Quantum behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice

Naomasa Ueki

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 rapidly and the effect of the kinetic part are investigated.

1. Introduction

Let

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

be the Landau Hamiltonian on L^{2}(ℝ^{2}) with the uniform magnetic field B > 0 subtracted B so that the lowest eigenvalue is 0, where i = \sqrt{−1}. Let V_{ξ}(x) = ∑_{q∈ℤ^{2}} u(x − q − ξ_{q}) 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

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

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). We will consider the random Schrödinger operators

(1.2)  H_{ξ} = H + V_{ξ},

and the restriction H_{ξ} of H to the complement of T (ξ) = \overline{⋃_{q∈ℤ^{2}} B(q + ξ_{q} : r_{0})} by the Dirichlet boundary condition, where r_{0} ∈ (0, ∞) and B(p : r) := {x ∈ ℝ^{2} : |x − p| < r} is the open ball with the center p and the radius r. V_{ξ} and T (ξ) are soft and hard obstacles, respectively. We will consider the integrated density of states (N(λ))_{λ≥0} and (N (λ))_{λ≥0} of H_{ξ} and H_{ξ}, respectively: (N(λ))_{λ≥0} is defined by

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

for any point of continuity of N(λ) 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)^{2} with the Dirichlet boundary condition. N(λ) exists as a deterministic increasing function (cf. [2], [13]). (N (λ))_{λ≥0} is similarly defined for H_{ξ}.

In this note, we first prove the following:

Theorem 1. (i) If

(1.4)  essinf_{|x|≤R} u(x) > 0

for some R > 0 and

(1.5)  \overline{lim}_{|x|→∞} |x|^{−2} log u(x) = −∞,

then we have

(1.6)  \underline{lim}_{λ↓0} ( log \frac{1}{λ})^{−(1+θ/2)} log N(λ) ≥ \frac{−2^{2+θ/2}π}{(θ + 1)(θ + 2)B^{1+θ/2}}

and

(1.7)  \overline{lim}_{λ↓0} ( log \frac{1}{λ})^{−1}( log log \frac{1}{λ})^{−n} log N(λ) = −∞

for any n ∈ ℕ. Moreover if θ > 4, then we have

(1.8)  \overline{lim}_{λ↓0} ( log \frac{1}{λ})^{−(1+(θ−4)/6)} log N(λ) ≤ \frac{−K}{B^{1+(θ−4)/6}} ,

where K is a finite constant independent of B. The same estimates hold for (N (λ))_{λ≥0}.

(ii) If (1.4) for any R ≥ 1 and

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

as |x| → ∞ with α = 2, then we have

(1.10)  \underline{lim}_{λ↓0} ( log \frac{1}{λ})^{−(1+θ/2)} log N(λ) ≥ \frac{−2π}{(θ + 1)(θ + 2)}(\frac{2}{B} + C_{0})^{1+θ/2}.

We next formulate the 3-dimensional problem by referring to the corresponding result [9] for the Poisson case. 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}

for arbitrarily fixed θ = (θ_{⊥}, θ_{3}) ∈ (0, ∞)^{2} and p ∈ [1, ∞]. Let V_{ξ}(x) = ∑_{q∈ℤ^{3}} u(x − q − ξ_{q}) be a random potential on ℝ^{3}, where u is a nonnegative function belonging to the Kato class K_{3} (cf. [2] p-53), ξ = (ξ_{q})_{q∈ℤ^{3}} is a collection of independently and identically distributed ℝ^{3}-valued random variables with the distribution

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

and Z(θ, p) is the normalizing constant. 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}}

be the direct sum of the Landau Hamiltonian on L^{2}(ℝ^{2}) subtracted the magnetic field B and the Laplacian on L^{2}(ℝ). As in the 2-dimensional case and as in [20], we will consider the integrated density of states (N(λ))_{λ≥0} of the random Schrödinger operator

(1.12)  H_{ξ} = H + V_{ξ},

and that (N (λ))_{λ≥0} of the restriction H_{ξ} of H to the complement of T (ξ) = \overline{⋃_{q∈ℤ^{3}} B(q + ξ_{q} : r_{0})} by the Dirichlet boundary condition, where r_{0} ∈ (0, ∞) and B(p : r) := \{x ∈ ℝ^{3} : |x − p| < r} is the open ball with the center p ∈ ℝ^{3} and the radius r.

For this we prove the following:

Theorem 2. We assume

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

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

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

We set

(1.15)  μ_{1}(α_{⊥}, θ) = \frac{3}{α_{⊥} − 2} + \frac{1}{2} + \frac{3θ_{⊥}}{2(α_{⊥} − 2)} ∨ \frac{θ_{3}}{2}

and

(1.16)  μ_{2}(α, θ) = \frac{2/α_{⊥}}{1 − 1/α_{3} − 2/α_{⊥}} + \frac{1}{2} + \frac{θ_{⊥}/α_{⊥}}{1 − 1/α_{3} − 2/α_{⊥}} ∧ \frac{θ_{3}}{2} .

Then we have

\underline{lim}_{λ↓0}λ^{μ_{1}(α_{⊥},θ)} log N(λ) > −∞, (1.17)

\overline{lim}_{λ↓0}λ^{μ_{2}(α,θ)} log N(λ) < 0 , (1.18)

(1.19)  \underline{lim}_{λ↓0} \frac{λ^{(1+θ_{3})/2}}{log(1/λ)} log N (λ) > −∞

and

(1.20)  \overline{lim}_{λ↓0}\sqrt{λ} log N (λ) < 0.

If supp u is compact, then (1.19) and (1.20) hold by replacing N (λ) by N(λ).

As in [20], the above results are extensions of the results in [7] and [8], where the same problem is considered in the case without magnetic fields. As is discussed in [7] and [8], 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 lattice ℤ^{2} as θ → ∞ by slightly modifying the definition as P_{θ}(ξ_{q} ∈ dx) = exp(−(1 + |x|)^{θ})dx/Z(θ), which brings no essential changes for our results. The results in [7] and [8] shows that the leading term of the integrated density of states also tends to those for the 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 [14]. The first result was given by Broderix, Hundertmark, Kirsch and Leschke [1]: they determined the leading term for the case where d = 2, u(x) = C_{0}|x|^{−α}(1 + o(1)) as |x| → ∞ is satisfied for some α > 2 and C_{0} > 0 and the point process {q + ξ_{q}}_{q∈ℤ^{2}} is replaced by the Poisson point process. As is discussed in [10] and [20], this leading term coincides with that of the classical integrated density of states, which depends only on the scalar potential, as in Pastur’s case [17] without magnetic fields. Then Erdős [5] treated the same 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 other precise informations on the single site potential as in Nakao’s case [15] without magnetic field referring to Donsker and Varadhan’s result [3]. On this behavior we may say that the quantum effect appears. The borderline between the classical and quantum behaviors was determined by Hupfer, Leschke and Warzel [10]. The borderline corresponds to the case of (1.9) with α = 2. They also determined the leading term for the case of (1.9) with α ∈ (0, 2). The leading term for the borderline case was determined by Erdős [6]. The leading term for the classical case was determined also in the 3-dimensional case by Hundertmark, Kirsch and Warzel [9]. For the 3-dimensional case, results appearing the quantum effect were obtained by Warzel [21], where general bounds and the leading order for special cases were obtained. In this note we try to extend the theory to our setting. We treat simple classical cases in [20] and remaining cases in this note. Our results in this note give only upper and lower estimates. By these upper estimates and Theorems 6.1 and 6.2 in [20], we see that the quantum effect appears in the following five cases: (i) (1.4) for any R ≥ 1 and (1.9) with α ≥ θ +2, (ii) (1.4) for any R ≥ 1 and (1.9) with α > 6 and θ > 4, (iii) supp u is a nonempty compact set, (iv) essinf_{|x|≤R} u(x) > 0 for any R ≥ 1 and (1.13) with (1.14), and (v) supp u is a nonempty compact set. We conjecture that the leading terms are close to our lower bounds in the 2-dimensional cases and are close to our upper bounds in the 3-dimensional case. One reason is that the bounds tend to the corresponding leading terms given in [5], [6] and [14] for the Poisson case as θ → 0. Thus the borderline between the classical and quantum behaviors is expected to be the case of Theorem 1 (ii) and the case of 2/α_{⊥} + 3/α_{3} = 1 in Theorem 2 as in the Poisson case.

The organization of this note is as follows. We prove Theorem 1 in Sections 2, 3 and 4: we prove the lower estimates in Section 2, the upper estimate (1.8) in Section 3 and the upper estimate (1.7) in Section 4. We next prove Theorem 2 in Sections 5 and 6: we prove the lower estimate in Section 5 and the upper estimate in Section 6.

2. Lower estimates for the 2-dimensional case

In this section we give lower estimates for Theorem 1. Let

Ñ(t) = ∫_{0}^{∞} e^{−tλ}dN(λ).

(1.10) is proven by the following, which we prove by referring to [1] and [10]:

Proposition 2.1. If (1.9) holds with α = 2, then we have

(2.1) \underline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)^{1+θ/2}} ≥ \frac{−2π}{(θ + 1)(θ + 2)}(\frac{2}{B} + C_{0})^{1+θ/2}.

The basic inequality for the proof is the following extension of (3.5) in [1]: Proposition 2.2. Ñ(t) ≥ Ñ_{1}(t)/(4πt), where

Ñ_{1}(t) = ∫_{Λ_{1}} dx E_{θ}[ exp ( − t ∫_{ℝ^{2}} ϕ_{B}(y)V_{ξ}(x + y)dy)]

and ϕ_{B}(x) = exp(−B|x|^{2}/2)B/(2π). Proof. In (2.4) of [20] we obtain

(2.2) lim_{Ω↓0} \frac{Ω^{2}t}{π}E_{θ}[Tr[exp(−t(H_{ξ} + Ω^{2}|x|^{2}))]] = Ñ(t).

Then the rest of the proof is same with that of (3.5) in [1]. □ Proof of Proposition 2.1. Ñ_{1}(t) in Proposition 2.2 is rewritten as

∫_{Λ_{1}} dxE_{θ}[ exp ( − t ∑_{q∈ℤ^{2}} u_{B}(x − q − ξ_{q}))],

where u_{B} = u ∗ ϕ_{B}. By Lemma 3.5 (ii) in [10], we have

lim_{|x|→∞} |x|^{−2} log u_{B}(x) = −1/(C_{0} + 2/B) =: −1/C_{B}.

By the same lower estimate of Section 3 in [20], we obtain

log Ñ_{1}(t) ≥ −t exp(−(1 − ε)^{4}R^{2}/C_{B}) − (R + 3ε + 1)^{θ+2} \frac{2π}{(θ + 1)(θ + 2)}

for large enough R. By setting R = \sqrt{C_{B}(log t)}(1 − ε)^{−2}, we can complete the proof. □ Proof of ”(1.10) implies (1.6)”. For any η > 0, we can take a single site potential u_{η} satisfying u_{η} ≥ u and (1.9) where α and C_{0} is replaced by 2 and η, respectively. The corresponding integrated density of states N_{η}(λ) satisfies N(λ) ≥ N_{η}(λ) and

\underline{lim}_{λ↓0} ( log \frac{1}{λ})^{−(1+θ/2)} log N_{η}(λ) ≥ \frac{−2π}{(θ + 1)(θ + 2)}(\frac{2}{B} + η)^{1+θ/2}.

Since η is arbitrary, we obtain (1.6). □

3. An upper estimate appearing quantum effects for the 2-dimensional case In this section we prove the following upper estimate which is enough for (1.8):

Proposition 3.1. If θ > 4 and u(x) = C_{0}1_{B(r_{0})}(x), then we have

(3.1)  \overline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)^{(2+θ)/6}} ≤ \frac{−C_{1}}{B^{(2+θ)/6}}

for some positive constant C_{1} independent of B, where B(r_{0}) := B(0 : r_{0}).

We reduce the proof to an estimate of the lowest eigenvalue of an operator with the Dirichlet boundary condition following Section 6 in Erdős [5]: for any β, ε_{1} ∈ (0, ∞),

(3.2)  \overline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)^{(2+θ)/6}} ≤ \overline{lim}_{t↑∞} \frac{1}{(log t)^{(2+θ)/6}} log ∫_{Λ_{1}} dzE_{θ}[ exp (\frac{ε_{1} − t}{4}λ_{1}(H_{ξ,B(z:s)}^{B+β}))].

In (3.2), s = \sqrt{(8/β) log t} and H_{ξ,B(z:s)}^{B+β} is H_{ξ,B(z:s)} where B is replaced by B + β. The proof of (3.2) can be given by the same method of the proof of Theorem 6.3 in Erdős [5], where we used the estimate

Tr [ exp ( − \frac{t}{4}H_{ξ,B(z:s)}^{B+β})] ≤ exp (\frac{ε_{1} − t}{4}λ_{1}(H_{ξ,B(z:s)}^{B+β}))\frac{cs^{2}}{ε_{1}} exp (\frac{ε_{1}(B + β)}{4})

instead of (3.4) in [5].

The obstacles are reduced to the hard obstacles by the following:

Lemma 3.1. If u(x) = C_{0}1_{B(2r_{0})}(x), then we have

(3.3)  λ_{1}(H_{ξ,B(z:s)}) ≥ \frac{C_{0}}{2} ∧ \frac{λ_{1}(H_{B(z:s)\T (ξ)})}{4(1 + cr_{0}^{−2})}.

Proof. We represent the Landau Hamiltonian by the creation and annihilation operators: H = A^{∗}A, where A = (i∂/(∂x_{1}) − Bx_{2}/2) + i(i∂/(∂x_{2}) + Bx_{1}/2). Thus the lowest eigenvalue has the representaion

λ_{1}(H_{ξ,B(z:s)}) = ∥Aφ_{0}∥^{2} + (V_{ξ}φ_{0}, φ_{0}),

where φ_{0} is a normalized ground state of H_{ξ,B(z:s)}. Since

(V_{ξ}φ_{0}, φ_{0}) ≥ C_{0}∥φ_{0}∥_{L^{2}(B(z:s)∩⋃_{q∈ℤ^{2}} B(q+ξ_{q}:2r_{0}))}^{2},

we have

∥φ_{0}∥_{L^{2}(B(z:s)∩⋃_{q∈ℤ^{2}} B(q+ξ_{q}:2r_{0}))}^{2} ≤ λ_{1}(H_{ξ,B(z:s)})/C_{0}.

On the other hand, we take a smooth function ϑ on ℝ^{2} such that ϑ = 0 on ⋃_{q∈ℤ^{2}} B(q + ξ_{q} : r_{0}), ϑ = 1 on (⋃_{q∈ℤ^{2}} B(q + ξ_{q} : 3r_{0}/2))^{c}, 0 ≤ ϑ ≤ 1 and |∇ϑ|^{2} ≤ 9/r_{0}^{2}. Then we have

\begin{matrix}∥Aφ_{0}∥^{2} ≥ ∥ϑAφ_{0}∥^{2} ≥ \frac{1}{2}∥Aϑφ_{0}∥^{2} − \frac{9}{r_{0}^{2}} ∥φ_{0}∥_{L^{2}(B(z:s)∩⋃_{q∈ℤ^{2}} B(q+ξ_{q}:3r_{0}/2))}^{2} \\ ≥ \frac{1}{2}λ_{1}(H_{B(z:s)\T (ξ)})∥ϑφ_{0}∥^{2} − \frac{9}{r_{0}^{2}} \frac{λ_{1}(H_{ξ,B(z:s)})}{C_{0}}. \end{matrix}

Moreover we have

∥ϑφ_{0}∥^{2} ≥ 1 − ∫_{B(z:s)∩⋃_{q∈ℤ^{2}} B(q+ξ_{q}:3r_{0}/2)} |φ_{0}(x)|^{2}dx ≥ (1 − \frac{λ_{1}(H_{ξ,B(z:s)})}{C_{0}})_{+}.

Therefore, if 1 − λ_{1}(H_{ξ,B(z:s)})/C_{0} ≥ 1/2, then we have

λ_{1}(H_{ξ,B(z:s)}) ≥ \frac{1}{4}λ_{1}(H_{B(z:s)\T (ξ)}) − \frac{9}{r_{0}^{2}} \frac{λ_{1}(H_{ξ,B(z:s)})}{C_{0}},

from which we obtain (3.3). □ To estimate λ_{1}(H_{B(z:s)\T (ξ)}), we develop Erdős’s isoperimetric inequality stating

(3.4)  λ_{1}(H_{D}) ≥ λ_{1}(H_{B(L)})

for any bounded domain D of ℝ^{2} with the area |D| = |B(L)| = πL^{2} and a smooth boundry [4]. The right hand side is dominated from below by exp(−BL^{2}(1 + ε)/2) for sufficiently large L, where ε is an arbitrarily small positive constant. We need more precise estimates for more complicated domains. In this note we prove the following:

Proposition 3.2. Let D be a bounded domain of ℝ^{2} with the area |D| = πL^{2} whose boundary is a finite union of smooth curves. Let ρ be the radius of the largest disk contained in D. Then we have

(3.5)  λ_{1}(H_{D}) ≥ \frac{C_{1}}{L^{2}} exp(−C_{2}BL(1 ∨ ρ)^{3}),

where C_{1} and C_{2} are universal positive constants.

Proof. By the same proof of the inequality (17) in [4], we have

λ_{1}(H_{D}) ≥ inf { ∫ |e^{−Λ(ψ(x))}∇e^{Λ(ψ(x))}ψ(x)|^{2}dx/ ∫ ψ(x)^{2}dx
: 0 ≤ ψ ∈ C^{∞}(\overline{D}) ∩ C^{ω}(D), ψ = 0 on ∂D,
Λ(c) = B ∫_{c}^{∞} dξ|{ψ > ξ}|( ∫_{{ψ=ξ}} |∇ψ(x)|L(dx))^{−1}},

where C^{ω}(D) is the set of all analytic functions on D and L(·) is the measure corresponding to the length of curves. Λ is a nonnegative, strictly monotone decreasing, continuous function on the range of ψ. It is also included in the set C_{∗}^{∞} of all real valued functions on ℝ which are smooth everywhere except for finitely many points. As in the equations (19) and (20) in [4], we rewrite the quatities in the infimum as follows:

∫ |e^{−Λ(ψ(x))}∇e^{Λ(ψ(x))}ψ(x)|^{2}dx = B ∫_{0}^{∞}(Θ′(b))^{2}e^{−2b}F (b)db

and

∫ ψ(x)^{2}dx = ∫_{0}^{∞} Θ(b)^{2}e^{−2b}F′(b)db,

where Θ(b) = Λ^{−1}(b)e^{b} and F (b) = |{x ∈ D : h(x) < b}| with h(x) = Λ(ψ(x)). We now apply Lemma 3.2 below. Then we have

L(\{x : h(x) = b\}) ≥ C_{1}(1 ∨ ρ)\sqrt{|\{x : h(x) < b\}|(1 − (\frac{\sqrt{|\{x : h(x) < b\}|} − ρ\sqrt{π}}{\sqrt{|\{x : h(x) < b\}|} + ρ\sqrt{π}} )_{+}^{2})^{−1}}

and

F′(b) ≥ \frac{C_{2}(1 ∨ ρ)^{2}}{B(1 − (^{\frac{\sqrt{F (b)}−ρ\sqrt{π}}{\sqrt{F (b)}+ρ\sqrt{π}}} )_{+}^{2})}.

Therefore (23) in [4] is rewritten as follows: λ_{1}(H_{D})

≥ inf {B ∫_{0}^{b_{0}}(Θ′(b))^{2}e^{−2b}F (b)db( ∫_{0}^{b_{0}} Θ(b)^{2}e^{−2b}F′(b)db)^{−1}

: F, Θ ∈ C_{∗}^{∞}, Θ ≥ 0, Θ is bounded, F is strictly monotone increasing,

F′(b) ≥ \frac{C_{2}(1 ∨ ρ)^{2}}{B(1 − (^{\frac{\sqrt{F (b)}−ρ\sqrt{π}}{\sqrt{F (b)}+ρ\sqrt{π}}} )_{+}^{2})} for a.a. 0 < b < b_{0},

F (0) = 0, F (b_{0}) = |D|, Θ(b_{0}) = 0}.

As in [4], if we set h^{∗}(r) = F^{−1}(πr^{2}), a(r) = (h^{∗})′(r) and q(r) = Θ(h^{∗}(r)) exp(−h^{∗}(r)), then q, q′ ∈ L^{2}((0, L), rdr) and q(L) = 0. (25) and (26) in [4] also hold:

2π ∫_{0}^{L}(q′(r) + a(r)q(r))^{2}rdr ≤ B ∫ (Θ′(b))^{2}e^{−2b}F (b)db

and

2π ∫_{0}^{L} q(r)^{2}rdr = ∫ (Θ(b))^{2}e^{−2b}F′(b)db.

(24) in [4] is rewritten as

0 ≤ a(r) = (h^{∗})′(r) = \frac{2πr}{F′(F^{−1}(πr^{2}))} ≤ C_{3}(1 ∨ ρ)^{2}Br(1 − (\frac{r − ρ}{r + ρ})_{+}^{2}) =: a_{ρ}(r).

Therefore we have

\begin{aligned}λ_{1}(H_{D}) ≥ & inf \{2π ∫_{0}^{L}(q′(r) + a(r)q(r))^{2}rdr(2π ∫_{0}^{L} q(r)^{2}rdr)^{−1} \\ & : q, q′ ∈ L^{2}((0, L), rdr), q(L) = 0, q(r) ≥ 0, 0 ≤ a(r) ≤ a_{ρ}(r)\}. \end{aligned}

By Lemma 3.1 in [4], we have

\begin{aligned}λ_{1}(H_{D}) ≥ & inf \{2π ∫_{0}^{L}(q′(r) + a_{ρ}(r)q(r))^{2}rdr(2π ∫_{0}^{L} q(r)^{2}rdr)^{−1} \\ & : q, q′ ∈ L^{2}((0, L), rdr), q(L) = 0, q(r) ≥ 0\}. \end{aligned} ; (3.6)

We rewrite q as q(r) = exp(−A_{ρ}(r))Q(r), where

A_{ρ}(r) = ∫_{0}^{r} a_{ρ}(s)ds.

Since A_{ρ}(r) is monotone increasing, by the uniform estimate, we have

λ_{1}(H_{D}) ≥ inf {2π ∫_{0}^{L} Q′(r)^{2} exp(−2A_{ρ}(r))rdr/2π ∫_{0}^{L} Q(r)^{2} exp(−2A_{ρ}(r))rdr
: Q, Q′ ∈ L^{2}((0, L), rdr), Q(L) = 0, Q(r) ≥ 0}
≥λ_{1}(−Δ_{B(L)}) exp(−2A_{ρ}(L)),

where λ_{1}(−Δ_{B(L)}) is the lowest eigenvalue of the Dirichlet Laplacian of B(L). By the scaling, we have λ_{1}(−Δ_{B(L)}) = λ_{1}(−Δ_{B(1)})/L^{2}. By a simple calculation, we have

(3.7)  A_{ρ}(r) ≤ C_{4}B(1 ∨ ρ)^{3}L.

Thus we obtain (3.5). □ The following is the estimate used in the proof of the last proposition:

Lemma 3.2. Suppose that the domain D is defined by B(R) \ \overline{⋃_{i∈ℕ} B(a_{i} : r_{0})} for some 0 < r_{0} < R < ∞ and {a_{i} : i ∈ ℕ} ⊂ ℝ^{2} and that ρ is the radius of the largest open disk contained in D. Then for any domain D contained in D such that its boundary ∂D is a finite union of rectifiable Jordan curves, we have

(3.8)  (1 + cρ)^{2}(1 − (\frac{\sqrt{L(∂D)} − ρ\sqrt{π}}{\sqrt{L(∂D)} + ρ\sqrt{π}} )^{2}) ≥ \frac{4π|D|}{L(∂D)}^{2},

where c is a finite constant depending only on r_{0}.

Proof. This estimate has its origin in the classical isoperimetric inequality stating L(∂D)^{2} ≥ 4π|D| for any domain D in ℝ^{2} bounded by a finite union of rectifiable Jordan curves. This inequality has been improved for more complicated domains. Many such inequalities are known as Bonnesen type isoperimetric inequalities (see Ossermann [16] and references therein). Among them we apply the following inequality by Ossermann: if D is a domain of ℝ^{2} bounded by a rectifiable Jordan curve, then we have

(3.9)  L(∂D)^{2} − 4π|D| ≥ L(∂D)^{2}(\frac{R − ρ}{R + ρ})^{2},

where R is the radius of the smallest disc including D and ρ is the radius of the largest disc included in D ((23) in Ossermann [16]). To apply (3.9), D should be simply connected. Now for any domain D in D = B(R) \ \overline{⋃_{i∈ℕ} B(a_{i} : r_{0})}, we classify its holes to two groups: let \{H_{k}\}_{k=1}^{K} and \{Ĥ_{i}\}_{i∈I} be simply connected closed domains such that

H_{k} ∩ \overline{⋃_{i∈ℕ} B(a_{i} : r_{0})} ≠ ∅, Ĥ_{i} ∩ \overline{⋃_{i∈ℕ} B(a_{i} : r_{0})} = ∅

and D + ⋃_{k=1}^{K} H_{k} + ⋃_{i∈I} Ĥ_{i} is a simply connected domain. Therefore {H_{k}}_{k=1}^{K} are holes intersecting the holes of D and {Ĥ_{i}}_{i∈I} are holes apart from the holes of D. We may erase the holes {Ĥ_{i}}_{i∈I} to replace D by the domain D̂ := D + ⋃_{i∈I} Ĥ_{i}, since the radius of the largest disc included in D̂ is still ρ. For the holes {H_{k}}_{k=1}^{K}, we have K ≤ L(∂D)/(2πr_{0}), since L(∂D) ≥ 2πr_{0}. By erasing K numbers of line segments {C_{k}}_{k=1}^{K} of the length L(C_{k}) less than or equal to 2ρ, the domain D̂ becomes a disjoint union of finite number of simply connected domains {D_{j}}_{j=1}^{J}: D̂ \ ∑_{k=1}^{K} C_{k} = ∑_{j=1}^{J} D_{j}. Then Ossermann’s inequality (3.9) gives

L(∂D_{j}) ≥ \sqrt{4π|D_{j}|/(1 − (\frac{\sqrt{|D_{j}|} − ρ\sqrt{π}}{\sqrt{|D_{j}|} − ρ\sqrt{π}} ))}

by dominating also the radius of the smallest disc including D_{j} by \sqrt{|D_{j}|/π}. By using Lemma 3.3 below, we have

L(∂D̂) + 2 ∑_{k=1}^{K} L(c_{k}) ≥ \sqrt{4π|D̂|/(1 − (\frac{\sqrt{|D̂|} − ρ\sqrt{π}}{\sqrt{|D̂|} − ρ\sqrt{π}} ))},

from which we easily obtain (3.8). □

Lemma 3.3. Let H(F_{1}, ⋯, F_{n}) be a function of F_{1}, …, F_{n} ≥ 0 defined by

H(F_{1}, ⋯, F_{n}) = ∑_{i=1}^{n} \{\frac{F_{i}}{1 − (\frac{\sqrt{F_{i}}−β}{\sqrt{F_{i}}+β} )_{+}^{2}} \}^{1/2}

with β ∈ (0, ∞). Then we have

H(F_{1}, ⋯, F_{n}) ≥ H( ∑_{i=1}^{n} F_{i}, 0, ⋯, 0).

Proof. Let H_{1}(f) := H(F_{1} + f, F_{2} − f, F_{3}, ⋯, F_{n}). This function is increasing in small f ≥ 0. By using also the symmetry in (F_{1}, ⋯, F_{n}), we can complete the proof. □

Finally we complete the proof of Proposition 3.1. Proof of Proposition 3.1. We have only to show

\overline{lim}_{t↑∞} \frac{1}{(log t)^{(2+θ)/6}} log ∫_{Λ_{1}} dzE_{θ}[ exp (c_{1} \frac{ε_{1} − t}{s^{2}} exp(−c_{2}(B + β)ρ_{1}(z, s, ξ)^{3}s))]

≤ −K_{1}(θ)/B^{(2+θ)/6} (3.10)

for some β by (3.2), Lemma 3.1 and Proposition 3.2, where ρ_{1}(z, s, ξ) := 1 ∨ ρ(z, s, ξ), ρ(z, s, ξ) is the radius of the largest disk contained in B(z : s) \ T (ξ), c_{1}, c_{2} ∈ (0, ∞) and K_{1}(θ) ∈ (0, ∞) depends on θ. For this, we have only to show

(3.11)  \overline{lim}_{t↑∞} \frac{1}{(log t)^{(2+θ)/6}} log ∫_{Λ_{1}} dz ∫_{0}^{s^{−2}} dRe^{c_{3}(ε_{1}−t)R}P_{θ}(R̂ ≤ ρ_{1}(z, s, ξ))

is less than or equal to the right hand side of (3.10), where R̂ = {(log(R^{−1}s^{−2}))/(c_{2}(B + β)s)}^{1/3}. The qunatity in (3.11) is dominated from above by

(3.12)  − \underline{lim}_{t↑∞} \frac{1}{(log t)^{(2+θ)/6}} inf_{0<R≤s^{−2},z∈Λ_{1}}\{c_{3}(t − ε_{1})R − log P_{θ}(R̂ ≤ ρ_{1}(z, s, ξ))\}

The probability can be estimated as follows:

P_{θ}(R̂ ≤ ρ_{1}(z, s, ξ)) ≤ P_{θ}(B(a : R̂) ⊂ B(s) \ T (ξ) for some a ∈ B(s))

≤ ∑_{a∈B(s)∩(ε_{2}ℤ^{2})} P_{θ}(|ξ_{q}| ≥ R̂ + r_{0} − |q − a| − ε_{2} for ∀q ∈ B(a : R̂ − ε_{2}) ∩ ℤ^{2})
≤#{B(s) ∩ (ε_{2}ℤ^{2})} exp ( − (1 − ε_{3}) ∑_{q∈B(a:R̂−ε_{2})∩ℤ^{2}}(R̂ + r_{0} − |q − a| − ε_{2})_{+}^{θ} + c_{4}R̂^{2})
≤c_{5}s^{2} exp ( − (1 − ε_{3})\frac{2π(R̂ − 1 − ε_{2})_{+}^{θ+2}}{(θ + 1)(θ + 2)} + c_{4}R̂^{2}),

where ε_{2} and ε_{3} are arbitrarily small positive constants, and c_{4} and c_{5} are finite constants depending ε_{2} and ε_{3}. Thus the quantity (3.12) is less than or equal to

(3.13)  − \underline{lim}_{t↑∞} \frac{1}{(log t)^{(2+θ)/6}} inf_{0<R≤s^{−2}} \{c_{3}(t − ε_{1})R + c_{6}((\frac{1}{s} log \frac{1}{Rs^{2}} )^{1/3} − c_{7})_{+}^{θ+2}\},

where c_{6} = (1−2ε_{3})2π{(θ+1)(θ+2)(c_{2}(B+β))^{(θ+2)/3}}^{−1} and c_{7} = (1+ε_{2})(c_{2}(B+β))^{1/3}. For an arbitrary positive number v, a sufficient condition for ((log(R^{−1}s^{−2}))/s)^{1/3} −c_{7} ≥ vc_{7} is R ≤ exp(−s(v +1)^{3}c_{7}^{3})/s^{2}. Then we have

(\frac{1}{s} log \frac{1}{Rs^{2}} )^{1/3} − c_{7} ≥ \frac{v}{v + 1}(\frac{1}{s} log \frac{1}{Rs^{2}} )^{1/3}

and

inf \{c_{3}(t − ε_{1})R + c_{6}((\frac{1}{s} log \frac{1}{Rs^{2}} )^{1/3} − c_{7})_{+}^{θ+2} : 0 < R ≤ exp(−s(v + 1)^{3}c_{7}^{3})/s^{2}\}

≥ inf{F (R) : 0 < R ≤ exp(−s(v + 1)^{3}c_{7}^{3})/s^{2}},

where

F (R) = c_{3}(t − ε_{1})R + c_{8}(\frac{1}{s} log \frac{1}{Rs^{2}} )^{(θ+2)/3}

with c_{8} = c_{6}(v/(v + 1))^{θ+2}. The infimum of F (R) is attained at R = R(t) satisfying

R(t) = \frac{c_{8}(θ + 2)}{3c_{3}(t − ε_{1})s^{(θ+2)/3}} ( log \frac{1}{R(t)s^{2}} )^{(θ−1)/3}.

Since R(t) = (1 + o(1))(log t)^{(θ−4)/6}/t as t → ∞, R(t) ≤ exp(−s(v + 1)^{3}c_{7}^{3})/s^{2} and

inf{F (R) : 0 < R ≤ exp(−s(v + 1)^{3}c_{7}^{3})/s^{2}} = F (R(t))

for sufficiently large t. It is easy to see that

lim_{t→∞} \frac{F (R(t))}{(log t)^{(2+θ)/6}} = c_{8}(\frac{β}{8} )^{(2+θ)/6}.

On the other hand, by only the effect of the first term, we have

inf\{c_{3}(t − ε_{1})R + c_{6}((\frac{1}{s} log \frac{1}{Rs^{2}} )^{1/3} − c_{7})_{+}^{θ+2} : exp(−s(v + 1)^{3}c_{7}^{3})/s^{2} ≤ R ≤ s^{−2}\}

≥ c_{3} \frac{t − ε_{1}}{s^{2}} exp(−s(v + 1)^{3}c_{7}^{3}) ≥ c_{9}\sqrt{t}.

Thus the quantity (3.13) is less than or equal to −c_{8}(β/8)^{(2+θ)/6}. By taking the limit ε_{3} ↓ 0 and v → ∞, we see that the quantity (3.13) is less than or equal to

\frac{−2π}{(θ + 1)(θ + 2)}(\frac{β}{8c_{2}^{2}(B + β)^{2}} )^{(2+θ)/6}.

This becomes the optimal value, the right hand side of (3.10), when β = B. □

4. A general upper estimate for the 2-dimensional case

In this section we prove the following upper estimate which is enough for (1.7):

Proposition 4.1. If u(x) = C_{0}1_{B(r_{0})}(x) with C_{0}, r_{0} ∈ (0, ∞), then we have

(4.1)  \overline{lim}_{t↑∞} \frac{log Ñ(t)}{(log t)(log log t)^{n}} = −∞ for any n ∈ ℕ.

To prove this proposition, we modify Erdős’s upper estimate for the Poisson case [5]. For the estimate, he applied his method of enlargement of obstacles referring to Sznitman’s theory (cf. [19]). As in [5], we assume 0 < r_{0} < 1. We fix β and ε > 0 arbitrarily, and take s = \sqrt{(8/β) log t}, ℓ = 10\sqrt{(log t)/B} and b > 10r_{0} specified later. In his theory, the points {q + ξ_{q}} corresponding to the centers of obstacles are classified to two groups: ℓΛ_{1}(m) ∋ q + ξ_{q} is defined to be ”good” if

\begin{matrix}|(ℓΛ_{1}(m)) ∩ \overline{B(q + ξ_{q} : 10^{k+1}b)} ∩ ⋃_{q′+ξ_{q′} ∈ℓΛ_{1}(m)} \overline{B(q′ + ξ_{q′} : b)}| \\ ≥ \frac{ε}{9}|(ℓΛ_{1}(m)) ∩ \overline{B(q + ξ_{q} : 10^{k+1}b)}| \end{matrix}

for any k ∈ ℤ_{+} ∩ [0, (log(ℓ/(2b)))/(log 10)], and the other ℓΛ_{1}(m) ∋ q + ξ_{q} is defined to be ”bad”. Then bad obstacles B(q + ξ_{q} : r_{0}) are erased and good obstacles B(q + ξ_{q} : r_{0}) are enlarged to B(q + ξ_{q} : b). In his theory, b is fixed to be a constant. We now take b as an increasing function of t. A sufficient condition for his theory to be generalized to this setting is

(4.2)  lim_{t→∞} b^{2}(\frac{ℓ}{b})^{−k(ε,b)} = 0,

where k(ε, b) = (4 log 10)^{−1} log{(1 − c(b)p(ε))^{−1}}, c(b) = P (inf{t : r(t) = r_{1}/b} < inf{t : r(t) = 6}, r(t) is a 2-dimensional Bessel process starting at 2, r_{1} ∈ (0, 1) and p(ε) is a [0, 1]-valued function decaying as ε ↓ 0 (cf. Lemma 7.5 in [5]). Indeed the condition (4.2) is sufficient to obtain (7.23) in [5] from the last estimate in the proof of Lemma 7.5 (i) in [5]. By a result on 1-dimensional diffusion process, we can rewrite

c(b) = (log 3)/ log \frac{6b}{r_{1}}

(cf. Theorem VI-3.1 in [11], [12]). Then we see that b = (log log t)^{γ} satisfies (4.2) for any γ > 0. The rest of the proof is same.

5. Lower estimates for the 3-dimensional case

In this section we prove (1.17). We first assume (1.13) with α ∈ (0, ∞)^{2}. For R_{⊥} and R_{3} ∈ ℕ, we consider the event

{|q_{⊥}| ≤ 3R_{⊥} and |q_{3}| ≤ 3R_{3} ⇒ |q_{⊥} + ξ_{q,⊥}| ≥ 2R_{⊥} or |q_{3} + ξ_{q,3}| ≥ 2R_{3},

”|q_{⊥}| > 3R_{⊥} or |q_{3}| > 3R_{3}” and (\frac{|q_{⊥}|}{R_{⊥}})^{α_{⊥}} ≥ (\frac{|q_{3}|}{R_{3}})^{α_{3}} ⇒ |ξ_{q,⊥}| ≤ K_{⊥}|q_{⊥}|,

”|q_{⊥}| > 3R_{⊥} or |q_{3}| > 3R_{3}” and (\frac{|q_{⊥}|}{R_{⊥}})^{α_{⊥}} < (\frac{|q_{3}|}{R_{3}})^{α_{3}} ⇒ |ξ_{q,3}| ≤ K_{3}|q_{3}|\},

(5.1)

where K_{⊥} = (1 − 3^{−(1∧(α_{3}/α_{⊥}))})/2 and K_{3} = (1 − 3^{−(1∧(α_{⊥}/α_{3}))})/2. We have

(5.2)  V_{ξ}(x) ≤ c_{1}R_{⊥}^{2}R_{3}(R_{⊥}^{α_{⊥}} ∧ R_{3}^{α_{3}})^{−1}

for x ∈ Λ_{2R_{⊥}} × I(R_{3}) on this event. Here and in the following, I(R) = (−R, R) for any R > 0. To prove (5.2) we divide the summation as

\begin{aligned}V_{ξ}(x) = & ∑_{q∈ℤ^{3}∩(B(3R_{⊥})×I(3R_{3}))} u(x − q − ξ_{q}) \\ & + ∑_{q∈ℤ^{3}\(B(3R_{⊥})×I(3R_{3})):(|q_{⊥}|/R_{⊥})^{α_{⊥}} ≥(|q_{3}|/R_{3})^{α_{3}}} u(x − q − ξ_{q}) \\ & + ∑_{q∈ℤ^{3}\(B(3R_{⊥})×I(3R_{3})):(|q_{⊥}|/R_{⊥})^{α_{⊥}} <(|q_{3}|/R_{3})^{α_{3}}} u(x − q − ξ_{q}). \end{aligned} ; (5.3)

The first term in the right hand side is easily dominated from above by R_{⊥}^{2}R_{3}/(R_{⊥}^{α_{⊥}} ∧ R_{3}^{α_{3}}). Since ∥x∥α ; p̃ ≥ |x_{⊥}|^{α_{⊥}}, the second term is dominated by

∑_{q∈ℤ^{3}\(B(3R_{⊥})×I(3R_{3})):(|q_{⊥}|/R_{⊥})^{α_{⊥}} ≥(|q_{3}|/R_{3})^{α_{3}}} ((1 − K_{⊥})|q_{⊥}| − R_{⊥})^{−α_{⊥}}.

By taking the summaton with respect to q_{3} and replacing the summation by the integration, this is dominated by

∫_{q_{⊥}∈ℝ^{2}:|q_{⊥}|>3^{1∧(α_{3}/α_{⊥})}R_{⊥}} \frac{R_{3}(|q_{⊥}|/R_{⊥})^{α_{⊥}/α_{3}}}{((1 − K_{⊥})|q_{⊥}| − R_{⊥})^{α_{⊥}}} dq_{⊥}.

Since 1 > 2/α_{⊥} + 1/α_{3} and (1 − K_{⊥})3^{1∧(α_{3}/α_{⊥})} − 1 > 0, this is dominated by R_{3}R_{⊥}^{2−α_{⊥}}. Simirarly the third term in the right hand side of (5.3) is dominated by R_{⊥}^{2}R_{3}^{1−α_{3}}. Therefore we obtain (5.2).

The probability is estimated as

(5.4)  log P_{θ}( the event (5.1) occurs ) ≥ −c_{2}R_{⊥}^{2}R_{3}(R_{⊥}^{θ_{⊥}} ∨ R_{3}^{θ_{3}}).

Indeed we have

log P_{θ}( the event (5.1) occurs )

= ∑_{q∈ℤ^{3}∩(B(3R_{⊥})×I(3R_{3}))} log P_{θ}(|q_{⊥} + ξ_{q,⊥}| ≥ 2R_{⊥} or |q_{3} + ξ_{q,3}| ≥ 2R_{3})

+ ∑_{q∈ℤ^{3}\(B(3R_{⊥})×I(3R_{3})):(|q_{⊥}|/R_{⊥})^{α_{⊥}} ≥(|q_{3}|/R_{3})^{α_{3}}} log P_{θ}(|ξ_{q,⊥}| ≤ K_{⊥}|q_{⊥}|)

+ ∑_{q∈ℤ^{3}\(B(3R_{⊥})×I(3R_{3})):(|q_{⊥}|/R_{⊥})^{α_{⊥}} <(|q_{3}|/R_{3})^{α_{3}}} log P_{θ}(|ξ_{q,3}| ≤ K_{3}|q_{3}|).

(5.5)

The first term in the right hand side is dominated from below by

R_{⊥}^{2}R_{3} log P_{θ}(|ξ_{0,⊥}| ≥ 5R_{⊥} or |ξ_{0,3}| ≥ 5R_{3}).

Since the probability is rewritten as P_{θ}(|ξ_{0,⊥}| ≥ 5R_{⊥} or |ξ_{0,3}| ≥ 5R_{3})

=R_{⊥}^{2}R_{3} ∫_{|y_{⊥}|≥5 or |y_{3}|≥5} exp(−∥|R_{⊥}y_{⊥}|^{θ_{⊥}}, |R_{3}y_{3}|^{θ_{3}}∥_{p})\frac{dy}{Z(θ, p)}
=R_{⊥}^{2}R_{3}P_{θ}(|ξ_{0,⊥}| ≥ 5 or |ξ_{0,3}| ≥ 5)
× E[ exp(−(∥|R_{⊥}ξ_{0,⊥}|^{θ_{⊥}}, |R_{3}ξ_{0,3}|^{θ_{3}}∥_{p} − ∥ξ∥_{p}^{θ})||ξ_{0,⊥}| ≥ 5 or |ξ_{0,3}| ≥ 5],

this probability is dominated from below by exp(−c_{3}(R_{⊥}^{θ_{⊥}} ∨ R_{3}^{θ_{3}})). Therefore the first term in the right hand side of (5.5) is dominated from below by −R_{⊥}^{2}R_{3}(R_{⊥}^{θ_{⊥}} ∨R_{3}^{θ_{3}}). The second and the third terms in the right hand side (5.5) is dominated from below by −R_{3} exp(−c_{4}R_{⊥}^{θ_{⊥}}) and −R_{⊥}^{2} exp(−c_{5}R_{3}^{θ_{3}}), respectively, by using log(1 − X) ≥ −2X for 0 ≤ X ≤ 1/2. Therefore we obtain (5.4).

We now recall

N(λ) ≥ \frac{1}{8R_{⊥}^{2}R_{3}} P_{θ}(λ_{1}(H_{B(R_{⊥})}) + λ_{1}( − (\frac{d^{2}}{dx^{2}} )_{I(R_{3})})

+ ∫ dx|ψ_{R_{⊥}}(x_{⊥})|^{2}ϕ_{R_{3}}(x_{3})^{2}V_{ξ}(x) ≤ λ

and the event (5.1) occurs), (5.6)

where H_{B(R_{⊥})} and (d^{2}/(dx^{2}))_{I(R_{3})} are the restrictions to the disk B(R_{⊥}) and the interval I(R_{3}), respectively, of H and the d^{2}/(dx^{2}), respectively, by the Dirichlet boundary condition, and ψ_{R_{⊥}} and ϕ_{R_{3}} are the normalized eigenfunctions corresponding to H_{B(R_{⊥})} and (d^{2}/(dx^{2}))_{I(R_{3})}, respectively (cf. Theorem (5.25) in [18]). By Erdős’s bound

λ_{1}(H_{B(R_{⊥})}) ≤ exp ( − \frac{B}{2} R_{⊥}^{2}(1 − ε_{1})) and λ_{1}( − (\frac{d^{2}}{dx^{2}} )_{(−R_{3},R_{3})}) = (\frac{π}{2R_{3}} )^{2},

we have

log N( exp ( − \frac{B}{2} R_{⊥}^{2}(1 − ε_{1})) + (\frac{π}{2R_{3}} )^{2} + \frac{c_{1}R_{⊥}^{2}R_{3}}{R_{⊥}^{α_{⊥}} ∧ R_{3}^{α_{3}}} ) ≥ −c_{6}R_{⊥}^{2}R_{3}(R_{⊥}^{θ_{⊥}} ∨ R_{3}^{θ_{3}}),

where ε_{1} is an arbitrarily fixed positive constant.

By specifying as R_{⊥} = R_{3}^{β} with β ∈ (0, ∞), we have

log N(c_{7}/R_{3}^{2∧{(α_{⊥}−2)β−1}∧(α_{3}−1−2β)}) ≥ −c_{8}R_{3}^{2β+1+(βθ_{⊥})∨θ_{3}}

and log N(λ) ≥ −c_{9}/λ^{f(β;α,θ)}, where

f(β; α, θ) = \frac{2β + 1 + (βθ_{⊥}) ∨ θ_{3}}{2 ∧ \{(α_{⊥} − 2)β − 1\} ∧ (α_{3} − 1 − 2β)}.

The function f(β; α, θ) attains μ_{1}(α_{⊥}, θ) at β = 3/(α_{⊥} − 2). Therefore we obatin (1.17).

For the operator H_{ξ}, we consider a simpler event, (5.1) with α_{⊥} = α_{3}. On this event we have T (ξ) ∩ (Λ_{2R_{⊥}} × I(R_{3})) = ∅ for large R_{⊥} and R_{3}. Thus we have

log N ( exp ( − \frac{B}{2} R_{⊥}^{2}(1 − ε_{1})) + (\frac{π}{2R_{3}} )^{2}) ≥ −c_{6}R_{⊥}^{2}R_{3}(R_{⊥}^{θ_{⊥}} ∨ R_{3}^{θ_{3}}).

By taking R_{⊥} = 2\sqrt{(log R_{3})/(B(1 − ε_{1}))} so that exp(−BR_{⊥}^{2}(1 − ε_{1})/2) ≤ R_{3}^{−2}, we have

log N (c_{7}/R_{3}^{2}) ≥ −c_{8}R_{3}^{1+θ_{3}} log R_{3}

and log N (λ) ≥ −c_{9}λ^{−(1+θ_{3})/2} log(1/λ).

6. Upper estimates for the 3-dimensional case

In this section we prove (1.18). We may assume that u is continuous since the essential condition is only (1.13). Following Proposition 5.11 in Warzel [21], we have

(6.1)  Ñ(t) ≤ \frac{B}{2π(1 − e^{−2Bt})} ∫_{Λ_{1}} dxE_{θ}[exp(−t(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·)))(x_{3}, x_{3})],

where exp(−t(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·)))(x_{3}, x′_{3}) is the integral kernel of the heat semigroup generated by the Schrödinger operator −∂_{3}^{2} + V_{ξ}(x_{⊥}, ·) on L^{2}(ℝ) and Λ_{R} := (−R/2, R/2)^{3} for any R > 0. By the Neumann-bracketing, the right hand side is dominated by

\frac{B}{2π(1 − e^{−2Bt})R_{3}} ∫_{Λ_{1}} dx_{⊥}E_{θ}[Tr[exp(−t(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N})]]

≤\frac{c_{1}}{R_{3}} ∫_{Λ_{1}} dx_{⊥}E_{θ}[exp(−(t − ε_{1})λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N})],

(6.2)

where (−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} is the restriction to the interval I(R_{3}/2) by the Neumann boundary condition and ε_{1} is an arbitrary small positive constant.

The eigenvalue is estimated as

\begin{matrix}λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} \\ ≥ & inf \{λ_{1}( − ∂_{3}^{2} + ∑_{j=1}^{M} u(x − b_{j}))_{I(R_{3}/2)}^{N} : b_{1}, …, b_{M} ∈ Λ_{R_{⊥}} × I(R_{3}/2)\} \\ ≥ & inf \{λ_{1}( − \frac{d^{2}}{dt^{2}} + \frac{c_{2}M}{∥R_{⊥}^{α_{⊥}}, |t − b|^{α_{3}}∥p̃})_{I(R_{3}/2)}^{N} : b ∈ I(R_{3}/2)\}, \end{matrix} ; (6.3)

where M = #{b ∈ ℤ^{3} ∩ (Λ_{R_{⊥}} × I(R_{3}/2)) : b + ξ_{b} ∈ Λ_{R_{⊥}} × I(R_{3}/2)}. We take small positive numbers ε_{2}, ε_{3} and ε_{⊥}. Then

(6.4)  ε_{2}R_{⊥}^{2}R_{3} ≥ #{b ∈ ℤ^{3} ∩ (Λ_{R_{⊥}} × I(R_{3}/2)) : |ξ_{b,⊥}|_{∞} > ε_{⊥}R_{⊥}/2 or |ξ_{b,3}| > ε_{3}R_{3}/2}

implies

(6.5)  M > {(1 − ε_{⊥})^{2}(1 − ε_{3}) − ε_{2}}R_{⊥}^{2}R_{3}.

Indeed since M ≥ #{b ∈ ℤ^{3} ∩ (Λ_{(1−ε_{⊥})R_{⊥}} × I((1 − ε_{3})R_{3}/2)) : |ξ_{b,⊥}|_{∞} ≤ ε_{⊥}R_{⊥}/2 and |ξ_{b,3}| ≤ ε_{3}R_{3}/2}, the right hand side of (6.4) is less than or equal to

\begin{matrix}#\{b ∈ ℤ^{3} ∩ (Λ_{(1−ε_{⊥})R_{⊥}} × I((1 − ε_{3})R_{3}/2)) : |ξ_{b,⊥}|_{∞} > ε_{⊥}R_{⊥}/2 or |ξ_{b,3}| > ε_{3}R_{3}/2\} \\ > (1 − ε_{⊥})^{2}(1 − ε_{3})R_{⊥}^{2}R_{3} − M. \end{matrix}

(6.5) implies the right hand side of (6.3) is greater than or equal to

inf \{λ_{1}( − \frac{d^{2}}{dt^{2}} + \frac{c_{3}R_{⊥}^{2}R_{3}}{∥R_{⊥}^{α_{⊥}}, |t − b|^{α_{3}}∥p̃})_{I(R_{3}/2)}^{N} : b ∈ I(R_{3}/2)\}.

This equals

(6.6)  \frac{1}{h^{2}} inf \{λ_{1}( − \frac{d^{2}}{dt^{2}} + \frac{c_{3}R_{⊥}^{2}R_{3}}{h^{α_{3}−2}∥R_{⊥}^{α_{⊥}}/h^{α_{3}}, |t − b|^{α_{3}}∥p̃})_{I(R_{3}/(2h))}^{N} : b ∈ I(R_{3}/(2h))\}

for any h ∈ (0, ∞) by changing the variables. Referring λ_{1}(−d^{2}/(dt^{2}) + 1_{[0,1]}/R)_{[0,R]}^{N} ≥ c_{4}/R^{2} for any R ≥ 2 and the condition (1.14), we let h = (R_{⊥}R_{3})^{2/(α_{3}−1)} and

(6.7)  c_{5} ≤ R_{⊥}^{α_{⊥}(α_{3}−1)/(2α_{3})−1} ≤ R_{3}

so that h^{α_{3}−2}/(R_{⊥}^{2}R_{3}) = R_{3}/h ≥ 2 and R_{⊥}^{α_{⊥}}/h^{α_{3}} ≤ 1. Thus the quantity in (6.6) is dominated from below by R_{3}^{−2} and we obtain λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} ≥ c_{6}/R_{3}^{2}. Therefore we have

\begin{matrix}P_{θ}(λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} < c_{6}/R_{3}^{2}) \\ ≤ & P_{θ}(#\{b ∈ ℤ^{3} ∩ (Λ_{R_{⊥}} × I(R_{3}/2)) : |ξ_{b,⊥}|_{∞} > ε_{⊥}R_{⊥}/2 or |ξ_{b,3}| > ε_{3}R_{3}/2\} > ε_{2}R_{⊥}^{2}R_{3}). \end{matrix}

Since the event in the probability on the right hand side implies

∑_{b∈ℤ^{3}∩(Λ_{R_{⊥}} ×I(R_{3}/2))} ∥ξ_{b}∥_{p}^{θ} ≥ ε_{2}R_{⊥}^{2}R_{3}\{(\frac{ε_{⊥}R_{⊥}}{2} )^{θ_{⊥}} ∧ (\frac{ε_{3}R_{3}}{2} )^{θ_{3}}\},

we have

(6.8)  P_{θ}(λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} < c_{6}/R_{3}^{2}) ≤ exp(−c_{7}R_{⊥}^{2}R_{3}(R_{⊥}^{θ_{⊥}} ∧ R_{3}^{θ_{3}})).

We take β ∈ (0, ∞) to specify as R_{⊥} = R_{3}^{β}. Then the condition (6.7) becomes β ≤ (2/α_{⊥})/(1 − 2/α_{⊥} − 1/α_{3}) =: \overline{β} and the right hand side of (6.8) becomes exp(−c_{7}R_{3}^{2f(β;θ)}), where f(β; θ) = β +1/2+((βθ_{⊥})∧ θ_{3})/2. The function f(β; θ) attains its maximum μ_{2}(α, θ) defined in (1.16) at β = \overline{β}. Thus our optimal estimate is

P_{θ}(λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(R_{3}/2)}^{N} < c_{6}/R_{3}^{2}) ≤ exp(−c_{7}R_{3}^{2μ_{2}(α,θ)}).

By (6.2), we have

Ñ(t) ≤ \frac{c_{1}}{R_{3}} exp(−c_{7}R_{3}^{2μ_{2}(α,θ)}) + \frac{c_{1}}{R_{3}} exp ( − \frac{(t − ε_{1})c_{6}}{R_{3}^{2}} ).

Since

Ñ(t) = t ∫_{0}^{∞} dλe^{−tλ}N(λ) ≥ t ∫_{η/R_{3}^{2}}^{∞} dλe^{−tλ}N(\frac{η}{R_{3}^{2}} ) = exp ( − t\frac{η}{R_{3}^{2}} )N(\frac{η}{R_{3}^{2}} ),

we have N(η/R_{3}^{2}) ≤ exp(−c_{8}R_{3}^{2μ_{2}(α,θ)}) for large R_{3} by taking η = c_{6}/2 and t = c_{7}R_{3}^{2+2μ_{2}(α,θ)}/(2η). By taking λ = η/R_{3}^{2}, we obtain (1.18).

For N and the case that supp u is compact, we may assume u = C_{0}1_{\overline{B(r_{⊥})}×\overline{I(r_{3})}} with 0 < r_{⊥}, r_{3} < 1/2. Then we apply a standard Brownian estimate to reduce to an estimate of the eigenvalue of the operator with the Dirichlet boundary condition:

Ñ(t) ≤\frac{B}{2π(1 − e^{−2Bt})} ∫_{Λ_{1}} dx_{⊥}E_{θ}[Tr[exp(−t(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(t)}]] + e^{−c_{0}t}
≤c_{1} ∫_{Λ_{1}} dx_{⊥}E_{θ}[exp(−(t − ε_{1})λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(t)})] + e^{−c_{0}t}

(cf. [12] Section 1.7). Then we use Theorem 3.1 in the page 123 in [19] to have

λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(t)} ≥ π^{2}/(sup_{k} |I_{k}| + c_{2})^{2},

where {I_{k}}_{k} are the random open disjoint intervals such that

∑_{k} I_{k} = I(t) \ ⋃_{q∈ℤ^{3}:q_{⊥}+ξ_{q,⊥}∈\overline{B(x_{⊥}:r_{⊥})}}[q_{3} + ξ_{q,3} − r_{3}, q_{3} + ξ_{q,3} + r_{3}]

and |I_{k}| is the length of I_{k}. If sup_{k} |I_{k}| ≥ s, then there exists p ∈ ℤ ∩ I(t) such that

[p, p + s − 2] ∩ ⋃_{q∈ℤ^{3}:q_{⊥}+ξ_{q,⊥}∈\overline{B(x_{⊥}:r_{⊥})}}[q_{3} + ξ_{q,3} − r_{3}, q_{3} + ξ_{q,3} + r_{3}] = ∅.

Then we have (0, 0, q) + ξ_{(0,0,q)} ∉ \overline{B(x_{⊥} : r_{⊥})} × [p − r_{3}, p + s − 2 + r_{3}] for any q ∈ [p − r_{3}, p + s − 2 + r_{3}] ∩ ℤ. Thus the probability of this event is estimated as

P_{θ}(sup_{k} |I_{k}| ≥ s) ≤ ∑_{p∈ℤ∩I(t)} ∏ P_{θ}(ξ_{(0,0,q_{3}),⊥} ∉_{q∈[p−r_{3}+1,p+s−3+r_{3}]∩ℤ} \overline{B(x_{⊥} : r_{⊥})} or |ξ_{(0,0,q_{3}),3}| > 1)

≤ 2t exp(−c_{3}(s + 2r_{3} − 4))

if s ≥ 4. Therefore we have

P_{θ}(λ_{1}(−∂_{3}^{2} + V_{ξ}(x_{⊥}, ·))_{I(t)} ≥ π^{2}/(s + c_{2})^{2}) ≤ 2t exp(−c_{3}(s + 2r_{3} − 4))

for s ≥ 4. The rest of the proof is same.

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