Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice^{*}

Naomasa Ueki^{1} 1Graduate School of Human and Environmental Studies, Kyoto University Kyoto 606-8501, Japan. E-mail: ueki@math.h.kyoto-u.ac.jp

Abstract. For the Schrödinger operators on L^{2}(ℝ^{2}) and L^{2}(ℝ^{3}) with the uniform magnetic field and the scalar potentials located at all sites of a randomly perturbed lattice, the asymptotic behavior of the integrated density of states at the infimum of the spectrum is investigated. The randomly perturbed lattice is the model considered by Fukushima and this describes an intermediate situation between the ordered lattice and the Poisson random field. In this paper the scalar potentials are assumed to decay slowly and its effect to the leading term of the asymptotics are determined explicitly. As the perturbed lattice tends to the Poisson model, the determined leading term tends to that for the Poisson model.

Keywords: Lifshitz tails, magnetic Schrödinger operator AMS Subject Classification: 34L40, 82B44

1. Introduction Let

H = A^{∗}A = (i\frac{∂}{∂x_{1}} − \frac{Bx_{2}}{2} )^{2} + (i\frac{∂}{∂x_{2}} + \frac{Bx_{1}}{2} )^{2} − B  (1.1)

be the smaller component of the Pauli Hamiltonian ℝ^{2} with the uniform magnetic field B > 0, where i = \sqrt{−1} and

A = (i\frac{∂}{∂x_{1}} − \frac{Bx_{2}}{2} ) + i(i\frac{∂}{∂x_{2}} + \frac{Bx_{1}}{2} ).  (1.2)

Let

V_{ξ}(x) = ∑_{q∈ℤ^{2}} u(x − q − ξ_{q})  (1.3)

be a random potential on ℝ^{2}, where ξ = (ξ_{q})_{q∈ℤ^{2}} is a collection of independently and identically distributed ℝ^{2}-valued random variables with the distribution

P_{θ}(ξ_{q} ∈ dx) = exp(−|x|^{θ})dx/Z(θ),  (1.4)

*This work was partially supported by KAKENHI (21540175)

2 Naomasa Ueki

|x| = \sqrt{x_{1}^{2} + x_{2}^{2}}, θ ∈ (0, ∞), Z(θ) is the normalizing constant, and u is a nonnegative function belonging to the Kato class K_{2} (cf. [2] p-53). Let H_{ξ,Λ_{R}} be the self-adjoint restriction of the random Schrödinger operator

H_{ξ} = H + V_{ξ},  (1.5)

to the L^{2} space on the cube Λ_{R} := (−R/2, R/2)^{2} with the Dirichlet boundary condition and N_{ξ,R}(λ) be its number of eigenvalues not exceeding λ. It is well-known that there exists a deterministic increasing function N such that

R^{−2}N_{ξ,R}(λ) ⟶ N(λ) as R → ∞ (1.6)

for any point of continuity of N and almost all ξ under the condition sup |x|^{α}u(x) < ∞ for some α > 2 (cf. [2], [7], [10]). This function (N(λ))_{λ≥0} is called as the integrated density of states for the random Schrödinger operator (1.5). For this function, we show the following: Theorem 1. (i) If

essinf_{|x|≤R} u(x) > 0  (1.7)

holds for any R ≥ 1 and

u(x) = C_{0}|x|^{−α}(1 + o(1))  (1.8)

as |x| → ∞ for some C_{0} ∈ (0, ∞) and α ∈ (2, ∞), then we have

lim_{λ↓0} λ^{κ} log N(λ) = \frac{−κ^{κ}}{(κ + 1)^{κ+1}} \{ ∫_{ℝ^{2}} dq inf_{y∈ℝ^{2}} (\frac{C_{0}}{|q + y|^{α}} + |y|^{θ})\}^{κ+1},  (1.9)

where κ = (2 + θ)/(α − 2).

(ii) If (1.7) holds for any R ≥ 1 and

u(x) = exp (\frac{−|x|^{α}}{C_{0}} (1 + o(1)))  (1.10)

as |x| → ∞ for some C_{0} ∈ (0, ∞) and α ∈ (0, 2), then we have

lim_{λ↓0}(log(1/λ))^{−(2+θ)/α} log N(λ) = \frac{−2πC_{0}^{(2+θ)/α}}{(θ + 1)(θ + 2)}.  (1.11)

We also consider the 3-dimensional problem in the following formulation referring to [8] and [17]. Let

H = (i\frac{∂}{∂x_{1}} − \frac{Bx_{2}}{2} )^{2} + (i\frac{∂}{∂x_{2}} + \frac{Bx_{1}}{2} )^{2} − B − \frac{∂^{2}}{∂x_{3}^{2}}  (1.12)

be the smaller component of the Pauli Hamiltonian on ℝ^{3} with the constant magnetic field B parallel to the x_{3}-axis. We write any element x of ℝ^{3} as (x_{⊥}, x_{3}) ∈ ℝ^{2} × ℝ and set

∥x∥_{p}^{θ} := \begin{cases}∥|x_{⊥}|^{θ_{⊥}}, |x_{3}|^{θ_{3}}∥_{p} = (|x_{⊥}|^{θ_{⊥}p} + |x_{3}|^{θ_{3}p})^{1/p} if p ∈ [1, ∞), \\ |x_{⊥}|^{θ_{⊥}} ∨ |x_{3}|^{θ_{3}} if p = ∞, \end{cases}  (1.13)

for arbitrarily fixed θ = (θ_{⊥}, θ_{3}) ∈ (0, ∞)^{2} and p ∈ [1, ∞]. Let ξ = (ξ_{q})_{q∈ℤ^{3}} be a collection of independently and identically distributed ℝ^{3}-valued random variables with the distribution

P_{θ}(ξ_{q} ∈ dx) = exp(−∥x∥_{p}^{θ})dx/Z(θ, p),  (1.14)

where Z(θ, p) is the normalizing constant. Let u be a nonnegative function belonging to the Kato class K_{3} (cf. [2] p-53) and satisfying

u(x) = \frac{C_{0}}{∥x∥α ; p̃} (1 + o(1))  (1.15)

as |x| → ∞ for some C_{0} ∈ (0, ∞), p̃ ∈ [1, ∞] and α = (α_{⊥}, α_{3}) ∈ (0, ∞)^{2} such that

\frac{2}{α_{⊥}} + \frac{1}{α_{3}} < 1.  (1.16)

Since

∫_{q∈ℝ^{3}:|q|≥1} \frac{dq}{∥q∥_{p}^{α}} < ∞  (1.17)

under the condition (1.16), we can show

V_{ξ}(x) = ∑_{q∈ℤ^{3}} u(x − q − ξ_{q})  (1.18)

belong to the local Kato class K_{3,loc} by the same method in Lemma 7.1 in [7]. As in the 2-dimensional case we will consider the integrated density of states (N(λ))_{λ≥0} of the random Schrödinger operators

H_{ξ} = H + V_{ξ} :  (1.19)

it is a deterministic increasing function satisfying

R^{−3}N_{ξ,R}(λ) ⟶ N(λ)  as R → ∞  (1.20)

for any point of continuity of N and almost all ξ, where N_{ξ,R}(λ) is the number of eigenvalues not exceeding λ of the self-adjoint operator H_{ξ,Λ_{R}} on the L^{2} space on the cube Λ_{R} := (−R/2, R/2)^{3} with the Dirichlet boundary condition (cf. [2], [7], [10]). Our result is the following:

4 Naomasa Ueki

Theorem 2. We assume

essinf_{|x|≤R} u(x) > 0  (1.21)

holds for any R ≥ 1 and (1.15) as |x| → ∞ for some C_{0} ∈ (0, ∞) and

\frac{2}{α_{⊥}} + \frac{3}{α_{3}} > 1.  (1.22)

We set

κ(α, θ) = \frac{\frac{θ_{⊥}}{α_{⊥}} ∨ \frac{θ_{3}}{α_{3}} + \frac{2}{α_{⊥}} + \frac{1}{α_{3}}}{1 − \frac{2}{α_{⊥}} − \frac{1}{α_{3}}} ,  (1.23)

and

\begin{matrix}C(α, θ, C_{0}) \\ = & ∫_{ℝ^{3}} dq inf_{y=(y_{⊥},y_{3})∈ℝ^{3}} (\frac{C_{0}}{∥q + y∥α ; p̃} + ∥1_{\frac{θ_{⊥}}{α_{⊥}} ≥ \frac{θ_{3}}{α_{3}}} |y_{⊥}|^{θ_{⊥}}, 1_{\frac{θ_{⊥}}{α_{⊥}} ≤ \frac{θ_{3}}{α_{3}}} |y_{3}|^{θ_{3}} ∥_{p}) \end{matrix}  (1.24)

(cf. (1.17) ). Then we have

lim_{λ↓0} λ^{κ(α,θ)} log N(λ) = \frac{−κ(α, θ)^{κ(α,θ)}}{(1 + κ(α, θ))^{1+κ(α,θ)}} C(α, θ, C_{0})^{1+κ(α,θ)}.  (1.25)

These results are extensions of the results in [6] and [7], where the same problem is considered in the case without magnetic fields. As is discussed in [6] and [7], our model describes an intermediate situation between a completely ordered situation and a completely disordered situation since the point process {q + ξ_{q}}_{q∈ℤ^{2}} converges weakly to the Poisson point process with the intensity 1 as θ → 0 and converges weakly to the complete lattice ℤ^{2} as θ → ∞ by slightly modifying the definition as

P_{θ}(ξ_{q} ∈ dx) = exp(−(1 + |x|)^{θ})dx/Z(θ),  (1.26)

which brings no essential changes for our results. The results in [6] and [7] show that the leading term of the integrated density of states for each case also tends to that for the corresponding Poisson case as θ → 0 and decays as θ → ∞ which reflects that the infimum of the spectrum is strictly positive if the perturbations ξ_{q} of sites are all bounded. In the case with uniform magnetic fields the asymptotics of the integrated density of states has been investigated mainly for the Poisson case. For this topic and the relation with other topics, refer to a recent survey by Kirsch and Metzger [11]. The first result was given by Broderix, Hundertmark, Kirsch and Leschke [1]: they determined the leading term for the case corresponding to Theorem 1 (i) in this paper where the point process {q + ξ_{q}}_{q∈ℤ^{2}} is replaced by the Poisson point process. As in (1.9), this leading term was determined mainly by the classical effect of the

magnetic Lifshitz tails 5

scalar potential as in Pastur’s case [14] without magnetic fields. In fact these asymptotics coincide with those of the corresponding classical integrated density of states defined by

N_{c}(λ) = E_{θ}[|{(x, p) ∈ Λ_{R} × ℝ^{2} : H_{ξ,c}(x, p) ≤ λ}|](2πR)^{−2}  (1.27)

for any R ∈ ℕ, where | · | is the 4-dimensional Lebesgue measure and

\begin{matrix}H_{ξ,c}(x, p) \\ = & \{(p_{1} − \frac{Bx_{2}}{2} ) + i(p_{2} + \frac{Bx_{1}}{2} )\}^{∗}\{(p_{1} − \frac{Bx_{2}}{2} ) + i(p_{2} + \frac{Bx_{1}}{2} )\} \\ + V_{ξ}(x) \\ = & (p_{1} − \frac{Bx_{2}}{2} )^{2} + (p_{2} + \frac{Bx_{1}}{2} )^{2} + V_{ξ}(x) \end{matrix}  (1.28)

is the classical Hamiltonian (cf. [9]). Therefore we may say that only the classical effect from the scalar potential determines the leading term. Then Erdös [4] treated the case where the single site potential u is replaced by a function with a compact support and he determined the corresponding leading term of the integrated density of states, which depends only on the magnetic field and the intensity of the point process and is independent of the precise informations on the single site potential as in Nakao’s case [13] without magnetic field referring to Donsker and Varadhan’s result [3]. This behavior is different from that of the classical integrated density of states. Thus we may say that the quantum effect appears in this behavior. The borderline between the classical and quantum behaviors was determined by Hupfer, Leschke and Warzel [9]. The borderline corresponds to the case of α = 2 in Theorem 1 (ii) in this paper. They also determined the leading term for the case of Theorem 1 (ii) in this paper for the Poisson case. The leading term for the borderline case was determined by Erdös [5]. The leading term for the case that only the classical effect appears was determined also in the 3-dimensional case by Hundertmark, Kirsch and Warzel [8]. Results appearing the quantum effect in 3-dimensional cases were obtained by Warzel [17], where general bounds and the leading order for special cases were obtained. Now our problem is to consider the same problem in our setting. As is discussed in [16] we conjecture that the borderline between the classical and quantum behaviors is the case of α = 2 in Theorem 1 (ii) and the case of 2/α_{⊥} + 3/α_{3} = 1 in Theorem 2 as in the Poisson case. In this paper we treat only the classical case and we will treat the quantum case in [16]. Now Theorems 1 and 2 show that our leading terms coincide with those of the corresponding classical integrated densities of states. They also tend to the corresponding leading term for the Poisson case as θ → 0.

6 Naomasa Ueki

The leading term we obtained coincides with that for the classical case without magnetic fields obtained in [7]. The classical case without magnetic fields is that of (1.8) with 2 < α < 4 if the dimension is 2. Theorem 1 shows that this condition for only the classical effect to appear is weakened by the magnetic field, since the effect of the Landau Hamiltonian to the asymptotics of the integrated density of states is weaker than that of the usual Laplacian. The 3-dimensional classical case without magnetic fields is that of (1.15) with 4/α_{⊥} + 1/α_{3} > 1 and (1.16). This result was obtained for the Poisson case by Kirsch and Warzel [12], where the results for the single site potential with an anisotropic decay are summarized. Theorem 2 shows that this condition for only the classical effect to appear is weakened in the direction perpendicular to the magnetic field and is strengthen in the direction parallel to the magnetic field.

The organization of this paper is as follows. We first prove Theorem 1 in Sections 2 and 3: we give the upper estimate in Section 2 and the lower estimate in Section 3. We next prove Theorem 2 in Sections 4 and 5: we give the upper estimate in Section 4 and the lower estimate in Section 5. In Section 6 we give the leading terms of the low energy asymptotics of the classical integrated densities of states defined in a general setting.

2. 2-dimensional upper estimate In this section we prove upper estimates necessary to prove Theorem 1. We denote the Laplace-Stieltjes

transform of the integrated density of states N(λ) by Ñ(t):

Ñ(t) = ∫_{0}^{∞} e^{−tλ}dN(λ).

Then the results are the following: : Proposition 2.1. (i) Under the conditions of Theorem 1 (i) we have

\overline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{(2+θ)/(α+θ)}} ≤ − ∫_{ℝ^{2}} dq inf_{y∈ℝ^{2}} (\frac{C_{0}}{|q + y|^{α}} + |y|^{θ}).  (2.1)

(ii) Under the conditions of Theorem 1 (ii) we have

\overline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)^{(2+θ)/α}} ≤ \frac{−2πC_{0}^{(2+θ)/α}}{(θ + 1)(θ + 2)}.  (2.2)

(i) is proven by the following proposition and the same proof of Proposition 4 in [7]. The following proposition is an extension of the basic inequality (3.6) in [1] for a ℝ^{2}-stationary random potential to a ℤ^{2}- stationary random potential, which is proven by applying the basic inequality in [1] to the ℝ^{2}-stationary random potential

V_{ξ,ζ}(x) = ∑_{q∈ℤ^{2}} u(x − q − ξ_{q} − ζ_{q}),  (2.3)

where ζ = (ζ_{q})_{q∈ℤ^{2}} is an independent family of random vectors uniformly distributed on Λ_{1} :

Proposition 2.2.

Ñ(t) ≤ \frac{B}{2π(1 − e^{−2Bt})}Ñ_{1}(t),

where

Ñ_{1}(t) = ∫_{Λ_{1}} dxE_{θ}[exp(−tV_{ξ}(x))].  (2.4)

Proof of Proposition 2.1 (ii). By replacing the summation by the integration, we have

log Ñ_{1}(t) ≤ ∫_{ℝ^{2}} dq log E_{θ}[ exp ( − t inf_{x∈Λ_{2}} u(x − q − ξ_{0}))].

We restrict the integration to |q| ≤ L for some finite L. For any ε_{1} > 0, there exists R_{1} such that u(x) ≥ exp(−(1 + ε_{1})|x|^{α}/C_{0}) for any |x|_{∞} ≥ R_{1}, where |x|_{∞} = |x_{1}| ∨ |x_{2}|. Thus the right hand side is dominated by

∫ dq_{|q|≤L} log \{ ∫ \frac{dy}{Z(θ)}_{|q+y|_{∞}≥R_{1}+1} exp ( − t inf_{x∈Λ_{2}} exp ( − (1 + ε_{1})\frac{|x − q − y|^{α}}{C_{0}} ) − |y|^{θ})

+ exp ( − t inf_{Λ_{2R_{1}+4}} u)}.

By changing the variables, this equals

(\frac{C_{0} log t}{1 + ε_{1}})^{2/α} ∫_{|q|≤L} dq log \{Ñ_{2}(t, q) + exp ( − t inf_{Λ_{2R_{1}+4}} u)\},

where L = L((1 + ε_{1})/(C_{0} log t))^{1/α},

Ñ_{2}(t, q) =(\frac{C_{0} log t}{1 + ε_{1}})^{2/α} ∫_{|q+y|_{∞}≥(R_{1}+1)((1+ε_{1})/(C_{0} log t))^{1/α}} \frac{dy}{Z(θ)}

× exp ( − t^{1−s(t,q,y)} − (\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}|y|^{θ}),

and

s(t, q, y) = sup_{x∈Λ_{2((1+ε_{1})/(C_{0} log t))^{1/α}}} |x − q − y|^{α}.

We take L as an arbitrary positive constant less than 1 and independent of t. Taking ε_{2} ∈ (0, 1) arbitrarily, we dominate Ñ_{2}(t, q) by exp(−Ñ_{3}(t, q))ε_{2}^{−2/θ}, where

Ñ_{3}(t, q) = inf \{t^{1−s(t,q,y)} + (1 − ε_{2})(\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}|y|^{θ} : y ∈ ℝ^{2}\}.

We use the polar coordinate to express as

Ñ_{3}(t, q) = inf \{t^{1−(s(t,q)+r)^{α}} + (1 − ε_{2})(\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}r^{θ} : r > 0\},  (2.5)

8 Naomasa Ueki

where

s(t, q) = sup_{x∈Λ_{2((1+ε_{1})/(C_{0} log t))^{1/α}}} |x − q|.

We take small ε_{3} > 0 and estimate the infimum in (2.5) as

Ñ_{3}(t, q) ≥ inf \{(1 − ε_{2})(\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}r^{θ} : r > 1 − s(t, q) − ε_{3}\}
∧ inf {t^{1−(s(t,q)+r)^{α}} : 0 < r ≤ 1 − s(t, q) − ε_{3}}
=\{(1 − ε_{2})(\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}(1 − s(t, q) − ε_{3})^{θ}\} ∧ t^{1−(1−ε_{3})^{α}}

The right hand side is bounded from below by

(1 − ε_{2})(\frac{C_{0} log t}{1 + ε_{1}})^{θ/α}(1 − |q| − 2ε_{3})^{θ}

for sufficiently large t. By taking L as 1 − 3ε_{3} and using also the positivity assumption (1.7), we obtain

\overline{lim}_{t↑∞} \frac{log Ñ_{1}(t)}{(log t)^{(2+θ)/α}} ≤ −(1 − ε_{2})(\frac{C_{0}}{1 + ε_{1}} )^{(2+θ)/α} ∫_{|q|≤1−3ε_{3}}(1 − |q| − 2ε_{3})^{θ}dq.

Since ε_{1}, ε_{2}, ε_{3} are arbitrary, we have

\overline{lim}_{t↑∞} \frac{log Ñ_{1}(t)}{(log t)^{(2+θ)/α}} ≤ −C_{0}^{(2+θ)/α} ∫_{|q|≤1}(1 − |q|)^{θ}dq.

□

3. 2-dimensional lower estimates

In this section we prove the following lower estimates necessary to prove Theorem 1: Proposition 3.1. (i) If (1.8) holds, then we have

\underline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{(2+θ)/(α+θ)}} ≥ − ∫_{ℝ^{2}} dq inf_{y∈ℝ^{2}} (\frac{C_{0}}{|q + y|^{α}} + |y|^{θ}).  (3.1)

(ii) If (1.10) holds with some α ∈ (0, 2), then we have

\underline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)^{(2+θ)/α}} ≥ \frac{−2πC_{0}^{(2+θ)/α}}{(θ + 1)(θ + 2)}.  (3.2)

Proof. We use the bound

Ñ(t) ≥ (2R)^{−2} exp(−tλ_{1}(H_{B(R)}))Ñ_{1}(t),  (3.3)

for any R ∈ ℕ, where

Ñ_{1}(t) = E_{θ}[ exp ( − t ∫ dx|ψ_{R}(x)|^{2}V_{ξ}(x))],

magnetic Lifshitz tails 9

and ψ_{R} is a normalized eigenfunction for the lowest eigenvalue λ_{1}(H_{B(R)}) of the operator H restricted to the disk B(R) = {x ∈ ℝ^{2} : |x| < R} by the Dirichlet boundary condition. This is proven by the same method as for the corresponding bound in Theorem (9.6) in [15] for the ℝ^{2}-stationary random field.

λ_{1}(H_{B(R)}) ≤ exp ( − \frac{B}{2} R^{2}(1 − ε_{1}))  (3.4)

is proven by Erdös [4] for sufficiently large R for an arbitrarily taken ε_{1} > 0.

If (1.8) holds, then we can show (3.1) by the same proof of Proposition 5 in [7], where R is taken as ε_{0}t^{1/(α+θ)} with an arbitrarily small ε_{0} > 0.

We now assume (1.10).

log Ñ_{1}(t) = ∑_{q∈ℤ^{2}} Ñ_{2}(t, q),

where

Ñ_{2}(t, q) = log E_{θ}[ exp ( − t ∫ dx|ψ_{R}(x)|^{2}u(x − q − ξ_{0}))].

If |q| is sufficiently large and ξ_{0} is less than |q|/2, then we have |x − q − ξ_{0}| ≥ |q|/2 − R and

Ñ_{2}(t, q) ≥ −t exp ( − \frac{1}{C_{0}} (\frac{|q|}{2} − R)^{α}(1 − ε_{2})) + log P_{θ}(|ξ_{0}| ≤ |q|/2),

where ε_{2} is an arbitrarily small positive constant. By a simple estimate using log(1 − X) ≥ −2X for 0 ≤ X ≤ 1/2, we have

∑_{q∈ℤ^{2}\B(R^{1+ε_{3}} )} Ñ_{2}(t, q) ≥ −c_{1}t exp(−c_{2}R^{(1+ε_{3})α}) − c_{3} exp(−c_{4}R^{(1+ε_{3})θ})  (3.5)

for large enough R, where ε_{3} is an arbitrarily small positive constant, and c_{1}, …, c_{4} are positive finite constants. We dominate the other part as

∑_{q∈ℤ^{2}∩B(R^{1+ε_{3}} )} Ñ_{2}(t, q)

≥ ∑_{q∈ℤ^{2}∩B(R^{1+ε_{3}} )} log sup_{z∈ℝ^{2}} ∫_{|y|≤ε_{4}} \frac{dy}{Z(θ)}

× exp ( − t ∫ dx|ψ_{R}(x)|^{2}u(x − q − y − z) − |y + z|^{θ})

≥c_{5}R^{2(1+ε_{3})} log \frac{πε_{4}^{2}}{Z(θ)}

− ∑_{q∈ℤ^{2}∩B(R^{1+ε_{3}} )} inf_{z∈ℝ^{2}} sup_{|y|≤ε_{4}} (t ∫ dx|ψ_{R}(x)|^{2}u(x − q − y − z)

+ |y + z|^{θ}),

(3.6)

10 Naomasa Ueki

where c_{5} is a positive finite constant and ε_{4} is an arbitrarily small positive constant. We take z = 0 for q ∈ ℤ^{2} satisfying |q| ≥ (1 + s)R + ε_{4} and z = (−1 + ((1 + s)R + ε_{4})/|q|)q for other q ∈ ℤ^{2}, where s is a finite positive number specified later. Then the second term of the right hand side of (3.6) is bounded from below by

− ∑_{q∈ℤ^{2}∩(B(R^{1+ε_{3}} )\B((1+s)R+ε_{4}))} \{t exp ( − \frac{1 − ε_{2}}{C_{0}} (|q| − R − ε_{4})^{α}) + ε_{4}^{θ}\}

− ∑_{q∈ℤ^{2}∩B((1+s)R+ε_{4})} \{t exp ( − \frac{1 − ε_{2}}{C_{0}} (sR)^{α}) + ((1 + s)R + 2ε_{4} − |q|)^{θ}\}

≥ − c_{6}(1 + s)^{2}R^{2}t exp ( − \frac{1 − ε_{2}}{C_{0}} (sR)^{α}) − c_{7}ε_{4}^{θ}R^{2(1+ε_{3})}

− (1 + ε_{5}) ∫_{q∈ℝ^{2}:|q|≤(1+s)R+ε_{4}}((1 + s)R + 2ε_{4} − |q|)^{θ}dq

for large enough R, where ε_{5} is an arbitrarily small positive constant, and c_{6} and c_{7} are positive finite constants. By changing the variables, the last term is bounded from below by

−(1 + ε_{5})((1 + s)R + 2ε_{4})^{2+θ} \frac{2π}{(θ + 1)(θ + 2)}

for large enough R. The above estimate is summarized as follows:

log Ñ(t)

≥ − t exp ( − \frac{B}{2} R^{2}(1 − ε_{1}))

− c_{6}(1 + s)^{2}R^{2}t exp ( − \frac{1 − ε_{2}}{C_{0}} (sR)^{α}) − 2 log(2R)

− c_{1}t exp(−c_{2}R^{(1+ε_{3})α}) − c_{3} exp(−c_{4}R^{(1+ε_{3})θ})

+ c_{5}R^{2(1+ε_{3})} log \frac{πε_{4}^{2}}{Z(θ)} − c_{7}ε_{4}^{θ}R^{2(1+ε_{3})}

− (1 + ε_{5})((1 + s)R + 2ε_{4})^{2+θ} \frac{2π}{(θ + 1)(θ + 2)}

(3.7)

for large enough R. The first term and the second term change the role according to the value of α.

When α < 2, we take 1/s as an arbitrarily small positive constant and set R̂ = sR. Then the right hand side of (3.7) is bounded from below by

−c_{6}t exp ( − \frac{1 − 2ε_{2}}{C_{0}} R̂^{α}) − (1 + 2ε_{5})R̂^{2+θ} \frac{2π}{(θ + 1)(θ + 2)}

for large enough R̂. We now take

R̂ = (\frac{C_{0} log t}{1 − 2ε_{2}})^{1/α}.

magnetic Lifshitz tails 11

Then we obtain

log Ñ(t) ≥ −c_{6} − (1 + 2ε_{5})(\frac{C_{0} log t}{1 − 2ε_{2}})^{(2+θ)/α} \frac{2π}{(θ + 1)(θ + 2)}

for large enough t, from which we obtain (3.2). □

4. 3-dimensional upper estimate In this section we prove an upper estimates necessary to prove Theorem 2. We denote the Laplace-

Stieltjes transform of the integrated density of states N(λ) by Ñ(t):

Ñ(t) = ∫_{0}^{∞} e^{−tλ}dN(λ).

Then the result is the following: Proposition 4.1. Under the conditions of Theorem 2, we have

\overline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{κ̃(α,θ)}} ≤ −C(α, θ, C_{0}),  (4.1)

where

κ̃(α, θ) = \frac{\frac{θ_{⊥}}{α_{⊥}} ∨ \frac{θ_{3}}{α_{3}} + \frac{2}{α_{⊥}} + \frac{1}{α_{3}}}{\frac{θ_{⊥}}{α_{⊥}} ∨ \frac{θ_{3}}{α_{3}} + 1} .  (4.2)

To prove this proposition, we use the following simple extension of Proposition 2.2:

Proposition 4.2.

Ñ(t) ≤ \frac{B}{2π(1 − e^{−2Bt})\sqrt{4πt}}Ñ_{1}(t),

where

Ñ_{1}(t) = ∫_{Λ_{1}} dxE_{θ}[exp(−tV_{ξ}(x))].  (4.3)

Proof of Proposition 4.1. By replacing the summation by the integration, we have

log Ñ_{1}(t) ≤ ∫_{ℝ^{3}} dq log E_{θ}[ exp ( − t inf_{x∈Λ_{2}} u(x − q − ξ_{0}))].

We restrict the integration to |q_{⊥}|_{∞} ≤ L_{⊥} and |q_{3}| ≤ L_{3} for some finite L_{⊥} and L_{3}. For any ε_{1} > 0, there exists R_{1} such that u(x) ≥ C_{0}(1 − ε_{1})/∥x∥α ; p̃ for any |x|_{∞} ≥ R_{1}, where |x|_{∞} = max_{i} |x_{i}|. Thus the right hand side is dominated by

∫ dq log_{|q_{⊥}|_{∞}≤L_{⊥},|q_{3}|≤L_{3}}\{ ∫ \frac{dy}{Z(θ, p)}_{|q+y|_{∞}≥R_{1}+2} exp ( − t inf_{x∈Λ_{2}} \frac{C_{0}(1 − ε_{1})}{∥x − q − y∥α ; p̃} − ∥y∥_{p}^{θ})

+ exp ( − t inf_{Λ_{2R_{1}+6}} u)}.

12 Naomasa Ueki

By changing the variables (q_{⊥}, q_{3}) to (t^{η_{⊥}}q_{⊥}, t^{η_{3}}q_{3}) and (y_{⊥}, y_{3}) to (t^{η_{⊥}}y_{⊥}, t^{η_{3}}y_{3}) with η_{⊥}, η_{3} ∈ (0, ∞) satisfying η_{⊥}α_{⊥} = η_{3}α_{3}, we see that this equals

t^{2η_{⊥}+η_{3}} ∫_{|q|_{∞}≤L} dq log {Ñ_{2}(t, q) + exp ( − t inf_{Λ_{2R_{1}+6}} u)},  (4.4)

where Ñ_{2}(t, q)

=t^{2η_{⊥}+η_{3}} ∫ \frac{dy}{Z(θ, p)}_{\begin{matrix}|q_{⊥} + y_{⊥}|_{∞} ≥ (R_{1} + 2)t^{−η_{⊥}} \\ |q_{3} + y_{3}| ≥ (R_{1} + 2)t^{−η_{3}} \end{matrix}} exp ( − t^{1−η_{⊥}α_{⊥}} inf_{\begin{matrix}|x_{⊥}|_{∞} ≤ t^{−η_{⊥}} \\ |x_{3}| ≤ t^{−η_{3}} \end{matrix}} \frac{C_{0}(1 − ε_{1})}{∥x − q − y∥α ; p̃}

− t^{η_{∗}α_{∗}} ∥|y_{∗}|^{θ_{∗}}, \frac{|y_{∗∗}|^{θ_{∗∗}}}{t^{|η_{⊥}θ_{⊥}−η_{3}θ_{3}|}} ∥_{p}),

we take L as an arbitrary constant independent of t by taking (L_{⊥}, L_{3}) appropriately, and we take ∗ and ∗∗ as elements of {⊥, 3} such that η_{∗}θ_{∗} = (η_{⊥}θ_{⊥}) ∨ (η_{3}θ_{3}) and ∗∗ ≠ ∗. Moreover we take η_{⊥} and η_{3} as η_{∗} = 1/(θ_{∗}+α_{∗}) and η_{∗∗} = α_{∗}/{α_{∗∗}(θ_{∗}+α_{∗})} so that 1−η_{⊥}α_{⊥} = η_{∗}θ_{∗}. Then, taking ε_{2}, ε_{3} > 0 sufficiently small and using the positivity assumption, we can dominate Ñ_{2}(t, q) by exp(−t^{η_{∗}θ_{∗}} Ñ_{3}(t, q))ε_{2}^{−2/θ_{⊥}−1/θ_{3}} for large enough t, where

\begin{matrix}Ñ_{3}(t, q) \\ = & inf_{x∈Λ_{ε_{3}} ,y∈ℝ^{3}} \{\frac{C_{0}(1 − ε_{1})}{∥x − q − y∥α ; p̃} + (1 − ε_{2})∥|y_{∗}|^{θ_{∗}}, \frac{|y_{∗∗}|^{θ_{∗∗}}}{t^{|η_{⊥}θ_{⊥}−η_{3}θ_{3}|}} ∥_{p}\}. \end{matrix}  (4.5)

Therefore we obtain

\overline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{2η_{⊥}+η_{3}+η_{∗}θ_{∗}}} ≤ − ∫_{|q|_{∞}≤L} Ñ_{4}(q)dq,

where

Ñ_{4}(q) = inf_{x∈Λ_{ε_{3}} ,y∈ℝ^{3}} \{\frac{C_{0}(1 − ε_{1})}{∥x − q − y∥α ; p̃} + (1 − ε_{2})∥|y_{∗}|^{θ_{∗}}, 1_{θ_{⊥}η_{⊥}=θ_{3}η_{3}}|y_{∗∗}|^{θ_{∗∗}} ∥_{p}^{θ}\}.

Since ε_{1}, ε_{2}, ε_{3} and L are arbitrary, we can complete the proof. □

5. 3-dimensional lower estimate In this section we prove the following lower estimates necessary to prove Theorem 2:

Proposition 5.1. If (1.15) holds with (1.16) and (1.22), then we have

\underline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{κ̃(α,θ)}} ≥ −C(α, θ, C_{0})  (5.1)

with (4.2) and (1.24).

Proof. We use the bound

\begin{matrix}Ñ(t) \\ ≥ & (2R_{⊥})^{−2}(2R_{3})^{−1} exp \{ − tλ_{1}(H_{B(R_{⊥})}) \\ − tλ_{1}( − (\frac{d^{2}}{dx^{2}} )_{(−R_{3},R_{3})})\}Ñ_{1}(t), \end{matrix}  (5.2)

where R_{⊥}, R_{3} ∈ ℕ, H_{R_{⊥}} is the same symbol used in (3.3), (d^{2}/(dx^{2}))_{(−R_{3},R_{3})} is the restriction to the interval (−R_{3}, R_{3}) of the 1-dimensional Laplacian by the Dirichlet boundary condition,

Ñ_{1}(t) = E[ exp ( − t ∫ dxψ_{R_{⊥}}(x_{⊥})^{2}ϕ_{R_{3}}(x_{3})^{2}V_{ξ}(x))],

and ψ_{R_{⊥}} and ϕ_{R_{3}} are the normalized ground states of H_{B(R_{⊥})} and −(d^{2}/(dx^{2}))_{(−R_{3},R_{3})}, respectively. This can be proven by the same method as for the corresponding bound in (3.3) and Theorem (9.6) in [15] for ℝ^{d}-stationary random fields. We have Erdös’s bound

λ_{1}(H_{R_{⊥}}) ≤ exp ( − \frac{B}{2} R_{⊥}^{2}(1 − ε_{1}))  (5.3)

as in (3.4) and

λ_{1}( − (\frac{d^{2}}{dx^{2}} )_{(−R_{3},R_{3})}) = (\frac{π}{2R_{3}} )^{2},  (5.4)

where ε_{1} is an arbitrarily fixed positive constant. By replacing the summation by integration, we have

log Ñ_{1}(t) ≥ ∫_{ℝ^{3}} Ñ_{2}(t, q)dq,

where

Ñ_{2}(t, q) = log E[ exp ( − t ∫ dxψ_{R_{⊥}}(x_{⊥})^{2}ϕ_{R_{3}}(x_{3})^{2} sup_{z∈Λ_{1}} u(x − q − z − ξ_{0}))].

For any ε_{2} > 0, there exists R_{2} such that u(x) ≤ C_{0}(1 + ε_{2})/∥x∥α ; p̃ for any |x| ≥ R_{2} by the assumption (1.15). To use this bound in the above right hand side, we need inf{|x − q − z − ξ_{0}| : |x_{⊥}| < R_{⊥}, |x_{3}| < R_{3}, z ∈ Λ_{1}} ≥ R_{2}. A sufficient condition of this is |q_{⊥}| ≥ 2(R_{2} + R_{⊥} + 2) with |ξ_{0,⊥}| ≤ |q_{⊥}|/2 or |q_{3}| ≥ 2(R_{2} + R_{3} + 1) with |ξ_{0,3}| ≤ |q_{3}|/2. We fix β_{⊥}, β_{3} > 0 and take t large enough so that t^{β_{⊥}} ≥ 2(R_{2} + R_{⊥} + 2) and t^{β_{3}} ≥ 2(R_{2} + R_{3} + 1). Then, by using also log(1 − X) ≥ −2X for 0 ≤ X ≤ 1/2 we obtain

\begin{aligned}& ∫ dq_{|q_{⊥}|≥t^{β_{⊥}} ,|q_{3}|^{1/β_{3}} ≤|q_{⊥}|^{1/β_{⊥}}} Ñ_{2}(t, q) \\ ≥ & ∫ dq_{|q_{⊥}|≥t^{β_{⊥}} ,|q_{3}|^{1/β_{3}} ≤|q_{⊥}|^{1/β_{⊥}}} (\frac{−tC_{0}(1 + ε_{2})2^{α_{⊥}}}{(|q_{⊥}| − 2(R_{⊥} + 2))^{α_{⊥}}} \\ & + log P (|ξ_{0,⊥}| ≤ |q_{⊥}|/2)) \\ ≥ & − c_{1}t^{1+β_{3}−β_{⊥}(α_{⊥}−2)} − c_{2} exp(−c_{3}t^{β_{⊥}θ_{⊥}}) \end{aligned}  (5.5)

if

α_{⊥} > 2 + β_{3}/β_{⊥},  (5.6)

and

\begin{aligned}& ∫ dq_{|q_{3}|≥t^{β_{3}} ,|q_{⊥}|^{1/β_{⊥}} ≤|q_{3}|^{1/β_{3}}} Ñ_{2}(t, q) \\ ≥ & ∫ dq_{|q_{3}|≥t^{β_{3}} ,|q_{⊥}|^{1/β_{⊥}} ≤|q_{3}|^{1/β_{3}}} (\frac{−tC_{0}(1 + ε_{2})2^{α_{3}}}{(|q_{3}| − 2(R_{3} + 1))^{α_{⊥}}} \\ & + log P (|ξ_{0,3}| ≤ |q_{3}|/2)) \\ ≥ & − c_{4}t^{1+2β_{⊥}−β_{3}(α_{3}−1)} − c_{5} exp(−c_{6}t^{β_{3}θ_{3}}) \end{aligned}  (5.7)

if

α_{3} > 1 + 2β_{⊥}/β_{3}.  (5.8)

The other part is estimated as

∫ dq_{|q_{⊥}|≤t^{β_{⊥}} ,|q_{3}|≤t^{β_{3}}} Ñ_{2}(t, q)

≥ ∫ dq_{|q_{⊥}|≤t^{β_{⊥}} ,|q_{3}|≤t^{β_{3}}} log ∫ \frac{dy}{Z(θ, p)}_{|q_{⊥}+y_{⊥}|≥R_{2}+R_{⊥}+2 or |q_{3}+y_{3}|≥R_{2}+R_{3}+1}

× exp (\frac{−tC_{0}(1 + ε_{2})}{inf_{|x_{⊥}|<R_{⊥},|x_{3}|<R_{3},z∈Λ_{1}}\{∥x − q − z − y∥α ; p̃\}} − ∥y∥_{p}^{θ}).

(5.9)

By the same change of variables as in (4.4) and (4.5), we find that the right hand side equals

t^{2η_{⊥}+η_{3}} ∫ dq_{|q_{⊥}|≤t^{β_{⊥}−η_{⊥}} ,|q_{3}|≤t^{β_{3}−η_{3}}} log ∫ \frac{dyt^{2η_{⊥}+η_{3}}}{Z(θ, p)} ; \begin{matrix}|q_{⊥} + y_{⊥}| ≥ (R_{2} + R_{⊥} + 2)/t^{η_{⊥}} \\ or |q_{3} + y_{3}| ≥ (R_{2} + R_{3} + 1)/t^{η_{3}} \end{matrix} exp(−t^{η_{∗}θ_{∗}} Ñ_{3}(t, y, q)),

where

\begin{matrix}Ñ_{3}(t, q, y) \\ = & C_{0}(1 + ε_{2})/ inf\{|x − q − z − y|α ; p̃ : |x_{⊥}| ≤ R_{⊥}/t^{η_{⊥}}, |x_{3}| ≤ R_{3}/t^{η_{3}}, \\ |z_{⊥}|_{∞} ≤ 1/(2t^{η_{⊥}}), |z_{3}| ≤ 1/(2t^{η_{3}})\} \\ + ∥|y_{∗}|^{θ_{∗}}, \frac{|y_{∗∗}|^{θ_{∗∗}}}{t^{|η_{⊥}θ_{⊥}−η_{3}θ_{3}|}} ∥_{p}. \end{matrix}  (5.10)

Taking γ_{⊥}, γ_{3} ∈ (0, ∞) and w ∈ ℝ^{3}, we restrict the integration to B(w_{⊥}, t^{−γ_{⊥}}) × (w_{3} − t^{−γ_{3}}, w_{3} + t^{−γ_{3}}). Then we can bound the integrand with respect to q from below by

log \frac{2πt^{2(η_{⊥}−γ_{⊥})+(η_{3}−γ_{3})}}{Z(θ, p)} − t^{η_{∗}θ_{∗}} Ñ_{4}(t, q),  (5.11)

where Ñ_{4}(t, q)

= inf { sup_{y∈B(w_{⊥},t^{−γ_{⊥}})×(w_{3}−t^{−γ_{3}} ,w_{3}+t^{−γ_{3}} )} Ñ_{3}(t, q, y)

: w_{3} ∈ ℝ, w_{⊥} ∈ ℝ^{2}, d(B(w_{⊥}, t^{−γ_{⊥}}), −q_{⊥}) ≥ (R_{2} + R_{⊥} + 2)t^{−η_{⊥}}}

∧ inf { sup_{y∈B(w_{⊥},t^{−γ_{⊥}})×(w_{3}−t^{−γ_{3}} ,w_{3}+t^{−γ_{3}} )} Ñ_{3}(t, q, y)

: w_{⊥} ∈ ℝ^{2}, w_{3} ∈ ℝ, d((w_{3} − t^{−γ_{3}}, w_{3} + t^{−γ_{3}}), −q_{3}) ≥ (R_{2} + R_{3} + 1)t^{−η_{3}}}.

(5.12)

We now specify R_{⊥} and R_{3} as the integer parts of ε_{3}t^{η_{⊥}} and ε_{3}t^{η_{3}}, respectively, where ε_{3} is an arbitrarily fixed positive number. We take β_{⊥} and β_{3} as β_{⊥} = η_{⊥} + ζε_{4} and β_{3} = η_{3} + ε_{4}, respectively, where ε_{4} is also an arbitrarily fixed positive number and ζ is a number satisfying

\frac{1}{α_{⊥} − 2} < ζ < \frac{α_{3} − 1}{2}.

By taking ε_{4} small, we have 1 − 2η_{3}, 1 + β_{3} − β_{⊥}(α_{⊥} − 2), 1 + 2β_{⊥} − β_{3}(α_{3} − 1), 2β_{⊥} + β_{3} < 2η_{⊥} + η_{3} + η_{∗}θ_{∗} and (5.6) and (5.8). Thus we obtain

\underline{lim}_{t↑∞} \frac{log Ñ(t)}{t^{2η_{⊥}+η_{3}+η_{∗}θ_{∗}}} ≥ − \overline{lim}_{t↑∞} ∫_{|q_{⊥}|≤t^{β_{⊥}−η_{⊥}} ,|q_{3}|≤t^{β_{3}−η_{3}}} Ñ_{4}(t, q)dq.  (5.13)

16 Naomasa Ueki

When |q_{⊥}| ≤ t^{β_{⊥}−η_{⊥}} and |q_{3}| ≤ t^{β_{3}−η_{3}}, we have t^{−1} ≤ |q_{⊥}|^{−1/(β_{⊥}−η_{⊥})} and t^{−1} ≤ |q_{3}|^{−1/(β_{3}−η_{3})}. Thus, for large |q|, by taking w as 0, we can dominate Ñ_{4}(t, q) by

(|q_{⊥}| − 1)_{+}^{−α_{⊥}} ∧ (|q_{3}| − 1)_{+}^{−α_{3}} + |q_{⊥}|^{−γ_{⊥}θ_{⊥}/(β_{⊥}−η_{⊥})} ∧ |q_{3}|^{−γ_{3}θ_{3}/(β_{3}−η_{3})}.

This is integrable if we take γ_{⊥} and γ_{3} large enough so that γ_{⊥}θ_{⊥}/(β_{⊥} − η_{⊥}) > 3 and γ_{3}θ_{3}/(β_{3} − η_{3}) > 3. Thus, by the Lebesgue convergence theorem, we have

lim_{t↑∞} ∫_{|q_{⊥}|≤t^{β_{⊥}−η_{⊥}} ,|q_{3}|≤t^{β_{3}−η_{3}}} Ñ_{4}(t, q)dq

= ∫_{ℝ^{3}} dq inf \{\frac{C_{0}(1 + ε_{2})}{inf_{|x_{⊥}|,|x_{3}|≤ε_{3}} ∥x − q − y∥α ; p̃} + ∥|y_{∗}|^{θ_{∗}}, 1_{η_{3}θ_{3}=η_{∗}θ_{∗}}|y_{∗∗}|^{θ_{∗∗}}∥_{p}

: |y_{⊥} + q_{⊥}| ∨ |y_{3} + q_{3}| ≥ ε_{3}}.

Since ε_{2} and ε_{3} are arbitrary, this completes the proof. □

6. Classical integrated density of states In this section we summarize the results on the leading terms of the low energy asymptotics of the

classical integrated densities of states in a general setting.

The 2-dimensional results are the following: Theorem 3. (i) If (1.7) holds for any R ≥ 1 and (1.8) as |x| → ∞ for some C_{0} ∈ (0, ∞) and α ∈ (2, ∞), then the classical integrated density of states defined by (1.27) has the same leading term with (1.9):

lim_{λ↓0} λ^{κ} log N_{c}(λ) = \frac{−κ^{κ}}{(κ + 1)^{κ+1}} \{ ∫_{ℝ^{2}} dq inf_{y∈ℝ^{2}} (\frac{C_{0}}{|q + y|^{α}} + |y|^{θ})\}^{κ+1}.  (6.1)

(ii) If (1.7) holds for any R ≥ 1 and (1.10) as |x| → ∞ for some C_{0} ∈ (0, ∞) and α ∈ (0, 2 + θ), then the classical integrated density of states defined by (1.27) has the same leading term with (1.11):

lim_{λ↓0}(log(1/λ))^{−(2+θ)/α} log N_{c}(λ) = \frac{−2πC_{0}^{(2+θ)/α}}{(θ + 1)(θ + 2)}.  (6.2)

(iii) If (1.7) holds for any R ≥ 1 and (1.10) as |x| → ∞ for some C_{0} ∈ (0, ∞) and α ∈ (2 + θ, ∞), then we have

lim_{λ↓0}(log(1/λ))^{−1} log N_{c}(λ) = −1.  (6.3)

(iv) If (1.7) holds for any R ≥ 1 and (1.10) as |x| → ∞ for some C_{0} ∈ (0, ∞) and α = 2 + θ, then we have

lim_{λ↓0}(log(1/λ))^{−1} log N_{c}(λ) = −1 − \frac{2πC_{0}}{(θ + 1)(θ + 2)}.  (6.4)

magnetic Lifshitz tails 17

(v) If supp u is compact, then the classical integrated density of states defined by (1.27) satisfies the following:

N_{c}(λ) = \frac{K_{c}λ}{4π}(1 + o(1))  (6.5)

as λ ↓ 0, where

K_{c} = ∫_{Λ_{1}} dx ∏_{q∈ℤ^{2}} P_{θ}(x − q − ξ_{q} ∉ supp u).  (6.6)

(i)-(iv) are proven by the same methods in Sections 2 and 3, and (v) is proven by the Lebesgue convergence theorem.

Similarly by the methods in Sections 4 and 5, and the Lebesgue convergence theorem, we have the following 3-dimensional results:

Theorem 4. Let N_{c}(λ) be the classical integrated density of states of our 3-dimensional setting defined similarly as in (1.27).

(i) If (1.21) holds for any R ≥ 1 and (1.15) as |x| → ∞ for some C_{0} ∈ (0, ∞), p̃ ∈ [1, ∞] and α = (α_{⊥}, α_{3}) ∈ (0, ∞)^{2} satisfying (1.16), then N_{c}(λ) has the same leading term with (1.25):

lim_{λ↓0} λ^{κ(α,θ)} log N_{c}(λ) = \frac{−κ(α, θ)^{κ(α,θ)}}{(1 + κ(α, θ))^{1+κ(α,θ)}} C(α, θ, C_{0})^{1+κ(α,θ)}.  (6.7)

(ii) If (1.21) holds for any R ≥ 1 and

u(x) = exp (\frac{−∥x∥α ; p̃}{C_{0}} (1 + o(1)))  (6.8)

as |x| → ∞ for some C_{0} ∈ (0, ∞), p̃ ∈ [1, ∞] and α = (α_{⊥}, α_{3}) ∈ (0, ∞)^{2} satisfying

χ(α, θ) > 1,  (6.9)

then N_{c}(λ) satisfies

lim_{λ↓0}(log(1/λ))^{−χ(α,θ)} log N_{c}(λ) = −C_{0}^{χ(α,θ)}D(α, θ, p, p̃),  (6.10)

where

χ(α, θ) := \frac{2}{α_{⊥}} + \frac{1}{α_{3}} + (\frac{θ_{⊥}}{α_{⊥}} ∧ \frac{θ_{3}}{α_{3}} )  (6.11)

and

\begin{matrix}D(α, θ, p, p̃) \\ := ∫_{∥q∥α ; p̃≤1} dq inf_{y:∥q+y∥α ; p̃≥1} ∥1_{\frac{θ_{⊥}}{α_{⊥}} ≤ \frac{θ_{3}}{α_{3}}} |y_{⊥}|^{θ_{⊥}}, 1_{\frac{θ_{⊥}}{α_{⊥}} ≥ \frac{θ_{3}}{α_{3}}} |y_{3}|^{θ_{3}} ∥_{p}. \end{matrix}  (6.12)

18 Naomasa Ueki

(iii) If (1.21) holds for any R ≥ 1 and (6.8) as |x| → ∞ for some C_{0} ∈ (0, ∞), p̃ ∈ [1, ∞] and α = (α_{⊥}, α_{3}) ∈ (0, ∞)^{2} satisfying

χ(α, θ) < 1,  (6.13)

then N_{c}(λ) satisfies

lim_{λ↓0}(log(1/λ))^{−χ(α,θ)} log N_{c}(λ) = −\frac{3}{2}.  (6.14)

(iv) If (1.21) holds for any R ≥ 1 and (6.8) as |x| → ∞ for some C_{0} ∈ (0, ∞), p̃ ∈ [1, ∞] and α = (α_{⊥}, α_{3}) ∈ (0, ∞)^{2} satisfying

χ(α, θ) = 1,  (6.15)

then N_{c}(λ) satisfies

lim_{λ↓0}(log(1/λ))^{−1} log N_{c}(λ) = −\frac{3}{2} − C_{0}D(α, θ, p, p̃).  (6.16)

(v) If supp u is compact, then N_{c}(λ) satisfies the following:

N_{c}(λ) = \frac{K_{c}λ^{3/2}}{6π^{2}} (1 + o(1))  (6.17)

as λ ↓ 0, where

K_{c} = ∫_{Λ_{1}} dx ∏_{q∈ℤ^{3}} P_{θ}(x − q − ξ_{q} ∉ supp u).  (6.18)

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magnetic Lifshitz tails 19

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