Simple examples of Lifschitz tails in Gaussian random magnetic fields

by

Naomasa UEKI

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan

—————————————-

Abstract. – This is a first attempt to investigate the asymptotic behavior of the integrated density of states at the infimum of the spectrum for Schrödinger operators with magnetic fields which are Gaussian random fields. In simple examples it is shown that the integrate density of states decays exponentially. These examples shall give a hint to consider in more general framework.

Key words: Lifschitz tails, random Schrödinger operators, magnetic fields, Gaussian random fields

Résumé. – C’est un premier essai pour rechercher le comportement asymptotique, á la borne inférieure du spectre, de la densité d’état intégrée pour l’opérateur de Schrödinger avec champ magnétique qui est un champ aléatoire gaussiens. Nous allons montrer que la densité d’état intégrée décroissons exponentiellement en exemples simples. Ces exemples donneront une indication pour considérer en cadres plus généralaux.

—————————————-

1. INTRODUCTION

This is a first attempt to consider the asymptotic behavior of the integrated density of states N(λ) of the Schrödinger operator L(b_{ω}) with the magnetic field db_{ω} at the infimum of the spectrum, where

L(b_{ω}) = \frac{1}{2} ∑_{j=1}^{2} (i\frac{∂}{∂x^{j}} − b_{ω}^{j}(x))^{2} (i = \sqrt{−1}) (1.1)

and b_{ω} = ∑_{j=1}^{2} b_{ω}^{j}(x)dx^{j} (ω ∈ Ω, x ∈ ℝ^{2}) is a 1-form valued Gaussian random field on a probability space Ω that is stationary and ergodic with respect to the shift in the variable x ∈ ℝ^{2}. We show that N(λ) decays exponentially in the following two cases:

Case I. b_{ω}^{1}(x) = 0 and b_{ω}^{2}(x) = g_{ω}(x^{1}) depends only on the first coordinate.

Case II. b_{ω}(x) = d^{∗}f_{ω}(x), where f_{ω}(x) is a scalar valued Gaussian random field.

The main results are stated in Theorems 3.1 and 4.1 below. These simple examples shall give a hint to consider in more general framework.

The asymptotics of the integrated density of states has been studied mainly for the operator

H(V_{ω}) = −\frac{1}{2} ∑_{j=1}^{d} (\frac{∂}{∂x^{j}} )^{2} + V_{ω}(x)  (1.2)

on ℝ^{d}, with a real valued random field V_{ω}, as other spectral properties. For this aspect, we refer Carmona-Lacroix [4] and Pastur-Figotin [21]. In particular, the exponential decay of the integrated density of states conjectured by Lifschitz [16], which is governed by both the kinetic energy and the potential energy, was first proved by the large deviation theory of Donsker and Varadhan in the case that

V_{ω}(x) = ∫ u(x − y)μ_{ω}(dy),  (1.3)

where μ_{ω}(dy) is a Poisson random measure and u is a nonnegative integrable function on ℝ^{d} such that u(x) = o(|x|^{−d−2}) as |x| → ∞ [6], [20]. For the operator

\frac{1}{2} ∑_{j=1}^{d} (i\frac{∂}{∂x^{j}} − b_{ω}^{j}(x))^{2} + V_{ω}(x),

Matsumoto [18] and the author [28] studied simple cases where the result is not affected by the magnetic field. The most studied case is that db_{ω} is a deterministic constant and V_{ω} is as in (1.3) [2], [7], [10]. This case is related with the quantum Hall effect. Broderix, Hundertmark, Kirsch and Leschke [2] showed that the integrated density of states decays exponentially and determined the leading term which are governed solely by V_{ω}, when u decays polynomially. Erdös [7] showed that the integrated density of states decays polynomially and determined the leading order which is governed also by the magnetic field when u is of compact support. Hupfer, Leschke and Warzel [10] studied the case that u decays exponentially and showed that the situation on the dependence of the magnetic field changes when u decays as exp(−C|x|^{2}). Form these results the effect of the vector potential is weaker than that of the scalar potential. Therefore our problem in this paper is important mathematically to make clear the effect of the vector potential. On the other hand, the operator with random vector potential having the same form with (1.1) is studied also in the nonlinear electromagnetic field theory (see e.g. [1], [19] and the references therein). In that case, the field b_{ω} is a distribution valued random field and the analysis is much harder than ours.

Our problem is equaivalent, by the Tauberian theory, to investigate the asymptotic behavior as t ↑ ∞ of the Laplace-Stieltjes transform

\begin{matrix}Ñ(t) := & ∫_{0}^{∞} e^{−tλ}dN(λ) \\ = & \frac{1}{2πt}E^{w×b}[ exp (i ∑_{j=1}^{2} ∫_{0}^{t} b_{ω}^{j}(w(s)) ∘ dw^{j}(s))|w(t) = 0], \end{matrix}  (1.4)

where E^{w×b} is the expectation with respect to the 2-dimensional Wiener process w(s) = (w^{1}(s), w^{2}(s)), s ≥ 0, starting at 0 and the random field b_{ω}. The difficult points not appearing for the operator (1.2) are that the right hand side of (1.4) is an oscillatory integral and that the phase functional is given by a stochastic integral. Therefore our problem is one of the problems on the asymptotics of the stochastic oscillatory integrals (cf.[11],[12],[17]). However in Case I these difficulties disappear. In fact in this case the expectation with respect to w^{2}(s) in the right hand side of (1.4) can be computed easily and our problem is reduced to the same problem for the operator (1.2) on ℝ with

V_{ω}(x) = \frac{1}{2}g_{ω}(x)^{2}  (1.5)

We here recall a work by Figotin [8]: he determined the leading term of the asymptotics of the integrated density of states for the operator (1.2) with

V_{ω}(x) = \frac{1}{2} ∫_{ℝ^{d}} u(x − y)g_{ω}(x)^{2}dx,  (1.6)

where u is as in (1.3), by modifying the method by Donsker and Varadhan [6], [20]. By his method, we can obtain a lower estimate of the integrated density of states. However, to obtain an upper estimate, the techniques in [8] are not enough: in [8], the function u(x) plays an important role to make the potential V_{ω}(x) good enough to apply the large deviation theory by Donsker-Varadhan for the occupation measure by the Wiener process in the weak topology (see also [5], [6], [20], [21], [29]). On the other hand, Kirsch and Martinelli give a general upper estimate of the integrated density of states for the operator (1.2) with

V_{ω}(x) = v(g_{ω}(x)),  (1.7)

where v(x) is a locally bounded positive real function on ℝ, polynomially bounded at infinity, by a functional analytic approach [15]. From their estimate, we can obtain an upper estimate of the integrated density of states. For Case II, we can give some upper and lower bounds which are treated samely as in Case I.

The organization of this paper is as follows. In Section 2, we follow Figotin [8] and Kirsch and Martinelli [15] to show that the integrated density of states for the operator H(V_{ω}) in (1.2) with the potential V_{ω} as that in (1.5) decays exponentially at the infimum of the spectrum. In Section 3, we consider Case I and in Section 4, we consdier Case II.

2. SQUARE OF GAUSSIAN RANDOM FIELDS

In this section we follow Figotin [8] (see also Pastur-Figotin [21]) and Kirsch and Martinelli [15] to show that the integrated density of states N(λ) for the operator H(V_{ω}) in (1.2) with the potential

V_{ω}(x) = \frac{1}{2} ∑_{j=1}^{q} g_{ω}^{j}(x)^{2}  (2.1)

decays exponentially at the infimum of the spectrum, where {(g_{ω}^{j}(x))_{j=1,2,··· ,q}, x ∈ ℝ^{d}} is the ℝ^{q}-valued stationary ergodic Gaussian random field with mean zero and the covariance β_{jk}(x) = E[g_{ω}^{j}(x)g_{ω}^{k}(0)], j, k = 1, 2, ⋯, q. In the following, we set

β̂_{jk}(ξ) = ∫_{ℝ^{d}} e^{−iξ·x}β_{jk}(x)dx,

we use the polar coordinate ξ = (ζ, θ), ζ > 0, θ ∈ S^{d−1}, and we denote the eigenvalues of any nonnegative q × q matrix A by μ_{1}(A) ≤ μ_{2}(A) ≤ ⋯ ≤ μ_{q}(A).

We introduce the following conditions: for α > d and τ > 0,

(P1, α, d) (i) sup_{ξ} |ζ^{α}β̂_{jk}(ξ)| < ∞;

(ii) ζ^{α}β̂_{jk}(ξ) → β^{∘}_{jk}(θ) as ζ → ∞ for any θ ∈ S^{d−1} and some β^{∘}_{jk}(θ) ∈ ℝ;

(iii) sup_{x} |x|^{m}|β_{jk}(x)| < ∞ for some m > d + 2 + 2d/(α − d); (iv) vol {θ ∈ S^{d−1}; μ_{q}(β^{∘}(θ)) ≠ 0} is positive, where vol is the volume measure on S^{d−1}.

(E1, τ, d) (i) There exist σ > 0, 1 ≤ q_{∗} ≤ q and functions R_{j}(ξ), 1 ≤ j ≤ q_{∗} such that sup_{θ} |μ_{j}(β̂(ξ)) exp(σζ^{τ} + R_{j}(ξ)) − 1| → 0 as ζ → ∞, where sup_{θ} |R_{j}(ξ)|ζ^{−τ} → 0 as ζ → ∞ for any 1 ≤ j ≤ q_{∗} and the support of μ_{j}(β̂(ξ)) is compact for any j > q_{∗}; (ii) sup_{x} |x|^{m}|β_{jk}(x)| < ∞ for some m > d + 2.

(C1, d) (i) The support of β̂_{jk}(ξ) is compact; (ii) sup_{x} |x|^{m}|β_{jk}(x)| < ∞ for some m > d + 2; (iii) β_{jj}(0) > 0 for some j.

Then the result is the following:

Theorem 2.1. – (i) We assume the condition (P1, α, d) for some α > d. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−(α−d+2)/{2(αd^{−1}−1)}} ≤ log N(λ) ≤ −c′λ^{−d/2} log λ^{−1}

for any 0 < λ < ε.

(ii) We assume the condition (E1, τ, d) for some τ > 0. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−d/2} log^{(d+τ)/τ} λ^{−1} ≤ log N(λ) ≤ −c′λ^{−d/2} log λ^{−1}

for any 0 < λ < ε.

(iii) We assume the condition (C1, d). Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−d/2} log λ^{−1} ≤ log N(λ) ≤ −c′λ^{−d/2} log λ^{−1}

for any 0 < λ < ε.

Remark 2.1. – In our methods, the leading term in Theorem 2.1 (iii) is not determined. In fact, if we assume d = q = 1, β(x) ≥ 0 and the condition (C1, 1), then our method gives

\underline{lim}_{λ↓0} \frac{\sqrt{λ} log N(λ)}{log λ^{−1}} ≥ −\frac{3^{2}}{2^{5/2}π} | supp β̂| =: −c_{1}(β̂)

and

\overline{lim}_{λ↓0} \frac{\sqrt{λ} log N(λ)}{log λ^{−1}} ≥ −\frac{π}{3\sqrt{6}}β(0)/ ∫_{ℝ} |β(x)|dx =: −c_{2}(β̂).

For these bounds it holds that sup\{c_{2}(β̂)/c_{1}(β̂) : β satisfies (C1, 1) and β(x) ≥ 0\} = \frac{2π}{27\sqrt{3}} < 1.

This theorem is obtained from the corollary of Theorem 2.2 and Theorem 2.3 below. For Theorem 2.2 and its corollary, we follow Figotin [8] (see also Pastur-Figotin [21]). To state the theorem, we introduce the following condition:

(H1, d) (i) sup_{ξ} Tr[β̂(ξ)] ∏_{j=1}^{d}(1 + |ξ^{j}|^{1+γ}) < ∞ for some γ > 0; (ii) sup_{x} |x|^{m}|β_{jk}(x)| < ∞ for some m > d + 2 + 2γ^{−1}; (iii) lim_{t→∞} Φ(ct)/Φ(t) = c^{κ} for any c > 0 and some κ ∈ [0, 1), where

Φ(t) := \frac{1}{(2π)^{d}} ∫_{ℝ^{d}} dξ Tr[log(I + tβ̂(ξ)].  (2.2)

Let η(t), t > 0, be the solution of

η^{d/2}(t)Φ (\frac{t}{η^{d/2}(t)}) = \frac{t}{η(t)}.  (2.3)

Then we have the following:

Theorem 2.2. – Under the condition (H1, d), we have

\underline{lim}_{t→∞} \frac{η(t)}{t} log Ñ(t) > −∞.  (2.4)

From this theorem, we have the following:

Corollary. – (i) Under the condition (P1, α, d) for some α > d, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{(α−d+2)/\{α(d+2)/d−d\}}} > −∞.  (2.5)

(ii) Under the condition (E1, τ, d) for some τ > 0, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{d/(d+2)} log^{\{2(d+τ)\}/\{τ(d+2)\}} t} > −∞.  (2.6)

(iii) Under the condition (C1, d), we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{d/(d+2)} log^{2/(d+2)} t} > −∞.  (2.7)

Theorem 2.3 is a special case of a theorem obtained by Kirsch and Martinelli [15]. To state the theorem, we introduce the following condition:

(H2, d) (i) β_{jj}(x) ∈ L_{1}(ℝ^{d}); (ii) β_{jj}(x) is Riemannian approximable as lim_{ε↓0} ∑_{y∈ℤ^{d}} ε^{d}β_{jj}(εy) = ∫_{ℝ^{d}} β_{jj}(x)dx.

Then we have the following:

Theorem 2.3. – We assume the condition (H2, d). Then for any ε > 0 and 0 < s < 1, there exist positive constants c and λ_{0} independent of the random field g_{ω} such that

N(λ) ≤ cq exp \{ − (\frac{2^{1−d/2}π^{d}d^{d/2}}{B(d + 2)^{1+d/2}} − ε)\frac{1}{λ^{d/2}} log \sqrt{\frac{q min_{1≤j≤q} β_{jj}(0)}{λ^{s}}}\} (2.8)

for any 0 < λ < λ_{0}, where

B = max_{1≤j≤q} ∫_{ℝ^{d}} |β_{jj}(x)|dx/β_{jj}(0).

In particular we have

\overline{lim}_{λ↓0} \frac{λ^{d/2} log N(λ)}{log λ^{−1}} ≤ −\frac{π^{d}d^{d/2}}{B(d + 2)^{1+d/2}2^{d/2}} .  (2.9)

Proof of Theorem 2.3. – As in (21) in [15], we have

N(λ) ≤ k(1 + α)^{d/2}λ^{d/2}P (λ_{1}(−Δ_{Λ}^{N} + V_{ω}) < 2λ),

where k is a positive depending only on the dimension d, Λ = [−L/2, L/2]^{d}, L = π(1 + α)^{−1/2}(1 + η)^{−1/2}(2λ)^{−1/2}, α is a positive constant specified later, η is an arbitrarily small positive constant and λ_{1}(−Δ_{Λ}^{N} + V_{ω}) is the least eigenvalue of the operator −Δ_{Λ}^{N} + V_{ω} with the Neumann condition on Λ. We dominate the probability as

P (λ_{1}(−Δ_{Λ}^{N} + V_{ω}) < 2λ) ≤ ∑_{j=1}^{q} P (λ_{1}(−Δ_{Λ}^{N} + qg_{ω}^{j}(x)^{2}) < 2λ).

We estimate the each probability in the right hand side as in the proof of Theorem 6 in [15]. Then we have

P (λ_{1}(−Δ_{Λ}^{N} + qg_{ω}^{j}(x)^{2}) < 2λ) ≤ exp ( − \frac{|Λ|}{Q_{j}}G_{j}(α, λ)),

where |Λ| = L^{d} is the volume,

Q_{j} = ∫_{ℝ^{d}} |β_{jj}(x)|dx/β_{jj}(0),
G_{j}(α, λ) = sup_{t≥0} \{\frac{t}{1 + α} − log ( exp (\frac{t}{α})P (qg_{ω}^{j}(0)^{2} ≤ (2λ)^{s})

+ exp (\frac{2tλ}{(2λ)^{s} + 2αλ})P (qg_{ω}^{j}(0)^{2} > (2λ)^{s}))\}

=\frac{(2λ)^{s} + 2αλ}{(1 + α)(2λ)^{s}} log P (qg_{ω}^{j}(0)^{2} ≤ (2λ)^{s})^{−1}

+ (\frac{α((2λ)^{s} + 2αλ)}{(1 + α)(2λ)^{s}} − α(2λ)^{1−s}) log P (qg_{ω}^{j}(0)^{2} > (2λ)^{s})^{−1}

+ G_{0}(α, λ)

and G_{0}(α, λ) is a function depending only on s such that \overline{lim}_{λ↓0}G_{0}(α, λ) > −∞. We estimate as P (qg_{ω}^{j}(0)^{2} > (2λ)^{s}) ≤ 1 and

P (qg_{ω}^{j}(0)^{2} ≤ (2λ)^{s}) ≤ \sqrt{\frac{2(2λ)^{s}}{qπβ_{jj}(0)}}.

Then for any ε > 0, there exist λ_{1} > 0 independent of g_{ω} such that

G_{j}(α, λ) ≥ (\frac{α}{1 + α} − ε) log \sqrt{\frac{qβ_{jj}(0)}{λ^{s}}}

for any 0 < λ < λ_{1}. Thus for any ε > 0, there exist λ_{2} > 0 independent of g_{ω} such that

N(λ) ≤ k(1 + α)^{d/2}q exp ( − (\frac{π^{d}α}{B(1 + α)^{1+d/2}(1 + η)^{d/2}2^{d/2}} − ε)\frac{1}{λ^{d/2}} log \sqrt{\frac{q min_{j} β_{jj}(0)}{λ^{s}}})

for any 0 < λ < λ_{2}. By taking the minimum with respect to α, we obtain (2.8).

To prove Theorem 2.2, we prepare several lemmas:

Lemma 2.1. – The function Φ(t) defined in (2.2) satisfies the following:

(i) Φ(t) is well defined as a positive function of t > 0.

(ii) Φ is concave and monotone increasing. (iii) Φ(t)/t is monotone decreasing. (iv) for any ε > 0, there exists C(ε) > 0 such that Φ(t) ≤ C(ε)t^{1/(1+γ)+ε} for any t ≥ 1,

where γ is the constant in the condition (H1, d) (i).

Proof. – (iv) is proven by (H1, d) (i) and the induction on the dimension d.

Lemma 2.2. – There exists a positive function η(t), t > 0, such that the equation (2.3) is satisfied identically. This function satisfies the following:

(i) the functions η(t) and t/η^{d/2+1}(t) are monotone increasing and tend to ∞ as t → ∞. (ii) lim_{t→∞} t/η^{δ}(t) = 0 for any δ > d/2 + (1 + γ)/γ. (iii) lim_{t→∞}(η(t)/t) log η(t) = 0.

Proof. – There exists a positive function T (t), t > 0, such that Φ(T (t))/T (t) = (T (t)/t)^{2/d} by Lemma 2.1 (iii). Then the function η(t) := (t/T (t))^{2/d} satisfies (2.3). The rest of this proposition is also proven by using Lemma 2.1.

We next consider the functional

Φ(ϕ, t) = −\frac{η(t)}{t} log E^{g}[ exp ( − \frac{t}{2} ∫_{ℝ^{d}} ∑_{j=1}^{q}(g_{ω}^{j}(\sqrt{η(t)}x))^{2}ϕ(x)dx)]  (2.10)

in 0 ≤ ϕ ∈ L_{1}(ℝ^{d}) and t > 0, where E^{g} is the expectation with respect to g_{ω}:

Lemma 2.3. – (i) For any t > 0, let B(t) be an operator on L_{2}(ℝ^{d} → ℝ^{q}) defined by

\{B(t)φ\}^{j}(x) = ∫_{ℝ^{d}} ∑_{k=1}^{q} β_{jk}(\sqrt{η(t)}(x − y))φ^{k}(y)dy.

Then, for any 0 ≤ ϕ ∈ L_{1}(ℝ^{d}) and t > 0, Φ(ϕ, t) is represented as

Φ(ϕ, t) = \frac{η(t)}{2t} Tr[log(I + t\sqrt{ϕ}B(t)\sqrt{ϕ})],  (2.11)

where Tr[·] is the trace on the space L_{2}(ℝ^{d} → ℝ^{q}).

(ii) Φ(ϕ, t) is continuous in ϕ in the norm of L_{1}(ℝ^{d}).

(iii) Φ(ϕ_{1}, t) ≤ Φ(ϕ_{2}, t) if ϕ_{1} ≤ ϕ_{2}.

(iv) Φ(ϕ(· + a), t) = Φ(ϕ, t) for any a ∈ ℝ^{d}.

Proof. – For (i), referring that \sqrt{ϕ}B(t)\sqrt{ϕ} is a Hilbert-Schmidt operator, we take a complete orthonormal system {φ_{n}}_{n≥1} of L_{2}(ℝ^{d} → ℝ^{q}) consisting of the eigenfunctions of \sqrt{ϕ}B(t)\sqrt{ϕ}. Let \{β_{n}\}_{n≥1} be the corresponding eigenvalues. Then we can write as

Φ(ϕ, t) = − lim_{N→∞} \frac{η(t)}{t} log E^{g×X}[ exp (i ∑_{n=1}^{N} X_{n}\sqrt{t}(g_{ω}(\sqrt{η(t)}x)\sqrt{ϕ(x)}, φ_{n}(x)))],

where (·, ·) is the inner product of the space L_{2}(ℝ^{d} → ℝ^{q}) and E^{g×X} is the expectation with respect to g_{ω} and a system {X_{n}}_{n≥1} of independent copies of the Gaussian random variable with mean zero and covariance 1. We first calculate the expectation with respect to g_{ω} and then calculate the expectation with respect to {X_{n}}. Then we have

Φ(ϕ, t) = \frac{η(t)}{2t} ∑_{n=1}^{∞} log(1 + tβ_{n}) = \frac{η(t)}{2t} Tr[log(I + t\sqrt{ϕ}B(t)\sqrt{ϕ})].

(ii) – (iv) are easily obtained from the expression (2.10) .

To treat the expression as (2.11), we prepare the following:

Lemma 2.4. – For any nonnegative operators B_{1} and B_{2} on a Hilbert space of trace class, we have the following:

(i) Tr[log(I + B_{1} + B_{2})] ≤ Tr[log(I + B_{1})] + Tr[log(I + B_{2})].

(ii) | Tr[log(I + B_{1})] − Tr[log(I + B_{2})]| ≤ ∥B_{1} − B_{2}∥(Tr[log(I + B_{1})] + Tr[log(I + B_{2})]) if Tr[B_{1}] = Tr[B_{2}], where ∥ · ∥ is the operator norm.

Proof. – (i) Since the operators B_{1} and B_{2} are of trace class, it is enough to show this lemma on a finite dimensional Hilbert space. If we set

φ(t) := Tr[log(I + B_{1})] + Tr[log(I + tB_{2})] − Tr[log(I + B_{1} + tB_{2})],

then we have

φ′(t) = Tr[{(I + tB_{2})^{−1} − (I + B_{1} + tB_{2})^{−1}}B_{2}].

Since (I + tB_{2})^{−1} ≥ (I + B_{1} + tB_{2})^{−1}, we have φ′(t) ≥ 0 and φ(t) ≥ 0.

(ii) By the same method in the proof of (i), we have

\frac{d}{dt} Tr[log(I + B(t)) − B(t)] = Tr[(B_{1} − B_{2})B(t)(I + B(t))^{−1}],

where B(t) := tB_{1} + (1 − t)B_{2} for 0 ≤ t ≤ 1. Then, by λ(1 + λ)^{−1} ≤ log(1 + λ) for any λ ≥ 0 and (i), we have

| Tr[log(I + B_{1}) − B_{1}] − Tr[log(I + B_{2}) − B_{2}]|
≤∥B_{1} − B_{2}∥(Tr[log(I + B_{1})] + Tr[log(I + B_{2})]).

(2.12)

To consider the asymptotics as t → ∞, we prepare the following:

Lemma 2.5. – Let

Φ_{R}(t) := R^{−d} Tr[log(I + tχ_{Λ(R)}Bχ_{Λ(R)})]  (2.13)

for R > 0, where Λ(R) = [−R/2, R/2]^{d} and B is an operator on L_{2}(ℝ^{d} → ℝ^{q}) defined by

{Bφ}^{j}(x) = ∫_{ℝ^{d}} ∑_{k=1}^{q} β_{jk}(x − y)φ^{k}(y)dy.

Then we have

lim_{R→∞} Φ_{R}(t) = Φ(t)

for any t ≥ 0, where Φ(t) is the function defined in (2.2).

Proof. – For any ℓ ∈ ℕ, we have

lim_{R→∞} R^{−d} Tr[(χ_{Λ(R)}Bχ_{Λ(R)})^{ℓ}] = \frac{1}{(2π)^{d}} ∫_{ℝ^{d}} dξ Tr[β̂(ξ)^{ℓ}]

by the Lebesgue convergence theorem, the condition (H1, d) (ii) and the Fourier transform. Then by using also the condition (H1, d) (i), (ii), we see that the measure

m_{R}(dλ) = R^{−d} ∑_{j} δ_{μ(j;R)}(λ)λdλ

on [0, ∞) converges weakly to

m_{∞}(dλ) = \frac{1}{(2π)^{d}} ∫_{ℝ^{d}} dξ ∑_{k} δ_{μ(k;ξ)}(λ)λdλ

as R → ∞, where, for each R > 0 and ξ ∈ ℝ^{d}, {μ(j; R)}_{j} and {μ(k; ξ)}_{k} are the eigenvalues of the operator χ_{Λ(R)}Bχ_{Λ(R)} and the matrix β̂(ξ), respectively. By taking a test function as λ^{−1} log(1 + tλ), we can complete the proof.

We now consider the asymptotics of Φ(ϕ, t) as t → ∞:

Lemma 2.6. – For any c > 0 and ρ > 0, we have

\overline{lim}_{t→∞}Φ(cχ_{Λ(ρ)}, t) ≤ \frac{1}{2}c^{κ}ρ^{d}.  (2.14)

Proof. – By Lemma 2.3 (i) and (2.3), we have

Φ(cχ_{Λ(ρ)}, t) = \frac{1}{2} \frac{Tr[log(I + ctχ_{Λ(ρ)}B(t)χ_{Λ(ρ)})]}{η^{d/2}(t)Φ(ct/η^{d/2}(t))} \frac{Φ(ct/η^{d/2}(t))}{Φ(t/η^{d/2}(t))}.  (2.15)

By the condition (H1, d) (iii) and Lemma 2.2 (i), we have

lim_{t→∞} \frac{Φ(ct/η^{d/2}(t))}{Φ(t/η^{d/2}(t))} = c^{κ}.

Then it is enough to show that

\overline{lim}_{t→∞} \frac{Tr[log(I + ctχ_{Λ(ρ)}B(t)χ_{Λ(ρ)})]}{η^{d/2}(t)Φ(ct/η^{d/2}(t))} ≤ ρ^{d}.  (2.16)

For any small ε > 0, we use Lemma 2.3 (iii) to estimate as

Tr[log(I + ctχ_{Λ(n(ρ+ε))}B(t)χ_{Λ(n(ρ+ε))})] ≥ Tr[log(I + ctΣ + ctΣ′)],  (2.17)

where Σ = ∑_{ℓ} χ_{ρ,ℓ}B(t)χ_{ρ,ℓ}, Σ′ = ∑_{ℓ≠k} χ_{ρ,ℓ}B(t)χ_{ρ,k} and χ_{ρ,ℓ}(x) = χ_{Λ(ρ)}(x − (ρ + ε)ℓ). By Lemma 2.4 (ii) and Tr[Σ′] = 0, the right hand side of (2.17) is dominated from below by

Tr[log(I + ctΣ)] − ct∥Σ′∥{Tr[log(I + ctΣ + ctΣ′)] + Tr[log(I + ctΣ)]}.

By the condition (H1, d) (ii), we have

|Σ′(x, y)| ≤ c_{1}χ_{\{|x−y|>ε\}}(\sqrt{η(t)}|x − y|)^{−m}.

Thus we have

Tr[log(I + ctχ_{Λ(n(ρ+ε))}B(t)χ_{Λ(n(ρ+ε))})]
≥\frac{1 − cc_{2}t/η^{m/2}(t)}{1 + cc_{2}t/η^{m/2}(t)} Tr[log(I + ct ∑_{ℓ} χ_{ρ,ℓ}B(t)χ_{ρ,ℓ})].

(2.18)

By the unitary transform ϕ(x) → η^{d/4}(t)ϕ(\sqrt{η(t)}x) and Lemma 2.3 (i),(iv), we rewrite as

Tr [ log (I + \frac{ct}{η^{d/2}(t)}χ_{Λ(n(ρ+ε)\sqrt{η(t)})}Bχ_{Λ(n(ρ+ε)\sqrt{η(t)})})]

≥\frac{1 − cc_{2}t/η^{m/2}(t)}{1 + cc_{2}t/η^{m/2}(t)} Tr[log(I + ctχ_{ρ}B(t)χ_{ρ})]n^{d}.

(2.19)

By using Lemma 2.5 (ii) to take the limit of n → ∞, we have

Φ(\frac{ct}{η^{d/2}(t)})(ρ + ε)^{d}η^{d/2}(t) ≥ \frac{1 − cc_{2}t/η^{m/2}(t)}{1 + cc_{2}t/η^{m/2}(t)} Tr[log(I + ctχ_{ρ}B(t)χ_{ρ})].

By Lemma 2.2 (ii), we obtain (2.16).

Proof of Theorem 2.2. – It is well known that

Ñ(t) = \frac{1}{(2πt)^{d/2}} E^{w×g}[ exp ( − \frac{1}{2} ∫_{0}^{t} ∑_{j=1}^{q}(g_{ω}^{j}(w(s)))^{2}ds)|w(t) = 0],

where E^{w×g} is the expectation with respect to the d-dimensional Wiener process w(s) and g_{ω}^{j}(x) (cf.[4],[21]). We use the scaling property of the Wiener process to rewrite as

\begin{aligned}Ñ(t) & = \frac{1}{(2πt)^{d/2}} E^{w×g}[ exp ( − \frac{η(t)}{2} ∫_{0}^{t/η(t)} ∑_{j=1}^{q}(g_{ω}^{j}(\sqrt{η(t)}w(s)))^{2}ds)|w(\frac{t}{η(t)}) = 0] \\ & = \frac{1}{η^{d/2}(t)}E^{g}[ exp ( − \frac{t}{η(t)}H(η(t)))(0, 0)], \end{aligned}

where exp(−tH(η))(x, y), t > 0, x, y ∈ ℝ^{d}, is the integral kernel of the heat semigroup generated by the operator

H(η) = \frac{1}{2}( − Δ + η ∑_{j=1}^{q}(g_{ω}^{j}(\sqrt{η}·))^{2}).

By Theorem (9.5) of [21] , we have

Ñ(t) ≥ \frac{1}{η^{d/2}(t)| supp ϕ|} exp ( − \frac{t}{η(t)}(\frac{∥∇ϕ∥^{2}}{2} + Φ(ϕ^{2}, t)))  (2.20)

for any ϕ ∈ C_{0}^{∞}(ℝ^{d}) such that ∥ϕ∥_{2} = 1, where ∥ · ∥_{2} and (·, ·) are the norm and the inner product of L_{2}(ℝ^{d}), respectively. By Lemma 2.6 and Lemma 2.2 (iii), we have

\underline{lim}_{t→∞} \frac{η(t)}{t} log Ñ(t) ≥ \frac{−1}{2}(∥∇ϕ∥_{2}^{2} + c^{2κ}R^{d}),  (2.21)

where c, R > 0 are taken as |ϕ| ≤ c and supp ϕ ⊂ Λ(R).

Corollary (i), (iii) of Theorem 2.2 are proven by Theorem 2.2 and the following:

Lemma 2.7. – (i) If Φ(t) = Ct^{A} log^{B} t(1 + o(1)) as t → ∞ for some B, C > 0 and 0 ≤ A < 1, then

η(t) = \frac{t^{A′}}{C′ log^{B′} t}(1 + o(1))  (2.22)

as t → ∞, where

A′ = \frac{2(1 − A)}{d(1 − A) + 2}, B′ = \frac{2B}{d(1 − A) + 2} and C′ = C^{2/(d(1−A)+2)}(\frac{2}{d(1 − A) + 2})^{B′}.

(ii) If

η(t) = \frac{t^{A}}{C log^{B} t}(1 + o(1))

and

\frac{η(t)}{t} log Ñ(t) = −D(1 + o(1))

as t → ∞ for some B, C, D > 0 and 0 < A < 1, then

log Ñ(t) = −CDt^{1−A} log^{B} t(1 + o(1))  (2.23)

as t → ∞.

This is easily proven by (2.3). For Corollary (ii), we use also the following:

Lemma 2.8. – For arbitrarily fixed σ, τ, R > 0, let

Φ_{σ,R}(t) := ∫_{R}^{∞} dζζ^{d−1} log(1 + te^{−σζ^{τ}}), t > 0. (2.24)

Then, for any fixed K > 0, we have

Φ_{σ,R}(Kt) = c log^{(d+τ)/τ} t(1 + o(1))  (2.25)

as t → ∞, where c = τ/{(d + τ)dσ^{d/τ}}.

Theorem 2.1 is obtained from Corollary of Theorem 2.2, Theorem 2.3 and the following: Lemma 2.9. – If

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{A} log^{B} t} ≥ −C

for some B, C > 0 and 0 < A < 1, then we have

\underline{lim}_{λ↓0} \frac{λ^{A′}}{log^{B′}(1/λ)} log N(λ) ≥ −C′,

where A′ = A/(1 − A), B′ = B/(1 − A) and C′ = C^{1/(1−A)}(1 − A)^{−B/(1−A)}.

3. CASE I

Let {g_{ω}(y), y ∈ ℝ} be the stationary ergodic Gaussian random process with mean zero and the covariance β(y) = E[g_{ω}(y)g_{ω}(0)]. We consider the operator L(b_{ω}) in (1.1) with b_{ω}^{1}(x) = 0 and b_{ω}^{2}(x) = g_{ω}(x^{1}). For this operator, we introduce the integrated density of states N(λ) by the same methods in [2], [7], [10], [18] and [28]. We use the condition (P1, α, 1), (E1, τ, 1), (C1, 1), (H1, 1) and (H2, 1) introduced in the last section.

Then the result is the following:

Theorem 3.1. – (i) We assume the condition (P1, α, 1) for some α > 1. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−(α+1)/{2(α−1)}} ≤ log N(λ) ≤ −c′λ^{−p/2} log λ^{−1}

for any 0 < λ < ε.

(ii) We assume the condition (E1, τ, 1) for some τ > 0. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−1/2} log^{(1+τ)/τ} λ^{−1} ≤ log N(λ) ≤ −c′λ^{−1/2} log λ^{−1}

for any 0 < λ < ε.

(iii) We assume the condition (C1, 1). Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−1/2} log λ^{−1} ≤ log N(λ) ≤ −c′λ^{−1/2} log λ^{−1}

for any 0 < λ < ε.

Remark 3.1. – The constants c and c′ in Theorem 3.1 are same as in Theorem 2.1. Thus the leading term is not determined in Theorem 3.1 (iii) as stated in Remark 2.1.

Theorem 3.2. – Under the condition (H1, 1), we have

\underline{lim}_{t→∞} \frac{η(t)}{t} log Ñ(t) > −∞.  (3.1)

Corollary. – (i) Under the condition (P1, α, 1) for some α > 1, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{(α−1)/(3α−1)}} > −∞.  (3.2)

(ii) Under the condition (E1, τ, p) for some τ > 0, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{1/3} log^{2(τ+1)/(3τ)} t} > −∞.  (3.3)

(iii) Under the condition (C1, p), we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{1/3} log^{2/3} t} > −∞.  (3.4)

Theorem 3.3. – Under the condition (H2, 1), we have

\overline{lim}_{λ↓0} \frac{λ^{1/2} log N(λ)}{log λ^{−1}} ≤ −\frac{π}{B3^{3/2}2^{1/2}} ,  (3.5)

where

B = ∫_{ℝ} |β(x)|dx/β(0).

Theorem 3.1 is obtained from the corollary of Theorem 3.2 and Theorem 3.3. For the proof of Theorems 3.2 and 3.3, we prepare the following:

Lemma 3.1. – (i) It holds that

Ñ(t) = \frac{1}{(2π)^{3/2}t^{1/2}} ∫_{ℝ}dξE^{w×g}[ exp ( − \frac{1}{2} ∫_{0}^{t}(g_{ω}(w(s)) + ξ)^{2}ds)|w(t) = 0], (3.6)

where E^{w×g} is the expectation with respect to the 1-dimensional Wiener process w and g_{ω}.

(ii) It holds that

Ñ(t) = \frac{1}{2πt}E^{w×g}[ exp ( − \frac{t}{2}V (g_{ω}, w, t))|w(t) = 0],  (3.7)

where

V (g_{ω}, w, t) = ∫_{0}^{t} \frac{ds}{t} (g_{ω}(w(s)) − ∫_{0}^{t} g_{ω}(w(r))\frac{dr}{t} )^{2}

is the sample variance.

(iii) It holds that

Ñ(t) ≥ \frac{1}{2πt}E^{w×g}[ exp ( − \frac{1}{2} ∫_{0}^{t} g_{ω}(w(s))^{2}ds)|w(t) = 0].  (3.8)

(iv) It holds that

Ñ(t) ≤ \frac{(1 + tβ(0))^{1/2}}{2πt} E^{w×g}[ exp ( − \frac{1}{2} ∫_{0}^{t} g_{ω}(w(s))^{2}ds)|w(t) = 0].  (3.9)

Proof. – (i) By Theorem 6.1 in Broderix, Hundertmark and Leschke [3], the integral kernel of the heat semigroup generated by the operator L(b_{ω}) is represented as

\begin{matrix}e^{−tL(b_{ω})}(x, y) = & E^{w}[ exp (i ∫_{0}^{t} g_{ω}(x^{1} + w^{1}(s))dw^{2}(s))|x + w(t) = y] \\ × \frac{1}{2πt} exp ( − \frac{|x − y|^{2}}{2t} ), \end{matrix}

where E^{w} is the expectation with respect to the 2-dimensional Wiener process w(s) = (w^{1}(s), w^{2}(s)) (cf.[11], [25], [26], [27]). Then by using the Fourier transform we have

e^{−tL(b_{ω})}(0, 0) =\frac{1}{(2π)^{2}t} ∫_{ℝ} dξ ∫_{ℝ} dy^{2} exp(iξy^{2})E^{w}[ exp (i ∫_{0}^{t} g_{ω}(w^{1}(s))dw^{2}(s))

|w^{1}(t) = 0, w^{2}(t) = y^{2}] exp ( − \frac{(y^{2})^{2}}{2t} ),

=\frac{1}{(2π)^{3/2}t^{1/2}} ∫_{ℝ} dξE^{w}[ exp (i ∫_{0}^{t}(g_{ω}^{j}(w^{1}(s)) + ξ)dw^{2}(s))|w^{1}(t) = 0].

By calculating the expectation with respect to w^{2}(s) and taking the expectation with respect to b_{ω}, we obtain (3.6).

(ii) and (iii) are easily obtained from (i) and (ii), respectively.

(iv) We rewrite (3.6) as

Ñ(t) = \frac{1}{(2π)^{3/2}t^{1/2}} ∫_{ℝ} dξE^{w}[exp(−tΦ(μ_{w}^{t}, t, ξ))|w(t) = 0],  (3.10)

where

Φ(μ, t, ξ) = −\frac{1}{t} log E[ exp ( − \frac{t}{2} ∫_{ℝ}(g_{ω}(x) + ξ)^{2}μ(dx))]  (3.11)

for any probability measure μ on ℝ and μ_{w}^{t} is the empirical distribution of the Wiener process: {w(s) : 0 ≤ s ≤ t} ↦ μ_{w}^{t}(B) := t^{−1}( the measure of {s : 0 ≤ s ≤ t, w(s) ∈ B}) for any Borel set B in ℝ. If we define an operator B(μ) on L_{2}(ℝ, dμ) by

{B(μ)φ}(x) = ∫_{ℝ} β(x − y)φ(y)μ(dy)

for each probability measure μ on ℝ, then we easily see that B(μ) is a Hilbert-Schmidt operator. Accordingly we take a complete orthonormal system {φ_{n}}_{n≥1} of L_{2}(ℝ, dμ) consisting of the eigenfunctions of B(μ). Let {β_{n}}_{n≥1} be the corresponding eigenvalues. Then by the same method as in the proof of Lemma 2.3 (i), we can write as

Φ(μ, t, ξ) = \frac{1}{2t} ∑_{n=1}^{∞} log(1 + tβ_{n}) + \frac{1}{2} ∑_{n=1}^{∞} \frac{α_{n}^{2}}{1 + tβ_{n}} ,  (3.12)

where α_{n} = (ξ, φ_{n})_{μ} and (·, ·)_{μ} is the inner product of the space L_{2}(ℝ, dμ). The first term is equal to Φ(μ, t, 0) and the second term is estimated as

Φ(μ, t, ξ) − Φ(μ, t, 0) ≥ \frac{1}{2(1 + tβ(0))} ∑_{n=1}^{∞} α_{n}^{2} = \frac{|ξ|^{2}}{2(1 + tβ(0))}

since the operator norm of B(μ) is estimated as ∥B(μ)∥ ≤ β(0). By applying this to (3.10), we obtain (3.9).

Now Theorem 3.2 and its corollary are easily obtained from Lemma 3.1 (iii), Theorem 2.2 and its corollary. To prove Theorem 3.3, we further use the following:

Lemma 3.2. – (i) If

\overline{lim}_{t→∞} \frac{log Ñ(t)}{t^{A} log^{B} t} ≤ −C  (3.13)

for some B, C > 0 and 0 < A < 1, then we have

\overline{lim}_{λ↓0} \frac{λ^{A′}}{log^{B′}(1/λ)} log N(λ) ≤ −C′,  (3.14)

where A′ = A/(1 − A), B′ = B/(1 − A) and C′ = C^{1/(1−A)}A^{A/(1−A)}(1 − A)^{1−B/(1−A)}. (ii) Conversely if (3.14) holds, then (3.13) also holds.

Proof. – The proof of (i) is almost same with that of Fukushima [9] as in Lemma 2.9. Now if (3.14) holds, then for any ε > 0, there exists λ_{0} > 0 such that N(λ) ≤ exp(−(C′ − ε)λ^{−A′} log^{B′} λ^{−1}) for any 0 < λ < λ_{0}. Acoordingly we devide as

Ñ(t) = ∫_{0}^{λ_{0}} e^{−tλ}dN(λ) + ∫_{λ_{0}}^{∞} e^{−tλ}dN(λ) =: Ñ_{1}(t) + Ñ_{2}(t)

and estimate as

Ñ_{2}(t) ≤ e^{−t(1−ε)λ_{0}} Ñ(εt) ≤ e^{−t(1−ε)λ_{0}}/(2πεt)

and

Ñ_{1}(t) ≤ t ∫_{0}^{λ_{0}} e^{−g(λ)}dλ + e^{−g(λ_{0})},

where g(λ) := tλ + (C′ − ε)λ^{−A′} log^{B′} λ^{−1}. For each t > 0, we see that min_{0≤λ≤λ_{0}} g(λ) = g(λ(t)) where λ(t) is the solution of

λ(t) = (C′ − ε)^{1/(A′+1)}t^{−1/(A′+1)}(B′ log^{B′−1} λ(t)^{−1} + A′ log^{B′} λ(t)^{−1})^{1/(A′+1)}. By the iteration, we have

min_{0≤λ≤λ_{0}} g(λ) ∼ t^{A} log^{B} tC(ε)

as t → ∞, where C(ε) = (C′ − ε)^{1/(A′+1)}(A′ + 1)^{−B}((A′)^{1/(A′+1)} + (A′)^{−A}). From this we obtain (3.13).

4. CASE II

Let {f_{ω}(x), x ∈ ℝ^{2}} be the real valued stationary ergodic Gaussian random field with mean zero and the covariance β(x) = E[f_{ω}(x)f_{ω}(0)]. Under the integrability of β(x) and β̂(ξ)(1 + |ξ|^{2}), {∇f_{ω}(x), x ∈ ℝ^{2}} is the stationary ergodic Gaussian random field with mean zero and the covariance

E[\frac{∂f_{ω}}{∂x^{j}} (x)\frac{∂f_{ω}}{∂x^{k}} (0)] = −\frac{∂^{2}β}{∂x^{j}∂x^{k}} (x)  (4.1)

for any j, k = 1, 2. Then, for the operator L(b_{ω}) in (1.1) with

b_{ω}^{1}(x) = \frac{∂f_{ω}}{∂x^{2}} (x) and b_{ω}^{2}(x) = −\frac{∂f_{ω}}{∂x^{1}} (x),  (4.2)

we have the integrated density of states N(λ) as in the last section.

We introduce the following conditions: for α > 4 and τ > 0,

(P2, α) β̂ ∈ C^{∞} and there exists R > 0 such that β̂(ξ) = |ξ|^{−α} for |ξ| ≥ R.

(E2, τ) β̂ ∈ C^{∞} and there exists R > 0 such that β̂(ξ) = exp(−|ξ|^{τ}) for |ξ| ≥ R.

(C2) 0 ≠ β̂ ∈ C_{0}^{∞}.

Then the result is the following:

Theorem 4.1. – (i) We assume the condition (P2, α) for some α > 4. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−(α−2)/(α−4)} ≤ log N(λ) ≤ −c′λ^{−1/2} log λ^{−1}

for any 0 < λ < ε.

(ii) We assume the condition (E2, τ) for some τ > 0. Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−1} log^{(τ+2)/τ} λ^{−1} ≤ log N(λ) ≤ −c′λ^{−1/2} log λ^{−1} for any 0 < λ < ε.

(iii) We assume the condition (C2). Then for small enough ε > 0, there exist c, c′ > 0 such that

−cλ^{−1} log λ^{−1} ≤ log N(λ) ≤ −c′λ^{−1/2} log λ^{−1}

for any 0 < λ < ε.

As in Theorems 2.1 and 3.1, this theorem is obtained from the corollary of Theorem 4.2 and Therem 4.3 below. Therefore the condtions (P2, α), (E2, τ) and (C2) can be weaken. To state Theorem 4.2, we introduce the following condition:

(H3) (i) sup_{ξ} β̂(ξ)(1 + |ξ|)^{γ+4} < ∞ for some γ > 0; (ii) β is C^{2} and sup_{x} |x|^{n}|\frac{∂^{2}β(x)}{∂x^{j}∂x^{k}} | < ∞ for any j, k = 1, 2 and some n > 4(γ^{−1} + 1);

(iii) lim_{t→∞} Φ(ct)/Φ(t) = c^{κ} for any c > 0 and some κ ∈ [0, 1),

where

Φ(t) := \frac{1}{(2π)^{2}} ∫_{ℝ^{2}} dξ Tr[log(I + tM(ξ, β̂)],  (4.3)

M(ξ, β̂) = (M_{jk}(ξ, β̂))_{1≤j,k≤2} and

\begin{matrix}M_{jk}(ξ, β̂) = & ∫_{ℝ^{2}} e^{−iξ·x}E[b_{ω}^{j}(x)b_{ω}^{k}(0)]dx \\ = & ( ∑_{ℓ>j,m>k} ξ^{ℓ}ξ^{m} + ∑_{ℓ<j,m<k} ξ^{ℓ}ξ^{m} − ∑_{ℓ>j,m<k} ξ^{ℓ}ξ^{m} − ∑_{ℓ<j,m>k} ξ^{ℓ}ξ^{m}) β̂(ξ). \end{matrix}

Let η(t), t > 0, be the solution of (2.3) where Φ(t) is replaced by that defined in (4.3). Then we have the following:

Theorem 4.2. – Under the condition (H3), we have

\underline{lim}_{t→∞} \frac{η(t)}{t} log Ñ(t) > −∞.

Corollary. – (i) Under the condition (P2, α) for some α > 4, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{(α−2)/(2α−6)}} > −∞.

(ii) Under the condition (E2, τ) for some τ > 0, we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{1/2} log^{(τ+2)/(2τ)} t} > −∞.

(iii) Under the condition (C2), we have

\underline{lim}_{t→∞} \frac{log Ñ(t)}{t^{1/2} log^{1/2} t} > −∞.

To state Theorem 4.3, we introduce the following condition: (H4) (i) β̂(ξ) is C^{4};

(ii) sup_{θ∈S^{1}} ∫_{0}^{∞} dζ(1 + ζ)^{3}|(\frac{∂}{∂ζ} )^{k}β̂(ζ, θ)| < ∞ for k = 0, 1, 2;

(iii) inf_{θ∈S^{1}} ∫_{0}^{∞} dζζ^{3}β̂(ζ, θ) > 0,

where (ζ, θ), ζ ≥ 0, θ ∈ S^{1}, is the polar coordinate of ξ. Then we have the following: Theorem 4.3. – Under the condition (H4), we have

\overline{lim}_{λ↓0} \frac{λ^{1/2} log N(λ)}{log λ^{−1}} ≤ −\frac{π}{3^{3/2}B} ,  (4.4)

where

B = sup_{θ∈S^{1}} ∫_{ℝ} dx^{1}| ∫_{0}^{∞} dζ cos(ζx^{1})ζ^{3}β̂(ζ, θ)|/ ∫_{0}^{∞} dζζ^{3}β̂(ζ, θ).

To prove Theorem 4.3, we prepare the following: Lemma 4.1. – Under the condition (H4), it holds that

Ñ(t) ≤ \frac{1}{2π} ∫_{S^{1}} Ñ_{θ}(t)dθ,  (4.5)

where

Ñ_{θ}(t) = E^{w×g}[ exp (i ∫_{0}^{t} g_{θ,ω}(w^{1}(s))dw^{2}(s))|w(t) = 0]\frac{1}{2πt}  (4.6)

is the Laplace-Stieltjes transform of the integrated density of states for the Schrödinger operator L(b_{ω}) in (1.1) with b_{ω}^{1} = 0 and b_{ω}^{2} = g_{θ,ω}(x^{1}), {g_{θ,ω}(x^{1}), x^{1} ∈ ℝ} is the stationary ergodic Gaussian random field with mean zero and the covariance

β_{θ}(x^{1}) = \frac{1}{2π} ∫_{0}^{∞} dζζ^{3}β̂(ζ, θ) cos(ζx^{1}),  (4.7)

and E^{w×g} is the expectation with respect to w(s) = (w^{1}(s), w^{2}(s)) and g_{θ,ω}(x^{1}).

Proof. – By Theorem 6.1 in Broderix, Hundertmark and Leschke [3] and [28], we have

Ñ(t) = E^{w×f}[e^{iΨ(w,ω)}|w(t) = 0]\frac{1}{(2πt)^{d/2}} ,  (4.8)

where

Ψ(w, ω) = ∫_{0}^{t} (\frac{∂f_{ω}}{∂x^{2}} (w(s))dw^{1}(s) − \frac{∂f_{ω}}{∂x^{1}} (w(s))dw^{2}(s))

and E^{w×f} is the expectation with respect to w(s) and f_{ω}. By calculating the expectation with respect to f_{ω}, we have

Ñ(t) = E^{w}[ exp ( − \frac{1}{2}E^{f}[Ψ(w, ω)^{2}])|w(t) = 0]\frac{1}{2πt},  (4.9)

where E^{f} is the expectation with respect to f_{ω}.

In terms of the covariance β, we can write as

E^{f}[Ψ(w, ω)^{2}] = \frac{1}{(2π)^{2}} ∫_{ℝ^{2}} dξβ̂(ξ)| ∫_{0}^{t} e^{iξ·w(s)}dξ ∧ w(s)|^{2},  (4.10)

where ξ ∧ w(s) = ξ^{2}w^{1}(s) − ξ^{1}w^{2}(s). In fact, if we take a division Δ of the interval [0, t] given by 0 = t_{0} < t_{1} < ⋯ < t_{n} = t, then we have I(Δ) = J(Δ), where

\begin{aligned}I(Δ) & := E^{f} [\{ ∑_{a=0}^{n−1} (\frac{∂f_{ω}}{∂x^{2}} (w(t_{a}))Δw^{1}(t_{a}) − \frac{∂f_{ω}}{∂x^{1}} (w(t_{a}))Δw^{2}(t_{a}))\}^{2}], \\ J(Δ) & := \frac{1}{(2π)^{2}} ∫_{ℝ^{2}} dξβ̂(ξ)| ∑_{a=0}^{n−1} e^{iξ·w(t_{a})}Δξ ∧ w(t_{a})|^{2} \end{aligned}

and Δw(t_{a}) = w(t_{a+1}) − w(t_{a}). By the definition of the Itô integral, we see that I(Δ) and J(Δ) converge the left and right hand sides of (4.10) respectively in L_{1}(P^{W} (·|w(t) = 0)) as |Δ| := max_{a} |t_{a+1} − t_{a}| → 0 where P^{W} is the Wiener measure.

We now use Jensen’s inequality in (4.9) as follows:

Ñ(t) ≤ \frac{1}{2π} ∫_{S^{1}} N̆_{θ}(t)dθ,  (4.11)

where

N̆_{θ}(t) := E[ exp ( − \frac{1}{2} ∫_{0}^{∞} dζ · ζβ̂(ζ, θ)| ∫_{0}^{t} e^{iξ·w(s)}dξ ∧ w(s)|^{2}|w(t) = 0]\frac{1}{2πt}.

By the O(2)-invariance, the process

(\frac{ξ · w(s)}{ζ}, \frac{ξ ∧ w(s)}{ζ})_{s}

is a 2-dimensional Wiener process independent of ζ. Thus we have

N̆_{θ}(t) = E[ exp ( − \frac{1}{2}I(θ, w))|w(t) = 0]\frac{1}{2πt},

where

I(θ, w) = \frac{1}{2π} ∫_{0}^{∞} dζ · ζ^{3}β̂(ζ, θ)| ∫_{0}^{t} e^{iζw^{1}(s)}dw^{2}(s)|^{2}.

By the same method as in (4.10), we have

E^{g}[G_{θ}(w, ω)^{2}] = I(θ, w)

for any θ ∈ S^{1}, where

G_{θ}(w, ω) = ∫_{0}^{t} g_{θ,ω}(w^{1}(s))dw^{2}(s)

and E^{g} is the expectation with respect to g_{θ,ω}. Now we see that N̆_{θ}(t) = Ñ_{θ}(t).

Proof of Theorem 4.3. – By Lemma 4.1 and (3.9), we have

Ñ(t) ≤ sup_{θ∈S^{1}} \sqrt{\frac{1 + tβ_{θ}(0)}{2πt}}Ñ_{0,θ}(t) =: N̆(t)

for any (j, k) ∈ S, where Ñ_{0,θ}(t) is the Laplace-Stieltjes transform of the integrated density of states for the operator H(V_{ω}) in (1.2) on L_{2}(ℝ) with V_{ω}(x) = g_{θ,ω}(x)^{2}/2. By (2.8) we see that, for any small ε > 0 and any 0 < s < 1, there exist c, λ_{0} > 0 such that

N_{0,θ}(λ) ≤ c exp \{ − (\frac{2^{1/2}π}{B3^{3/2}} − ε)\frac{1}{λ^{1/2}} log \sqrt{\frac{β_{0}}{λ^{s}}} \}

for any 0 < λ < λ_{0}, where N_{0,θ}(λ) is the integrated density of states for the operator H(V_{ω}) in (1.2) on L_{2}(ℝ) with V_{ω}(x) = g_{θ,ω}(x)^{2}/2 and β_{0} = sup_{θ∈S^{1}} β_{θ}(0). By the same method as in the proof of Lemma 3.2 (ii), we have

\overline{lim}_{t→∞} \frac{log N̆(t)}{t^{1/3} log^{2/3} t} ≤ −(\frac{π^{2}}{2} )^{1/3} \frac{1}{(3B)^{2/3}} .

From this and Lemma 3.2 (i), we obtain (4.4).

Proof of Theorem 4.2. – We use the scaling property of the Wiener process to obtain

e^{−tL(b_{ω})}(0, 0) = exp ( − \frac{t}{η(t)}L(\sqrt{η(t)}b_{ω}(\sqrt{η(t)}·)))(0, 0)\frac{1}{η(t)^{d/2}} ,

where η(t) is the function defined in (2.3). As in (2.20), by Theorem (9.5) of [21], we have

Ñ(t) ≥ \frac{1}{η(t)| supp ϕ|}E^{f} [ exp ( − \frac{t}{η(t)}(L(\sqrt{η(t)}b_{ω}(\sqrt{η(t)}·))ϕ, ϕ))]

for any ϕ ∈ C_{0}^{∞}(ℝ^{2} → ℂ). By restricting the function ϕ to be of real valued, we have

Ñ(t) ≥ \frac{1}{η(t)| supp ϕ|} exp \{ − \frac{t}{η(t)}(\frac{∥∇ϕ∥^{2}}{2} + Φ(ϕ^{2}, t))\}

for any ϕ ∈ C_{0}^{∞}(ℝ^{2} → ℝ), where

Φ(ϕ^{2}, t) = −\frac{η(t)}{t} log E^{f} [ exp ( − \frac{t}{2} ∫_{ℝ^{2}} |b_{ω}(\sqrt{η(t)}x)|^{2}ϕ^{2}(x)dx)].

Thus we can complete the proof as in the proof of Theorem 2.2.

REFERENCES

[1] S. Albeverio and H.Tamura, On the propagator of a scalar field in the presence of confining nonlinear electromagnetic force, preprint.

[2] K. Broderix, D. Hundertmark, W. Kirsch and H.Leschke, The fate of Lifschits tails in magnetic fields, J. Statist. Phys. 80 (1995), 1-22.

[3] K. Broderix, D. Hundertmark and H.Leschke, Continuity properties of Schrödinger semigroups with magnetic fields, preprint

[4] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990.

[5] J. -D. Deuschel and D. W. Stroock, Large deviations, Academic Press, Boston, 1989.

[6] M. D. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), 525–565.

[7] L. Erdös, Lifschitz tail in a magnetic field: the nonclassical regime, Probab. Theory Relat. Fields, 112 (1998), 321–371.

[8] A. L. Figotin, Eigenvalue distribution of random Schrödinger equations and asymptotic behavior of certain Wiener integrals, Dopovidi Akad. Nauk Ukr. SSR (6) (1981), 27–29.

[9] M. Fukushima, On the spectral distribution of a disordered system and the range of a random walk, Osaka J. Math. 11 (1974), 73–85.

[10] T. Hupfer, H. Leschke and S. Warzel, Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshitz tailing in magnetic fields, preprint

[11] N. Ikeda and S. Manabe, Asymptotic formulas for stochastic oscillatory integral, Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, Longman, New York,1993, 136– 155.

[12] N. Ikeda, I. Shigekawa and S. Taniguchi, The Malliavin calculus and long time asymptotics of certain Wiener integrals, Proc. Center for Math. Anal. Australian Nat. Univ. 9 Canberra, 1985, 46–113.

[13] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd. ed., Kodansha/North-Holland, Tokyo/Amsterdam, 1989.

[14] A. K. Klein, L. J. Landau and D. S. Shucker, Decoupling inequalities for stationary Gaussian processes, Ann. Probab., 10 (1982), 702–708.

[15] W. Kirsch and F. Martinelli, Large deviations and Lifschitz singularity of the integrated density of

states of random Hamiltonians, Commun. Math. Phys., 89 (1983), 27–40.

[16] I. M. Lifschitz, Energy spectrum structure and quantum states of disordered condensed systems, Usp. Fiz. Nauk 83 (1964), 617–663.

[17] P. Malliavin and S. Taniguchi, Analytic functions, Cauchy formula and stationary phase on a real abstract Wiener space, J. Funct. Anal. 143 (1997), 470–528.

[18] H. Matsumoto, On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials, J. Math. Soc. Japan 45 (1993), 197–214.

[19] K. Nakane, The Schrödinger operator with random vector potentials, Hokkaido Math. J. 25 (1996), 55–80.

[20] S. Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan J. Math. 3 (1977), 111–139.

[21] L. A. Pastur and A. L. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin, Heidelberg and New York, 1992.

[22] M. Reed and B. Simon, Methods of modern mathematical physics, II : Fourier analysis and selfadjointness, Academic Press, New York, 1975.

[23] M. Reed and B. Simon, Methods of modern mathematical physics, IV : Analysis of operators, Academic Press, New York, 1978.

[24] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, Berlin, Heidelberg and New York, 1991.

[25] B. Simon, Functional Integration and Quantum Physics, Academic Press, London, 1979.

[26] B. Simon, Maximal and minimal Schrödinger operators and forms, J. Operator Theory Appl. 1 (1979), 37–47.

[27] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.

[28] N. Ueki, On spectra of random Schrödinger operators with magnetic fields, Osaka J. Math. 31 (1994), 177–187.

[29] S. R. S. Varadhan, Large deviations and applications, Society for Industrial and Applied Mathematics, Philadelphia, 1984.