof. In the proof of Theorem 5.1 in [14], we require s satisfies

(p + bd)/h < s and s < θ. (3.14)

As in [14], we set

q_{L} = L^{−p}, t_{L} = L^{−s}, u_{L} = L^{−θ},
p_{L} = P (Λ_{L} is not (θ, E_{0})-suitable).

Let ℓ ∈ 6ℕ, ℓ > 6ρ, Y ∈ 2ℕ + 1 and L = Y ℓ. We set

Ξ_{L,ℓ} = Λ_{L} ∩ \frac{ℓ}{3}ℤ^{d}, C_{L,ℓ} = \{Λ_{ℓ}(y) : y ∈ Ξ_{L,ℓ}, Λ_{ℓ}(y) ⊏ Λ_{L}\}.

In the definition of the event F_{L,ℓ} in [14], we add the conditions on V_{−}^{ω} as follows:

F_{L,ℓ} = {ω : there exist two (θ, E_{0})-nonsuitable boxes
Λ_{ℓ_{1}}(y_{1}), Λ_{ℓ_{2}}(y_{2}) in C_{L,ℓ} such that
dist(Λ_{ℓ_{1}}(y_{1}), Λ_{ℓ_{2}}(y_{2})) > 2ρ}

∪{ω : dist(σ(H_{x,3ℓ}^{ω}), E_{0}) ≤ t_{L} for some x ∈ Ξ′_{L,ℓ}} ∪{ω : dist(σ(H_{0,L}^{ω}), E_{0}) ≤ t_{L}}

∪{ω : ess sup_{Λ_{L}} V_{−}^{ω} ≥ {2p(log L)^{2}/δ_{2}}^{1/β}}.

By Lemma 3.2, we have

{ω : Λ_{L} is not (θ, E_{0})-suitable} ⊂ F_{L,ℓ}

for L > Z_{1} = Z_{1}(d, β, δ_{1}, δ_{2}, θ, p, Y, s). By the assumptions (W), (F) and Lemma 3.4 below, we have

P (F_{L,ℓ}) ≤ \frac{(3Y − 4)^{2d}}{2} p_{ℓ}^{2} + \{(6Y)^{d} + 1\}QĨ_{0}L^{bd}t_{L}^{h} + δ_{1}L^{−2p}

≤ \frac{(3Y − 4)^{2d}}{2} p_{ℓ}^{2} + \frac{1}{2}q_{L}

for L > Z_{2} = Z_{2}(d, QĨ_{0}, β, δ_{1}, δ_{2}, b, h, θ, p, Y, s). The rest of the proof is same.

Lemma 3.4. Let Λ_{ℓ_{1}}(y_{1}), Λ_{ℓ_{2}}(y_{2}), …, Λ_{ℓ_{m}}(y_{m}) be finite number of boxes such that dist(Λ_{ℓ_{i}}(y_{i}), Λ_{ℓ_{j}}(y_{j})) > 2ρ for i ≠ j. Then, for any θ and E, the events {ω : Λ_{ℓ_{i}}(y_{i}) is (θ, E)-suitable}, i = 1, 2, …, m, are independent.

Proof. Let ζ_{i} ∈ C_{0}^{∞}(ℝ^{d} → [0, 1]) such that ζ_{i} ≡ 1 on Λ_{ℓ_{i}}(y_{i}) and ζ_{i} ≡ 0 on Λ_{ℓ_{i}+ρ/2}(y_{i})^{c}. We define

A_{j}^{[i],ω}(x) := \frac{−Γ(d/2)}{2π^{d/2}} ∫ ∑_{k≠j} \frac{x_{k} − y_{k}}{|x − y|^{d}} d(ζ_{i}A^{(2),ω})_{j,k}(y)ζ_{i}(y)dy.

Since A^{(2),ω} and dA^{(2),ω} are continuous in x, A^{[i],ω} is also continuous in x and we can integrate A^{(2),ω} and A^{[i],ω} along any C^{1} curves. Since dA^{[i],ω} = dA^{(2),ω} on Λ_{ℓ_{i}}(y_{i}), we can construct Φ^{[i],ω} ∈ C^{1}(Λ_{ℓ_{i}}(y_{i})) such that dΦ^{[i],ω} = A^{(2),ω} − A^{[i],ω} by the Stokes theorem. By the unitary operator exp(iΦ^{[i],ω}), the operators H_{ℓ_{i},y_{i}}^{ω} and H_{ℓ_{i},y_{i}}^{[i],ω} are unitary equivalent, where H_{ℓ_{i},y_{i}}^{[i],ω} is the operator obtained by replacing A^{(2),ω} by A^{[i],ω} in the definition of H_{ℓ_{i},y_{i}}^{ω}. Since the coefficients of H_{ℓ_{i},y_{i}}^{[i],ω} are determined by the random field {(A^{(1),ω}, dA^{(2),ω}, V^{ω}(x)) : x ∈ Λ_{ℓ_{i}+ρ/2}(y_{i})}, the desired independence follows from the assumption (IAD).

The rest of the theorems in Section 5 in [14] are also modified by using the assumptions (F) and (IAD) as above. We omit the detail.

Theorems 3.2 and 3.3 are proven similarly as in [14] by using Lemma 3.1 and the propositions in the last subsection.

Proposition 3.1 is proven by using the subsolution estimate (Theorem 8.17 in [19]) as in [28]. For this, we use Kato’s inequality (Theorem X.33 in [32]) to show (−Δ + (V^{ω} − E_{j,ω})_{−})|ϕ_{j,ω}(x)| ≤ 0 in the weak sense.

§3.4. The case of Section 2

In the case of Section 2, the assumptions (R), (S) and (IAD) hold. Moreover the assumption (K) in Proposition 3.1 is also satisfied. For any open interval Ĩ_{0} such that sup Ĩ_{0} < −(2∥w · ∇_{w}A∥_{∞}/n)^{2}), the assumption (W) with Ĩ_{0} is ensured by Theorem 2.1. The assumption (F) is ensured by the following lemma by Fischer, Leschke and Müller [12], which is based on Fernique’s theorem [10]:

Lemma 3.5 (Lemma 5.3 in [12]). There exists a positive finite constant L_{0} = L_{0}(R^{00}(0), ∥∇R^{00}∥_{∞}) such that

P ( sup_{x∈Λ_{L}} |W_{ω}^{0}(x)| ≥ η) ≤ 2^{2(d+1)} exp ( − \frac{η^{2}}{200R^{00}(0) log L})

for any L ≥ L_{0} and η ≥ 0.

The initial estimate (3.3) is also proven by using this lemma and a Combes-Thomas estimate, Lemma A.1 in [12] as in Lemma 5.5 in [12]:

Lemma 3.6. There exist positive constants c_{1} = c_{1}(d), c_{2} = c_{2}(d) and a universal finite constant c_{3} such that for any θ > 0, 12 ≤ L ∈ 6ℕ and E ≤ −𝔈(L), we have

P (Λ_{L} is (θ, E)-suitable) > 1 − 841^{−d},

where

𝔈(L) := {c_{3}L^{−1} log(c_{1}L^{θ+d+1})}^{2} + (c_{2}R^{00}(0) log L)^{n/2}.

Therefore the results in Subsection 3.1 hold for

E_{0} < \{−(2∥w · ∇_{w}A∥_{∞}/n)^{2}\} ∧ \{− inf_{θ>bd/h} 𝔈(\overline{L_{θ}})\}, (3.15)

where \overline{L_{θ}} is the constant given in Theorem 3.1.

These results are meaningful since the interval determined by (3.15) is included in the spectrum as is shown in the following:

Proposition 3.7 (Theorem 5.34 in [31]). We have

σ(H^{ω}) ⊃ \{inf σ(H(ℓ, \overline{w})) − (w^{0})^{n}

: ℓ > 0, w^{0} ∈ ℝ, \overline{w} ∈ supp P (\overline{W_{ω}}(0) ∈ ·|W_{ω}^{0}(0) = w^{0})\}

for a. e. ω, where H(ℓ, ω) is the self-adjoint operator on L^{2}(Λ_{ℓ}) obtained by restricting

∑_{j=1}^{d}(i∂_{j} + A_{j}(x, \overline{w}))^{2}

to Λ_{ℓ} by the Dirichlet boundary condition. In particular, if

sup\{|A(x, \overline{w})| : x ∈ Λ_{ℓ}, \overline{w} ∈ ℝ^{g}\} < ∞

for any ℓ > 0, then we have

σ(H^{ω}) = ℝ

for a. e. ω.

Proof. For any w^{0} > 0, \overline{w} ∈ supp P (\overline{W_{ω}}(0) ∈ ·|W_{ω}^{0}(0) = w^{0}), ℓ > 0 and |w^{0}| > ε > 0, we have

P ( sup_{x∈Λ_{ℓ}} |(\overline{W_{ω}}(x), W_{ω}^{0}(x)) − (\overline{w}, w^{0})| < ε) > 0.

If the event in this probability occurs, then by using

|A(x, \overline{W_{ω}}(x)) − A(x, \overline{w})| ≤ ε∥∇\overline{w}A∥_{∞},

we have

inf σ(H_{0,ℓ}^{ω})
≤ (1 + ε) inf σ(H(\overline{w}, ℓ)) + ε(1 + ε)∥∇\overline{w}A∥_{∞}^{2} − (w^{0} − ε)^{n}
=: \overline{W}(ε)

and

inf σ(H_{0,ℓ}^{ω})
≥ (1 − ε) inf σ(H(\overline{w}, ℓ)) − ε(1 − ε)∥∇\overline{w}A∥_{∞}^{2} − (w^{0} + ε)^{n}
=: \underline{W}(ε).

Since inf σ(H_{0,ℓ}^{ω}) ⊂ σ(H^{ω}), we have

P ((\underline{W}(ε), \overline{W}(ε)) ∩ σ(H^{ω}) ≠ ∅) > 0.

By the ergodicity, this probability is one. Therefore we obtain

1 = P ((\underline{W}(1/n), \overline{W}(1/n)) ∩ σ(H^{ω}) ≠ ∅ for any 1/w_{0} < n ∈ ℕ)
= P (inf σ(H(\overline{w}, ℓ)) − (w^{0})^{n} ∈ σ(H^{ω})).

By the same method we can prove this for w^{0} ≤ 0.

We next take an ℝ^{g}-valued continuous function \overline{w}(w^{0}) of w^{0} ∈ ℝ so that \overline{w}(w^{0}) ∈ supp P (\overline{W_{ω}}(0) ∈ ·|W_{ω}^{0}(0) = w^{0}) and \overline{w}(0) = 0. Then s(w^{0}, ℓ) := inf σ(H(\overline{w}(w^{0}), ℓ)) − (w^{0})^{n} is continuous in w_{0}. If

α_{ℓ} := sup\{|A(x, \overline{w})| : x ∈ Λ_{ℓ}, \overline{w} ∈ ℝ^{g}\} < ∞

for any ℓ > 0, then

s(w^{0}, ℓ) ≤ 2 inf σ(−Δ_{ℓ}) + 2α_{ℓ}^{2} − (w^{0})^{n},

where Δ_{ℓ} is the Dirichlet Laplacian on Λ_{ℓ}. By this we see that s(w^{0}, ℓ) tends to −∞ as w^{0} tends to ∞. Therefore we have (−∞, s(0, ℓ)) ⊂ σ(H^{ω}). By the diamagnetic inequality we have

s(0, ℓ) ≥ inf σ(−Δ_{ℓ}).

Then we see that s(0, ℓ) tends to ∞ as ℓ tends to 0. Therefore we have σ(H^{ω}) = ℝ.

Remark 3.1. In the case that A_{j} is independent of w and n = 1, Fischer, Leschke and Müller [12] proves the absence of the continuous spectrum in a low energy interval under more general conditions on the Gaussian random field.

Remark 3.2. The results in this subsection are extended to the cases stated in Remark 2.2.

§4. A characterization of the Anderson metal-insulator transport transition

In this section we extend the theory in Germinet and Klein [16]. As in the last section, we treat a random operator (3.1) with a general random field (A^{ω}(x), V^{ω}(x)). In [16], A^{ω} = 0, V^{ω} is relatively form bounded below with respect to the negative Laplacian and the Wegner estimate (W) holds with h = b = 1. In this section we remove this restrictions so that the operator in the preceding sections can be treated.

§4.1. Main theorem

We use the notations in the last section. We assume (R), (IAD), (F) and (W) on an open interval Ĩ_{0}. Moreover we assume the ergodicity:

(E) The random field (A^{(1),ω}(x), dA^{(2),ω}(x), V^{ω}(x)) is stationary and ergodic, with respect to the shift in the variable x ∈ ℝ^{d} or ℤ^{d}.

Then we have a nonrandom set Σ such that σ(H^{ω}) = Σ with probability one (cf. [31], [38]). Referring Theorem 3.1, we modify the definition of the multiscale analysis region Σ_{MSA} in [16] as

Σ_{MSA} := {E ∈ Σ : there exist θ > bd/h, a compact interval I_{0}

and L ∈ 6ℕ such that E ∈ I^{∘}_{0} ⊂ I_{0} ⊂ Ĩ_{0}, L ≥ \overline{L_{θ}} and

P (Λ_{L} : (θ, E) − suitable) > 1 − 841^{−d}, where \overline{L_{θ}} is the number given in Theorem 3.1\}.

As in [16], we consider moments defined as follows: for any n ≥ 0, f ∈ C_{0}^{∞}(ℝ → [0, ∞)), t ∈ ℝ and T ≥ 0, we set

M_{ω}(n, f, t) := |||⟨X⟩^{n/2} exp(−itH^{ω})f(H^{ω})χ_{0}|||_{2}^{2} (4.1) 𝕄(n, f, t) := E[M_{ω}(n, f, t)] (4.2)

and

M(n, f, T) := \frac{2}{T} ∫_{0}^{∞} e^{−2t/T} 𝕄(n, f, t)dt, (4.3)

where ⟨X⟩ := (1 + |X|^{2})^{1/2}. The strong insulator region Σ_{SI} is defined by

Σ_{SI} := {E ∈ Σ : there exists an open interval I including E

such that E[ sup_{t∈ℝ} M_{ω}(n, f, t)] < ∞

for any f ∈ C_{0}^{∞}(I → [0, ∞)) and n ≥ 0}.

By Theorem 3.2, we have Σ_{MSA} ∩ Ĩ_{0} ⊂ Σ_{SI}. Moreover, if E ∈ Σ_{SI} and I is an open interval appeared in the definition of E ∈ Σ_{SI}, then it is easy to see that

sup_{T ≥0} M(n, f, T) < ∞

for any n ≥ 0 and f ∈ C_{0}^{∞}(I → [0, ∞)). Conversely, we have the following, which is an extension of Theorem 2.11 in [16]:

Theorem 4.1. Let f ∈ C_{0}^{∞}(ℝ → [0, ∞)) such that f ≡ 1 on some open interval J ⊂ Ĩ_{0}, α ≥ 0 and

n > 2bdα/h + (2 + 9b/h)d. (4.4)

If

\underline{lim}_{T →∞} M(n, f, T)/T^{α} < ∞,

then J ∩ Σ ⊂ Σ_{MSA}.

We prove this theorem in the next subsection. In [16], many results are deduced from this result. Some of the results are stated in terms of an exponent defined in (4.8) below. To define the exponent in our case, we extend Proposition 3.1 in [16] as follows:

Proposition 4.1. Let f ∈ C_{0}^{∞}(ℝ → [0, ∞)) such that f(H^{ω}) ≠ 0 with probability one. Then we have

0 ≤ M_{ω}(0, f, 0) ≤ M_{ω}(n, f, t) (4.5)

≤ C(d, β, f, n, ω)⟨t⟩^{[[n+(5d+1)/2]]+2}

0 < 𝕄(0, f, 0) ≤ 𝕄(n, f, t) ≤ C(d, β, f, n)⟨t⟩^{[[n+(5d+1)/2]]+2} (4.6)

and

0 < 𝕄(0, f, 0) ≤ M(n, f, T) ≤ C′(d, β, f, n)⟨T ⟩^{[[n+(5d+1)/2]]+2} (4.7)

for any t ∈ ℝ and n, T ≥ 0, where [[m]] is the smallest integer greater than m for any m ≥ 0.

We prove this proposition in Subsection 4.3 below. In the situation of Proposition 3.1 in [16], the exponents of ⟨t⟩ and ⟨T ⟩ in the right hand sides of (4.5)–(4.7) are taken as [n + 3d/2] + 3 and the constant C(d, β, f, n, ω) in (4.5) is independent of ω.

As in [16] we define the n-th upper and lower transport exponents as follows: for f ∈ C_{0}^{∞}(ℝ → [0, ∞)) with f(H^{ω}) ≠ 0 with probability one, we set

β^{+}(n, f) := \overline{lim}_{T →∞} \frac{log M(n, f, T)}{n log T}

and

β^{−}(n, f) := \underline{lim}_{T →∞} \frac{log M(n, f, T)}{n log T}.

For f ∈ C_{0}^{∞}(ℝ → [0, ∞)) with f(H^{ω}) ≠ 0 with probability one, we set β^{±}(n, f) = 0. For any open interval I and E ∈ ℝ, we set

β^{±}(n, I) := sup{β^{±}(n, f)|f ∈ C_{0}^{∞}(I → [0, ∞))}

and

β^{±}(n, E) := inf{β^{±}(n, I)|I is an open interval including E}.

For these exponents we obtain the following by the entirely same proof of Proposition 3.2 in [16]:

Proposition 4.2. For any f ∈ C_{0}^{∞}(ℝ → [0, ∞)), any open interval I and any E ∈ ℝ, we have the following:

(i) β^{±}(n, f), β^{±}(n, I) and β^{±}(n, E) are monotone increasing in n > 0. (ii) 0 ≤ β^{±}(n, f), β^{±}(n, I), β^{±}(n, E) ≤ 1.

Therefore, as in [16], we can define

β^{±}(I) = lim_{n→∞} β^{±}(n, I) = sup_{n} β^{±}(n, I)

β^{±}(E) = lim_{n→∞} β^{±}(n, E) = sup_{n} β^{±}(n, E) (4.8)

and we have 0 ≤ β^{±}(I), β^{±}(E) ≤ 1 and β^{±}(E) = 0 for E ∉ Σ.

From Theorem 4.1 we have the following as in Theorem 2.10 in [16]:

Proposition 4.3. If E ∈ Ĩ_{0} and β^{−}(E) > 0, then β^{−}(n, E) ≥ h/(2bd)− (2h + 9b)/(2nb) for all n > (2h + 9b)d/h and β^{−}(E) ≥ h/(2bd).

Therefore the size of the jump of the exponent β^{−} at the transition point is greater than or equal to h/(2bd).

Theorems 2.8 and 4.2 in [16] are also extended appropriately.

§4.2. Proof of Theorem 4.1

Wegner estimates 47

To prove Theorem 4.1, we first modify Theorem A.5 in [16] (Theorem 2 in [15]) as follows:

Proposition 4.4. There exist finite constants C(d) and C such that |||χ_{x}f(H^{ω})χ_{y}||| (4.9)

≤ C(d)∥f∥_{(k+1)}(ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{(2k+d−3)/4} \frac{(Ck)^{k}}{⟨x − y⟩^{(2k−d−3)/2}}

for any f ∈ C^{∞}(ℝ → ℂ), (d + 5)/2 ≤ k ∈ ℕ, L ≥ 2 and x, y ∈ Λ_{L−1}, where R = 2\sqrt{d} + 4 and

∥f∥_{(n)} := ∑_{r=0}^{n} ∫_{ℝ} |f^{(r)}(u)|⟨u⟩^{r−1}du

for n ∈ ℕ.

The number R = 2\sqrt{d} + 4 can be replaced by any number greater than 2\sqrt{d} + 2 if L is large enough.

To prove this proposition, we first show the following: Lemma 4.1. There is a finite constant C(d) such that

|||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||

≤ \frac{C(d)}{η_{z}^{RL}}\{1 + ((η_{z} + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{1/2}/η_{z}) (4.10)

×((η_{z}^{RL} + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)/η_{z}^{RL})^{(d−1)/2}}

× exp ( − \frac{m_{z}^{RL}}{2 + \sqrt{η_{z}^{RL}}} |x − y|)

for any z ∈ ℂ − ℝ, L ≥ 2 and x, y ∈ Λ_{L−1}, where η_{z} := dist(z, σ(H^{ω})), η_{z}^{RL} := dist(z, σ(H_{RL}^{ω})) and

m_{z}^{RL} := \frac{η_{z}^{RL}}{32} (\frac{63}{ess sup_{Λ_{RL}} V_{−}^{ω} + η_{z}^{RL} + |z|})^{1/2}.

Proof. We take ϕ ∈ C_{0}^{∞}(Λ_{RL} → [0, 1]) so that ϕ ≡ 1 on Λ_{L} and supp ∇ϕ ⊂ Λ_{2(\sqrt{d}+1)L+3} − Λ_{2(\sqrt{d}+1)L+2}. Then we have

ϕ(H_{RL}^{ω} − z)^{−1} − (H^{ω} − z)^{−1}ϕ

= 2(H^{ω} − z)^{−1}(i∇ϕ)(i∇ + A^{ω})(H_{RL}^{ω} − z)^{−1} (4.11)

−(H^{ω} − z)^{−1}(Δϕ)(H_{RL}^{ω} − z)^{−1}

and

|||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||

≤ |||χ_{x}(H_{RL}^{ω} − z)^{−1}χ_{y}||| (4.12)

+(2|||χ_{x}(H^{ω} − z)^{−1}(i∇ϕ)(i∇ + A^{ω})ψ|||
+|||χ_{x}(H^{ω} − z)^{−1}(Δϕ)|||)|||ψ(H_{RL}^{ω} − z)^{−1}χ_{y}|||,

where ψ ∈ C_{0}^{∞}(Λ_{RL} → [0, 1]) such that ψ ≡ 1 on Λ_{2(\sqrt{d}+1)L+3} − Λ_{2(\sqrt{d}+1)L+2} and ψ ≡ 0 on (Λ_{2(\sqrt{d}+1)L+4} − Λ_{2(\sqrt{d}+1)L+1})^{c}. By the same proof of Corollary 1 of Theorem 1 in [15], we have

|||χ_{x_{0}}(H_{RL}^{ω} − z)^{−1}χ_{y_{0}}||| (4.13)

≤ \frac{9}{η_{z}^{RL}} exp (C\sqrt{d} − \frac{m_{z}^{RL}}{2 + \sqrt{η_{z}^{RL}}} |x_{0} − y_{0}|)

for some finite constant C and any x_{0}, y_{0} ∈ Λ_{RL−1}. Since |x_{0} − y| ≥ (\sqrt{d} + 1)L for any x_{0} ∉ Λ_{2(\sqrt{d}+1)L+1} and y ∈ Λ_{L−1}, we have

|||ψ(H_{RL}^{ω} − z)^{−1}χ_{y}|||

≤ \{ ∑_{x_{0}∈(Λ_{2(\sqrt{d}+1)L+4}−Λ_{2(\sqrt{d}+1)L+1})∩ℤ^{d}} |||χ_{x_{0}}(H_{RL}^{ω} − z)^{−1}χ_{y}|||^{2}\}^{1/2}

≤ \frac{C′′(d)}{η_{z}^{RL}}L^{(d−1)/2} exp ( − \frac{m_{z}^{RL}(\sqrt{d} + 1/2)L}{2 + \sqrt{η_{z}^{RL}}}) (4.14)

≤ \frac{C′(d)}{η_{z}^{RL}}(\frac{2 + \sqrt{η_{z}^{RL}}}{m_{z}^{RL}} )^{(d−1)/2} exp ( − \frac{m_{z}^{RL}\sqrt{d}L}{2 + \sqrt{η_{z}^{RL}}} )

≤ \frac{C(d)}{η_{z}^{RL}}\{(η_{z}^{RL} + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)/η_{z}^{RL}\}^{(d−1)/2}

× exp ( − \frac{m_{z}^{RL}\sqrt{d}L}{2 + \sqrt{η_{z}^{RL}}} ).

Since |x − y| ≤ \sqrt{d}L for any x, y ∈ Λ_{L−1}, we have

exp ( − \frac{m_{z}^{RL}\sqrt{d}L}{2 + \sqrt{η_{z}^{RL}}} ) ≤ exp ( − \frac{m_{z}^{RL}}{2 + \sqrt{η_{z}^{RL}}} |x − y|).

On the other hand we have

|||ψ(i∇ + A^{ω})(H^{ω} − z)^{−1}||| ≤ \frac{C}{η_{z}} (η_{z} + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{1/2} (4.15)

for some finite constant C. In fact, for any ϕ ∈ C_{0}^{∞}(ℝ^{d}), we have

∥ψ(i∇ + A^{ω})(H^{ω} − z)^{−1}ϕ∥^{2}
= ((H^{ω} − z)^{−1}ϕ, (i∇ + A^{ω})ψ^{2}(i∇ + A^{ω})(H^{ω} − z)^{−1}ϕ).

Since

(i∇ + A^{ω})ψ^{2}(i∇ + A^{ω})
= ψ^{2}(H^{ω} − z) + ψ^{2}(z − V^{ω}) + 2ψ(i∇ψ)(i∇ + A^{ω}),

and |||(H^{ω} − z)^{−1}||| ≤ 1/η_{z}, we have

∥ψ(i∇ + A^{ω})(H^{ω} − z)^{−1}ϕ∥^{2}

≤ \frac{1}{η_{z}} \{1 + \frac{1}{η_{z}} (|z| + ess sup_{Λ_{RL}} V_{−}^{ω} + \frac{1}{s} sup |∇ψ|^{2})\}∥ϕ∥^{2}

+s∥ψ(i∇ + A^{ω})(H^{ω} − z)^{−1}ϕ∥^{2}

for any s > 0. By taking s < 1, we obtain (4.15).

By applying (4.13), (4.14), (4.15), |||(H^{ω} − z)^{−1}||| ≤ 1/η_{z} and |||(H_{RL}^{ω} − z)^{−1}||| ≤ 1/η_{z}^{RL} to the right hand side of (4.12), we obtain (4.10).

Proof of Proposition 4.4. As in the proof of Theorem 2 in [15], we use the Helffer-Sjöstrand formula (cf. [8] Section 2.2):

f(H^{ω}) = \frac{1}{π} ∫_{ℝ^{2}} \frac{∂f̃_{k}(z)}{∂\overline{z}}(H^{ω} − z)^{−1}dudv,

where z = u + iv, k ∈ ℕ,

f̃_{k}(z) = \{ ∑_{r=0}^{k} \frac{1}{r!}f^{(r)}(u)(iv)^{r}\}σ(u, v),

σ(u, v) = τ(v/⟨u⟩) and τ ∈ C^{∞}(ℝ → [0, 1]) such that τ(s) = 1 on |s| ≤ 1 and τ(s) = 0 on |s| ≥ 2. Then we have

|||χ_{x}f(H^{ω})χ_{y}||| ≤ C(I_{1} + I_{2}),

where

I_{1} := ∑_{r=0}^{k} \frac{1}{r!} ∫_{⟨u⟩≤|v|≤2⟨u⟩} |f^{(r)}(u)|\frac{|v|^{r}}{⟨u⟩} |||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||dudv

50 Naomasa Ueki

and

I_{2} := \frac{1}{k!} ∫_{|v|≤2⟨u⟩} |f^{(k+1)}(u)||v|^{k}|||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||dudv.

As in [15], by (4.10) and |η_{z}^{RL}|, |η_{z}| ≥ |v|, we have |||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||

≤ \frac{C(d)}{|v|}\{1 + ((|v| + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{1/2}/|v|)

×((|v| + |z| + ess sup_{Λ_{RL}} V_{−}^{ω} + 1)/|v|)^{(d−1)/2}}

× exp ( − C \frac{|v||x − y|}{\sqrt{|v| + ess sup_{Λ_{RL}} V_{−} + |z|}(\sqrt{|v|} + 2)}).

If ⟨u⟩ ≤ |v| ≤ 2⟨u⟩, then we have |v| ≥ 1, |z| ≤ \sqrt{2}|v| and |||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||

≤ \frac{C(d)}{|v|}(ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{d/2} exp ( − C \frac{|x − y|}{\sqrt{ess sup_{Λ_{RL}} V_{−}^{ω} + 1}}).

Therefore we have

I_{1} ≤ C(d)(ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{d/2}( max_{0≤r≤k} \frac{2^{r}}{r · r!})∥f∥_{(k)}

× exp ( − C \frac{|x − y|}{\sqrt{ess sup_{Λ_{RL}} V_{−}^{ω} + 1}}).

If |v| ≤ 2⟨u⟩, then we have |z| ≤ 3⟨u⟩ and |||χ_{x}(H^{ω} − z)^{−1}χ_{y}|||

≤ \frac{C(d)}{|v|}\{1 + \frac{1}{\sqrt{|v|}}(⟨u⟩(ess sup_{Λ_{RL}} V_{−}^{ω} + 1)/|v|)^{d/2}\}

× exp ( − C \frac{|v||x − y|}{⟨u⟩\sqrt{ess sup_{Λ_{RL}} V_{−}^{ω} + 1}}).

Since

∫_{0}^{2⟨u⟩} v^{r} exp ( − C \frac{|v||x − y|}{⟨u⟩\sqrt{ess sup_{Λ_{RL}} V_{−}^{ω} + 1}})

≤ (2⟨u⟩)^{r+1}(\frac{Cr\sqrt{ess sup_{Λ_{RL}} V_{−}^{ω} + 1}}{⟨x − y⟩})^{r}

for any r ≥ 1, we have

I_{2} ≤ \frac{C(d)}{k!}(ess sup_{Λ_{RL}} V_{−}^{ω} + 1)^{(2n+d−3)/4} \frac{(Ck)^{k}}{⟨x − y⟩^{(2k−d−3)/2}} ∫ |f^{(k+1)}(u)|⟨u⟩^{k}du

for any k ≥ (d + 5)/2. Therefore we obtain (4.9)

We should also modify Lemma 6.4 in [16] as follows:

Lemma 4.2. There exists a finite constant C(d, h) satisfying the following: for any γ > bd/h, ϵ > 0, a > 0, E ∈ Ĩ_{0}, 0 < ε ≤ 1, 4 ≤ L ∈ 2ℕ and 0 < b_{1} < b_{2} < b_{3} ≤ 1 such that b_{2} − b_{1} > 1/4, we have

P (|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}χ_{0,b_{1}L}||| > a/4,

(4.16)  ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β})

≤ \frac{L^{γ}}{a} E[|||(χ_{0,b_{2}L} − χ_{0,b_{2}L−1})(H^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||

: ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}]

+QĨ_{0}C(d, h)(1 + |E|)^{h/2}(log RL)^{(ϵ+1)h/(2β)}/L^{γh−bd}

and

P (E ∉ σ(H_{L}^{ω}), |||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E)^{−1}χ_{0,b_{1}L}||| > a/2,

(4.17)  ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β})

≤ \frac{L^{γ}}{a} E[|||(χ_{0,b_{2}L} − χ_{0,b_{2}L−1})(H^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||

: ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}]

+QĨ_{0}C(d, h)(1 + |E|)^{h/2}(log RL)^{(ϵ+1)h/(2β)}/L^{γh−bd}

+QĨ_{0}(2\sqrt{ε/a})^{h}L^{bd}.

Proof. We take ϕ ∈ C_{0}^{∞}(Λ_{L} → [0, 1]) so that ϕ = 1 on Λ_{b_{1}L}, ϕ = 0 on Λ_{b_{2}L}^{c}, supp ∇ϕ ⊂ Λ_{b_{2}L} − \overline{Λ_{b_{2}L−1}}, |∇ϕ| ≤ 3\sqrt{d} and |∇^{2}ϕ| ≤ 7d. As in (4.11) and (4.12), we have

ϕ(H_{L}^{ω} − E + iε)^{−1} − (H^{ω} − E + iε)^{−1}ϕ
= 2(H^{ω} − E + iε)^{−1}(i∇ϕ)(i∇ + A)(H_{L}^{ω} − E + iε)^{−1}
−(H_{L}^{ω} − E + iε)^{−1}(Δϕ)(H_{L}^{ω} − E + iε)^{−1}

52 Naomasa Ueki

and

|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||
≤ (2|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H^{ω} − E − iε)^{−1}(i∇ + A)(∇ϕ)|||
+|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}(Δϕ)|||)
×|||(χ_{0,b_{2}L} − χ_{0,b_{2}L−1})(H^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||.

As in (4.15), we have |||(∇ϕ)(i∇ + A)(H_{L}^{ω} − E + iε)^{−1}|||

≤ \frac{C(d)}{dist(σ(H_{L}^{ω}), E)}(dist(σ(H_{L}^{ω}), E) + |E + iε| + ess sup_{Λ_{L}} V_{−}^{ω} + 1)^{1/2}.

Therefore we have |||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||

≤ \frac{C(d)}{dist(σ(H_{L}^{ω}), E)}(dist(σ(H_{L}^{ω}), E) + |E + iε| + ess sup_{Λ_{L}} V_{−}^{ω} + 1)^{1/2}

×|||(χ_{0,b_{2}L} − χ_{0,b_{2}L−1})(H^{ω} − E − iε)^{−1}χ_{0,b_{1}L}|||

and the right hand side of (4.16) is dominated by

P (|||(χ_{0,b_{2}L} − χ_{0,b_{2}L−1})(H^{ω} − E − iε)^{−1}χ_{0,b_{1}L}||| > a/L^{γ},

ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β})

+P ( dist(σ(H_{L}^{ω}), E) ≤ C(d)(|E + iε| + (log RL)^{(ϵ+1)/β} + 1)^{1/2}/L^{γ}). Then, by the Chebyshev inequality and the Wegner estimate, we obtain (4.16).

If E ∉ σ(H_{L}^{ω}), then (H_{L}^{ω} − E)^{−1} = (H_{L}^{ω} − E − iε)^{−1} − iε(H_{L}^{ω} − E)^{−1}(H_{L}^{ω} − E − iε)^{−1}

and

|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E)^{−1}χ_{0,b_{1}L}|||

≤ |||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}χ_{0,b_{1}L}||| + ε/ dist(σ(H_{L}^{ω}), E)^{2}. Therefore the right hand side of (4.17) is dominated by

P (|||(χ_{0,b_{3}L} − χ_{0,b_{2}L})(H_{L}^{ω} − E − iε)^{−1}χ_{0,b_{1}L}||| > a/4,

ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β})

+P ( dist(σ(H_{L}^{ω}), E) ≤ 2\sqrt{ε/a}).

Then, by (4.16) and the Wegner estimate, we obtain (4.17).

Proof of Theorem 4.1. Let I be a compact interval included in J and θ > bd/h specified later. For any E ∈ I and 36 < L ∈ 6ℕ, we set

P_{E,L} := P (E ∈ σ(H_{L}^{ω}))
+P (E ∉ σ(H_{L}^{ω}), |||Γ_{0,L}(H_{L}^{ω} − E)^{−1}χ_{0,L/3}||| > 1/(2L^{θ})).

If E ∉ σ(H_{L}^{ω}), then, by using the resolvent identity and χ_{0,L} = χ_{0,2L/3}+(χ_{0,L}− χ_{0,2L/3}) as in [16], we have

|||Γ_{0,L}(H_{L}^{ω} − E)^{−1}χ_{0,L/3}|||
≤ |||Γ_{0,L}(H_{L}^{ω} − E − iε)^{−1}χ_{0,L/3}|||
+ε|||Γ_{0,L}(H_{L}^{ω} − E)^{−1}χ_{0,2L/3}|||/ dist(σ(H_{L}^{ω}), E)
+ε|||(χ_{0,L} − χ_{0,2L/3})(H_{L}^{ω} − E − iε)^{−1}χ_{0,L/3}|||/ dist(σ(H_{L}^{ω}), E).

Therefore we have

P_{E,L} ≤ ∑_{j=1}^{5} I_{j},

where

I_{1} := P (|||Γ_{0,L}(H_{L}^{ω} − E − iε)^{−1}χ_{0,L/3}||| > 1/(4L^{θ}),

ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}),

I_{2} := P (E ∉ σ(H_{L}^{ω}), |||Γ_{0,L}(H_{L}^{ω} − E)^{−1}χ_{0,2L/3}||| > 1/2,

ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}),

I_{3} := P (|||(χ_{0,L} − χ_{0,2L/3})(H_{L}^{ω} − E − iε)^{−1}χ_{0,L/3}||| > 1/4,

ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}),

I_{4} := P (dist(σ(H_{L}^{ω}), E) ≤ 3εL^{θ}),

I_{5} := P (ess sup_{Λ_{RL}} V_{−}^{ω} > (log RL)^{(ϵ+1)/β})

and ϵ > 0 is taken arbitrarily. By the assumptions (W) and (F), we have

I_{4} := QĨ_{0}(3ε)^{h}L^{θh+bd}

and

I_{5} := δ_{1} exp(−δ_{2}(log RL)^{ϵ}).

By taking a = L^{−θ}, b_{1} = 1/3, b_{2} = (L − 3)/L and b_{3} = (L − 1)/L in Lemma 4.2, we have

I_{1} ≤ I′_{1} + QĨ_{0}C(I, d, h)(log L)^{(ϵ+1)h/(2β)}/L^{γh−bd},

where

I′_{1} := L^{γ+θ}E[|||Γ_{0,L−2}(H^{ω} − E − iε)^{−1}χ_{0,L/3}|||

: ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}]

and γ > bd/h is specified later. This is estimated as

I′_{1} ≤ I_{11} + I_{12},

where

I_{11} := L^{γ+θ}E[|||Γ_{0,L−2}(H^{ω} − E − iε)^{−1}f(H^{ω})χ_{0,L/3}|||]

and

I_{12} := L^{γ+θ}E[|||Γ_{0,L−2}(H^{ω} − E − iε)^{−1}(1 − f(H^{ω}))χ_{0,L/3}|||

: ess sup_{Λ_{RL}} V_{−}^{ω} ≤ (log RL)^{(ϵ+1)/β}].

As in [16], we have

I_{11} ≤ L^{γ+θ} ∑_{y∈ℤ^{d}∩Λ_{L/3}} E[|||Γ_{0,L−2}(H^{ω} − E − iε)^{−1}f(H^{ω})χ_{y}|||]

= L^{γ+θ} ∑_{y∈ℤ^{d}∩Λ_{L/3}} E[|||Γ_{−y,L−2}(H^{ω} − E − iε)^{−1}f(H^{ω})χ_{0}|||]

≤ L^{γ+θ}(\frac{L}{3})^{d}(\frac{L}{3} − \frac{3}{2})^{−n/2}E[|||⟨X⟩^{n/2}(H^{ω} − E − iε)^{−1}f(H^{ω})χ_{0}|||]

≤ \frac{12^{n/2}}{3^{d}L^{(n/2)−d−γ−θ}} O_{ε}(n, f, E)^{1/2},

where

O_{ε}(n, f, E) := E[|||⟨X⟩^{n/2}(H^{ω} − E − iε)^{−1}f(H^{ω})χ_{0}|||_{2}^{2}].

By using Proposition 4.4, we have

I_{12} ≤ C(c, f, k, I, β, ϵ)(log L)^{(ϵ+1)(2k+d−3)/(4β)}/L^{k−1−γ−θ−3d/2}

for any (d + 5)/2 ≤ k ∈ ℕ.

I_{2} and I_{3} are also estimated similarly by using Lemma 4.2 appropriately. Therefore we obtain

P_{E,L} ≤ ∑_{j=1}^{3} P_{E,L}^{j}

where

P_{E,L}^{1} := \frac{C(n, d)}{L^{(n/2)−d−γ−θ}} O_{ε}(n, f, E)^{1/2},

P_{E,L}^{2} := C(d, f, k, I, β, ϵ)(log L)^{(ϵ+1)(2k+d−3)/(4β)}/L^{k−1−γ−θ−3d/2}

+QĨ_{0}C(I, d, h)(log L)^{(ϵ+1)h/(2β)}/L^{γh−bd}
+δ_{1} exp(−δ_{2}(log RL)^{ϵ})

and

P_{E,L}^{3} := QĨ_{0}\{2^{h}\sqrt{ε^{h}L^{2bd}} + (3ε)^{h}L^{θh+bd}\}.

For any ε > 0 and 0 < p_{0} < 1, we set

L(ε, p_{0}) := [(\frac{p_{0}}{(3ε)^{h}} )^{1/(θh+bd)} ∧ (\frac{p_{0}}{ε^{h}} )^{1/(2bd)}]_{6ℕ}.

Then, since (3ε)^{h}L(ε)^{θh+bd}, ε^{h}L^{2bd} ≤ p_{0}, we have

P_{E,L(ε,p_{0})}^{3} ≤ QĨ_{0}(2^{h}\sqrt{p_{0}} + p_{0}).

By Proposition 6.1 in [16], we have

O := \underline{lim}_{ε↓0} ε^{1+α} ∫_{ℝ} O_{ε}(n, f, E)dE < ∞.

Then there exists a sequence {ε_{m}}_{m} such that ε_{m} ↘ 0 as m → ∞, L(ε_{m}, p_{0}) > 36 and

ε_{m}^{1+α} ∫_{ℝ} O_{ε_{m}}(n, f, E)dE ≤ 2O.

For any m ∈ ℕ and M > 0, we set

A_{m.M} := {E ∈ I : ε_{m}^{1+α}O_{ε_{m}}(n, f, E) ≤ M}.

Then we have |I − A_{m,M}| ≤ 2O/M. Since

ε_{m} ≥ C(θ, h, b, d)(\frac{p_{0}}{L(ε_{m}, p_{0})^{θh+bd}} )^{1/h},

56 Naomasa Ueki

we have

P_{E,L(ε_{m},p_{0})}^{1} ≤ \frac{C(d, n, θ, h, b)\sqrt{M}}{p_{0}^{(1+α)/(2h)}L(ε_{m}, p_{0})^{(n/2)−d−γ−θ−(1+α)(θh+bd)/(2h)}}

for any E ∈ A_{m,M}. Taking v > 0 arbitrarily, we set M(m) := 2OL(ε_{m}, p_{0})^{v}.

For any E′ ∈ I, there exists E ∈ A_{m,M(m)} such that |E−E′| ≤ L(ε_{m}, p_{0})^{−v}. Then the quantity

P′_{E′,L(ε_{m},p_{0})}

:= P (E′ ∈ σ(H_{L(ε_{m},p_{0})}^{ω}))
+P (E′ ∉ σ(H_{L(ε_{m},p_{0})}^{ω}),
|||Γ_{0,L(ε_{m},p_{0})}(H_{L(ε_{m},p_{0})}^{ω} − E′)^{−1}χ_{0,L(ε_{m},p_{0})/3}||| > L(ε_{m}, p_{0})^{−θ})

is dominated by

P_{E,L(ε_{m},p_{0})} + P (dist(E, σ(H_{L(ε_{m},p_{0})}^{ω})) < CL(ε_{m}, p_{0})^{−(v−θ)/2}). Therefore we obtain P′_{E′,L(ε_{m},p_{0})}

≤ \frac{C(n, d, θ, h, b)\sqrt{O}}{p_{0}^{(1+α)/(2h)}L(ε_{m}, p_{0})^{(n/2)−d−γ−θ−(1+α)(θh+bd)/(2h)−v/2}}

+C(d, f, k, I, β, ϵ)(log L(ε_{m}, p_{0}))^{(ϵ+1)(2k+d−3)/(4β)}/L(ε_{m}, p_{0})^{k−1−γ−θ−3d/2} +QĨ_{0}C(d, h, I)(log L(ε_{m}, p_{0}))^{(ϵ+1)h/(2β)}/L(ε_{m}, p_{0})^{γh−bd} +δ_{1} exp(−δ_{2}(log RL(ε_{m}, p_{0}))^{ϵ}) +QĨ_{0}(2^{h}\sqrt{p_{0}} + p_{0}) +QĨ_{0}C(h)/L(ε_{m}, p_{0})^{h(v−θ)/2−bd}.

If we can take n ≥ 0, γ, θ > bd/h, v > 0 and (d + 5)/2 ≤ k ∈ ℕ so that

n/2 > d + γ + θ + (1 + α)(θh + bd)/(2h) + v/2,

k > γ + θ + 3d/2 − 1, (4.18)

hv/2 > hθ/2 + bd,

then we have

\overline{lim}_{m→∞} P′_{E′,L(ε_{m},p_{0})} ≤ QĨ_{0}(2^{h}\sqrt{p_{0}} + p_{0})

and

\underline{lim}_{L→∞} P (E′ ∉ σ(H_{L}^{ω}), |||Γ_{0,L}(H_{L}^{ω} − E′)^{−1}χ_{0,L/3}||| ≤ L^{−θ}) = 1,

from which we have E′ ∈ Σ_{MSA}. For (4.18) to be satisfied, (4.4) is necessary and sufficient.

§4.3. Proof of Proposition 4.1

To prove Proposition 4.1, we use the following instead of Lemma A.4 in [16]:

Lemma 4.3. For L > 0 and a compact set B in ℝ, we set Θ^{ω} := 1 − (min B ∧ ess inf_{Λ_{2L}} V^{ω}). Then, for any d/2 < m ∈ ℕ, there exists a finite constant C(d, m) such that

Tr[χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}χ_{0,L}] ≤ C(d, m)L^{d}.

Proof. We take {ϕ_{j}}_{j∈ℕ} ⊂ C_{0}^{∞}(Λ_{2L} → [0, 1]) so that ϕ_{1} ≡ 1 on Λ_{L} and ϕ_{j+1} ≡ 1 on supp ϕ_{j} for any j ∈ ℕ. Then we have

E(B : H^{ω})(H^{ω} + Θ^{ω})^{−1}ϕ_{j}

= E(B : H^{ω})ϕ_{j}(H_{2L}^{ω} + Θ^{ω})^{−1} (4.19)

+E(B : H^{ω})(H^{ω} + Θ^{ω})^{−1}(i∇ + A_{ω})(2i∇ϕ_{j})(H_{2L}^{ω} + Θ^{ω})^{−1}
+E(B : H^{ω})(H^{ω} + Θ^{ω})^{−1}(Δϕ_{j})(H_{2L}^{ω} + Θ^{ω})^{−1}

for any j ∈ ℕ. By this, we have

χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}χ_{0,L}

= χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m+1}ϕ_{1}(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L} (4.20)

+χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}(i∇ + A_{ω})(2i∇ϕ_{1})
×(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L}
+χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}(Δϕ_{1})(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L}.

By using (4.19) again, the third term is rewritten as

χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}ϕ_{2}(H_{2L}^{ω} + Θ^{ω})^{−1}(Δϕ_{1})(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L}

+χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}(i∇ + A_{ω})(2i∇ϕ_{2})(H_{2L}^{ω} + Θ^{ω})^{−1}
×(Δϕ_{1})(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L}
+χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}(Δϕ_{2})(H_{2L}^{ω} + Θ^{ω})^{−1}
×(Δϕ_{1})(H_{2L}^{ω} + Θ^{ω})^{−1}χ_{0,L}.

We repeat this calculation also for the other terms of the right hand side of (4.20).

By the Feynman-Kac-Itô formula and the diamagnetic inequality, we have

Tr[exp(−t(H_{2L}^{ω} + Θ^{ω}))] ≤ e^{−t} Tr[exp(tΔ_{2L})]
≤ C(d)L^{d}e^{−t}/t^{d/2},

where exp(−t(H_{2L}^{ω} + Θ^{ω})) and exp(tΔ_{2L}) are the heat semigroups generated by H_{2L}^{ω} + Θ^{ω} and the negative Dirichlet Laplacian −Δ_{2L} on Λ_{2L}, respectively (cf. [3]). Thus, for any p > d, we have

|||(H_{2L}^{ω} + Θ^{ω})^{−1/2}|||_{p}^{p} ≤ C(d, p)L^{d}.

By applying this and

|||(i∇ + A_{ω})(H_{2L}^{ω} + Θ^{ω})^{−1/2}||| ≤ 1

to the expansion of χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}χ_{0,L}, we have

Tr[χ_{0,L}E(B : H^{ω})(H^{ω} + Θ^{ω})^{−m}χ_{0,L}]
≤ C′(d, m)|||(H_{2L}^{ω} + Θ^{ω})^{−1/2}|||_{2m}^{2m} ≤ C(d, m)L^{d}

for m > d/2.

Proof of Proposition 4.1. We can prove (4.7) and the first and the second inequalities in (4.5) and (4.6) by the same argument as in [16].

For the third inequalities of (4.5) and (4.6), we estimate as

M_{ω}(n, f, t) ≤ C(d, n) ∑_{a∈ℤ^{d}}⟨a⟩^{n}|||χ_{a}Y(H^{ω})χ_{0}||| × |||χ_{0}Y(H^{ω})χ_{a}|||_{1}

as in [16], where Y(u) = e^{−itu}f(u) for u ∈ ℝ. By Proposition 4.4, we have

|||χ_{a}Y(H^{ω})χ_{0}||| ≤ C(d, k, f)⟨t⟩^{k+1}( ess sup_{Λ_{RL(a)}} V_{−}^{ω} + 1)^{(2k+d−3)/4}⟨a⟩^{−(2k−d−3)/2}

for any (d + 5)/2 ≤ k ∈ ℕ, where L(a) = 2|a| + 1. On the other hand, we have

|||χ_{0}Y(H^{ω})χ_{a}|||_{1} ≤ 4 Tr[χ_{0,L(a)}|Y(H^{ω})|χ_{0,L(a)}]

≤ C(f, m)|Θ^{ω}|^{[[d/2]]} Tr[χ_{0,L(a)}E(supp f : H^{ω})(H^{ω} + Θ^{ω})^{−[[d/2]]}χ_{0,L(a)}],

where Θ^{ω} := 1 − (min supp f ∧ ess inf_{Λ_{2L(a)}} V^{ω}). Then we have

|||χ_{0}Y(H^{ω})χ_{a}|||_{1} ≤ C(f, d)( ess sup_{Λ_{2L(a)}} V_{−}^{ω} + 1)^{[[d/2]]}⟨a⟩^{d}

by Lemma 4.3. Therefore we obtain

M_{ω}(n, f, t) ≤ C(d, n, k, f)⟨t⟩^{k+1} ∑_{a∈ℤ^{d}}⟨a⟩^{(2n+3d+3−2k)/2}

×( ess sup_{Λ_{RL(a)}} V_{−}^{ω} + 1)^{(4[[d/2]]+2k+d−3)/4}.

For (4.5), we use Lemma 3.2 (ii) to obtain

ess sup_{Λ_{RL(a)}} V_{−}^{ω} + 1 ≤ C(β, ω)(log⟨a⟩ + 1)^{2/β}.

Then we have

M_{ω}(n, f, t) ≤ C(d, n, k, f, β, ω)⟨t⟩^{k+1} ∑_{a∈ℤ^{d}}⟨a⟩^{(2n+3d+3−2k)/2}

×(log⟨a⟩ + 1)^{(4[[d/2]]+2k+d−3)/(2β)}.

The right hand side is finite if k > (2n + 5d + 3)/2. Therefore, by taking k = [[n + (5d + 1)/2]] + 1, we obtain (4.5).

For (4.6), we use Lemma 3.2 (i) to obtain

E[( ess sup_{Λ_{RL(a)}} V_{−}^{ω} + 1)^{(4[[d/2]]+2k+d−3)/4}]

≤ C(d, k, β)(log⟨a⟩ + 1)^{(4[[d/2]]+2k+d−3)/(4β)}.

Then we have

𝕄(n, f, t) ≤ C(d, n, k, f, β)⟨t⟩^{k+1} ∑_{a∈ℤ^{d}}⟨a⟩^{(2n+d+3−2k)/2}

×(log⟨a⟩ + 1)^{(4[[d/2]]+2k+d−3)/(4β)}.

Therefore, by taking k = [[n + (5d + 1)/2]] + 1, we obtain (4.6).

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