LIFSCHITZ TAIL OF A RANDOM PAULI HAMILTONIAN WITH AN ANOMALOUS MAGNETIC MOMENT

Naomasa UEKI^{∗}

(Received 2001-2-2) (Revised 2001-10-22) (from Osaka Journal of Mathematics)

1. Introduction Let A = (A_{ω}^{j}(x))_{j=1}^{2} (ω ∈ Ω, x ∈ ℝ^{2}) be an ℝ^{2}-valued random field such that

(i) the sample path ℝ^{2} ∋ x ↦ A_{ω}(x) = (A_{ω}^{j}(x))_{j=1}^{2} is differentiable; (ii) B_{ω}(x) := ∂_{1}A_{ω}^{2}(x) − ∂_{2}A_{ω}^{1}(x) is a Gaussian random field; (iii) B_{ω}(x) is stationary and ergodic with respect to the shift in the variable x,

where ∂_{j} = ∂/(∂x^{j}). For each ω, regarding x ↦ A_{ω}(x) and x ↦ B_{ω}(x) as a vector potential and the magnetic field, respectively, we consider a Pauli Hamiltonian defined by

H_{ω}^{g,(±)} = ∑_{j=1}^{2}(i∂_{j} + A_{ω}^{j}(x))^{2} ± \frac{g}{2}B_{ω}(x) (i = \sqrt{−1}), (1.1)

where g > 2 is a fixed constant called the anomalous magnetic moment (cf. [2]). In this paper we show the following:

∗This work was supported by Grant-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan.

Running head: Lifschitz Tail of aRandom Pauli Hamiltonian

Theorem 1.1. The integrated density of states N^{g,(±)}(λ) of the random operator H_{ω}^{g,(±)} satisfies

lim_{λ↓−∞} \frac{1}{λ^{2}} log N^{g,(±)}(λ) = \frac{−2}{(g − 2)^{2}β(0)}, (1.2)

where β(x) = E[B(x)B(0)] − E[B(0)]^{2} is the covariance function.

Moreover we generalize this theorem to the case where the dimension is arbitrary and the conditions are assumed only on the magnetic field. The main theorem is Theorem 2.1 below.

The first related result was obtained by Pastur [22], Nakao [21] and Kirsch and Martinelli [15]: they showed that the integrated density of states N_{0}(λ) of the random Schrödinger operator − ∑_{j=1}^{d} ∂_{j}^{2} + V_{ω}(x) satisfies

lim_{λ↓−∞} \frac{1}{λ^{2}} log N_{0}(λ) = \frac{−1}{8γ(0)}, (1.3)

where V_{ω}(x) is the Gaussian random field which is stationary and ergodic in x ∈ ℝ^{d}, and γ(x) is its covariance function. This result was extended by Broderix, Heldt and Leschke [3] (see also Broderix, Hundertmark and Leschke [4]) to the operator ∑_{j=1}^{d}(i∂_{j} +A_{0}^{j}(x))^{2}+ V_{ω}(x), where A_{0} = (A_{0}^{j}(x))_{1≤j≤d} is a vector potential corresponding to a deterministic constant magnetic field. Moreover this result was extended by Matsumoto [18] and the author [26] to the operator ∑_{j=1}^{d}(i∂_{j} + A_{ω}^{j}(x))^{2} + V_{ω}(x) with a random magnetic field. In [18] and [26], A = (A_{ω}^{j}(x))_{1≤j≤d} and V_{ω}(x) were assumed to be statistically independent. Instead of this condition, the following condition on the uniform boundedness of the spectrum was also considered in [26]: there exists λ_{δ} for sufficiently small δ > 0 such that

P ( inf spec ∑_{j=1}^{d}(i∂_{j} + A_{ω}^{j}(x))_{{|x|<δ}}^{2} < λ_{δ}) = 1, (1.4)

where spec ∑_{j=1}^{d}(i∂_{j} +A_{ω}^{j}(x))_{{|x|<δ}}^{2} is the spectral set of the operator ∑_{j=1}^{d}(i∂_{j} +A_{ω}^{j}(x))^{2} restricted to {|x| < δ} by the Dirichlet boundary condition. This condition is satisfied, for example, when the magnetic field is bounded in ω and x. Under these conditions, the result (1.3) always holds and the leading term is determined only by the scalar potential V_{ω}(x). Such an asymptotics is called the classical asymptotics and is distinguished from the quantum asymptotics where the kinetic part also affects the asymptotics (cf. [23]).

Now the Pauli Hamiltonian H_{ω}^{g,(±)} is a typical operator for which the kinetic part and the scalar potential are closely related. Theorem 1.1 shows that the leading term is determined only by the difference ±{(g − 2)/2}B_{ω}(x) between H_{ω}^{g,(±)} and the Pauli Hamiltonian

H_{ω}^{2,(±)} = ∑_{j=1}^{2}(i∂_{j} + A_{ω}^{j}(x))^{2} ± B_{ω}(x)

with the nonanomalous magnetic moment g = 2. We prove this asymptotics by modifying the methods for the case (1.4) in [26] so that the Hamiltonian H_{ω}^{2,(±)} takes the place of the operator ∑_{j=1}^{d}(i∂_{j} + A_{ω}^{j}(x))^{2} in [26]. The key point is that the infimum of the spectrum of the Hamiltonian H_{ω}^{2,(±)} is nonnegative and small (cf. Lemma 4.1 below). Therefore the negative spectrum of H_{ω}^{g,(±)} appears only when g > 2, and its distribution is determined mainly by the effect from the part ±{(g − 2)/2}B_{ω}(x) in the Hamiltonian. In the proof we use only the techniques for the classical asymptotics.

For the problem on the Lifschitz tail, the case of 0 < g ≤ 2 seems to be more complicated. For the nonanomalous magnetic moment, g = 2, Casher and Neuberger [8] showed

N^{2,(±)}(λ) ∼ c\sqrt{λ} as λ ↓ 0

in the case where B_{ω} is a white noise, and Comtet, Georges and Le Doussal [9] represented N^{2,(±)}(λ) in terms of Bessel functions in the case where A_{ω}^{1}(x) is a white noise depending only on x^{1} and A_{ω}^{2} = 0, by physical arguments. In these cases, N^{2,(±)}(λ) decays slower than that for the free Hamiltonian − ∑_{j=1}^{2} ∂_{j}^{2}. This behavior is contrary to those of other random Schrödinger operators (cf.[7], [23]). To the knowledge of the author, there are no other results on the Lifschitz tail for both the cases g > 2, g = 2 and 0 < g < 2.

Our result is equivalent to a result on the large time asymptotics of a stochastic oscillatory integral. In fact, the Tauberian argument shows that the equation (1.2) is equivalent to

lim_{t↑∞} \frac{1}{t^{2}} log { \atop N^{g,(±)}}(t) = \frac{(g − 2)^{2}}{8} β(0), (1.5)

where

{ \atop N^{g,(±)}}(t) := ∫_{−∞}^{∞} e^{−tλ}dN^{g,(±)}(λ)

is the Laplace-Stieltjes transform of N^{g,(±)}(λ). This function { \atop N^{g,(±)}}(t) is represented by the following oscillatory integral:

{ \atop N^{g,(±)}}(t) = \frac{1}{4πt}E^{(ω,w)}[ exp ( − i ∑_{j=1}^{2} ∫_{0}^{2t} A_{ω}^{j}(w(s)) ∘ dw^{j}(s)

∓\frac{g}{4} ∫_{0}^{2t} B_{ω}(w(s))ds)|w(2t) = 0], (1.6)

where E^{(ω,w)} is the expectation with respect to ω and a two-dimensional Wiener process w = (w^{1}(t), w^{2}(t)), independent of ω, starting at 0, and ∘dw^{j}(s) is the Stratonovich stochastic differential (cf. [7], [14], [23]). Now the formula (1.5) indicates that the divergence as t → ∞ by the term

exp ( ∓ \frac{g}{4} ∫_{0}^{2t} B_{ω}(w(s))ds)

under the expectation on the right hand side of (1.6) is partially canceled by the effect of the oscillation of the term

exp ( − i ∑_{j=1}^{2} ∫_{0}^{2t} A_{ω}^{j}(w(s)) ∘ dw^{j}(s))

and that the leading term is same with that of the following integral of a nonnegative functional:

\frac{1}{4πt}E^{(ω,w)}[ exp ( ∓ \frac{g − 2}{4} ∫_{0}^{2t} B_{ω}(w(s))ds)|w(2t) = 0]. (1.7)

(1.7) is the Laplace-Stieltjes transform of the integrated density of states of the operator − ∑_{j=1}^{2} ∂_{j}^{2} ± {(g − 2)/2}B_{ω}, which is the operator considered in [22], [21] and [15].

The organization of this paper is as follows: in Section 2 we state the main theorem, Theorem 2.1, in the general framework. The proof of this theorem is reduced to those of two propositions, Propositions 2.1 and 2.2. In Sections 3 and 4 we prove Proposition 2.1 and in Section 5 we prove Proposition 2.2. In Section 3 we also prove the existence of the integrated density of states (Lemma 3.1) and in Section 5 we give several examples where the operator norm of the covariance appearing in Theorem 2.1 and Proposition 2.2 can be written explicitly.

2. Main results In this section we state our results in the general framework. Let d ≥ 2 be an integer and set D = d(d − 1)/2. Let 𝔹 = (B_{ω}^{(j,k)}(x))_{1≤j<k≤d} (ω ∈ Ω, x ∈

ℝ^{d}) be an ℝ^{D}-valued Gaussian random field such that (i) 𝔹 is stationary and ergodic with respect to the shift in the variable x; (ii) the sample path x ↦ 𝔹_{ω}(x) is continuous; (iii) the corresponding form is closed, that is, d{∑_{j<k} B_{ω}^{(j,k)}(x)dx^{j} ∧ dx^{k}} = 0 in the

sense of distributions.

We denote the mean by M^{(j,k)} = E[B_{ω}^{(j,k)}(0)] and the covariance function by

β_{(ℓ,m)}^{(j,k)}(x) = E[(B_{ω}^{(j,k)}(x) − M^{(j,k)})(B_{ω}^{(ℓ,m)}(0) − M^{(ℓ,m)})]. (2.1)

We can give a vector potential A_{ω} = (A_{ω}^{j}(x))_{1≤j≤d} by

A_{ω}^{j}(x) := ∫_{0}^{1} ∑_{k=1}^{d} B_{ω,j}^{k}(tx)tx^{k}dt (2.2)

(cf. (2.13.2) in [6]). In fact A_{ω}(x) is continuous in x and satisfies

d{ ∑_{j=1}^{d} A_{ω}^{j}(x)dx^{j}} = ∑_{j<k} B_{ω}^{(j,k)}(x)dx^{j} ∧ dx^{k}

in the sense of distributions, where, for each vector 𝕏 = (X^{(j,k)})_{1≤j<k≤d} in ℝ^{D}, X = (X_{k}^{j})_{1≤j,k≤d} is the corresponding real skew-symmetric matrix given by the relation

X_{k}^{j} = \begin{cases}X^{(j,k)} if j < k, \\ −X^{(k,j)} if j > k, \\ 0 if j = k. \end{cases} (2.3)

We always denote the index for the row by the superscript and the index for the column by the subscript.

Let γ_{1}, γ_{2}, …, γ_{d} be Hermitian matrices acting on a finite dimensional complex Hilbert space V and satisfying the commutation relation

γ_{j}γ_{k} + γ_{k}γ_{j} = \begin{cases}2I if j = k, \\ O if j ≠ k, \end{cases}

where I and O are the identity and the zero matrices, respectively (see e.g. [10] §12.2). Then the random Pauli Hamiltonian with the magnetic moment g > 2 and the magnetic field B_{ω} is a formal operator

H_{ω} = I ∑_{j=1}^{d}(i∂_{j} + A_{ω}^{j}(x))^{2} + \frac{g}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω}^{(j,k)}(x) (2.4)

acting on V -valued functions.

Remark 2.1. The operator H_{ω} in (2.4) is defined as a self-adjoint operator on L^{2}(ℝ^{d} → V) if we can take a vector potential Ã_{ω} = (Ã_{ω}^{j}(x))_{1≤j≤d} so that Ã_{ω}^{j} ∈ L_{loc}^{4}(ℝ^{d}) and d^{∗}(∑_{j=1}^{d} Ã_{ω}^{j}dx^{j}) ∈ L_{loc}^{2}(ℝ^{d}), where d^{∗} is the formal adjoint of d. In this case, the operator H̃_{ω} obtained by replacing A_{ω}^{j} by Ã_{ω}^{j} in (2.4) is essentially self-adjoint on C_{0}^{∞}(ℝ^{d} → V). In fact, by the conditions (i), (ii) of the random field 𝔹, Fernique’s theorem [12] and Borel-Cantelli’s lemma, we have

|B_{ω}^{(j,k)}(x)| ≤ c_{ε,ω}(1 + |x|^{ε})

for any x ∈ ℝ^{d} and ε > 0. Then the self-adjointness is proven by the theory of Leinfelder and Simader [17]. To the knowledge of the author, the operator H̃_{ω} without the condition d^{∗}(∑_{j=1}^{d} Ã_{ω}^{j}dx^{j}) ∈ L_{loc}^{2}(ℝ^{d}) is not defined as a self-adjoint operator. In (3.2) below, we take vector potentials satisfying this condition only on bounded regions. It is enough for our purpose.

We will consider its integrated density of states N(λ), λ ∈ ℝ, defined in (2.6) below: for each R > 0, let Λ(R) := (−R/2, R/2)^{d} and H_{ω,R} be the self-adjoint operator on the space L^{2}(Λ(R) → V) with the Dirichlet boundary condition: H_{ω,R} is the self-adjoint operator corresponding to the closure of the quadratic form

q_{ω}(Φ, Ψ) := ∑_{j=1}^{d}⟨(i∂_{j} + A_{ω}^{j}(x))Φ(x), (i∂_{j} + A_{ω}^{j}(x))Ψ(x)⟩_{V} (2.5)

+⟨Φ(x), \frac{g}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω}^{(j,k)}(x)Ψ(x)⟩_{V}

with the domain C_{0}^{∞}(Λ(R) → V), where ⟨·, ·⟩_{V} is the Hermitian inner product of the space L^{2}(Λ(R) → V). Then the spectra of H_{ω,R} are purely discrete. Let N_{ω,R}(λ) be its counting function, that is the number of eigenvalues of H_{ω,R} not exceeding λ. Then, by the usual method by Kirsch-Martinelli for the usual random Schrödinger operator (cf. [7], [15]), we can show the existence of a deterministic increasing function N(λ) such that

R^{−d}N_{ω,R}(λ) ⟶ N(λ) as R → ∞ (2.6)

for any point of continuity of N(λ) and almost all ω. Moreover we can show the existence of the Laplace-Stieltjes transform of N(λ). For the details, see Lemma 3.1 below.

Remark 2.2. Recently the definition of the integrated density of states of the random Schrödinger operator with magnetic field was proven to be independent of the boundary condition by Doi, Iwatsuka and Mine [11], Nakamura [20] and Hupfer, Leschke, Müller and Warzel [13] under several conditions on the scalar potential and the magnetic field. This independence also holds in our case: if we define an operator Ĥ_{ω,R} with a Neumann boundary condition by the corresponding self-adjoint operator to the quadratic form q_{ω}(Φ, Ψ) with the domain

W^{1,2}(Λ(R) → V) := {Φ ∈ L^{2}(Λ(R) → V) : ∂_{j}Φ ∈ L^{2}(Λ(R) → V) for j = 1, 2, …, d}

and denote the corresponding counting function of the spectra by N̂_{ω,R}(λ), then we have

R^{−d}N̂_{ω,R}(λ) ⟶ N(λ) as R → ∞ (2.7) for any point of continuity of N(λ) and almost all ω. In fact the only restriction in [11] is that the scalar potential is bounded below and that in [13] is that the magnetic field is constant. By combining the methods in [11] and [13], we can show (2.7).

To describe the asymptotics of N(λ), we introduce a norm for any D × D matrix α = (α_{(ℓ,m)}^{(j,k)})_{1≤j<k≤d,1≤ℓ<m≤d}, by

∥α∥_{(1←2)} := max{∥α(𝕏)∥_{(1)} : 𝕏 ∈ ℝ^{D}, ∥𝕏∥_{(2)} = 1}, (2.8)

where, for any 𝕏 = (X^{(j,k)})_{1≤j<k≤d} ∈ ℝ^{D}, α(𝕏) := (∑_{ℓ<m} α_{(ℓ,m)}^{(j,k)}X^{(ℓ,m)})_{1≤j<k≤d} ∈ ℝ^{D}, and ∥𝕏∥_{(1)} and ∥𝕏∥_{(2)} are the trace norm ∥X∥_{1} := Tr[\sqrt{X^{∗}X}] and the Hilbert-Schmidt norm ∥X∥_{2} := \sqrt{Tr[X^{∗}X]}, respectively, of the corresponding skew-symmetric matrix X = (X_{k}^{j})_{1≤j,k≤d} (cf. [24]). The Hilbert-Schmidt norm ∥𝕏∥_{(2)} = ∥X∥_{2} in (2.8) can also be replaced by \sqrt{2}|𝕏| where |𝕏| = \sqrt{∑_{j<k}(X^{(j,k)})^{2}} is the Euclidean norm.

Then the main theorem is the following:

Theorem 2.1. Let N(λ) be the integrated density of states defined by (2.6) of the random Pauli Hamiltonian H_{ω} formally defined by (2.4). Then it holds that

lim_{λ↓−∞} \frac{1}{λ^{2}} log N(λ) = − (\frac{2}{(g − 2)∥\sqrt{β(0)}∥_{(1←2)}} )^{2} , (2.9)

where \sqrt{β(0)} is the non-negative square root of the non-negative symmetric matrix β(0) = (β_{(ℓ,m)}^{(j,k)}(0))_{1≤j<k≤d,1≤ℓ<m≤d} determined by the covariance functions (2.1).

This theorem is deduced from the following two propositions:

Proposition 2.1. Let Ñ(t) be the Laplace-Stieltjes transform of the integrated density of states N(λ) defined by (2.6) of the random Pauli Hamiltonian H_{ω} formally defined by (2.4). Then it holds that

(2.10)  lim_{t↑∞} \frac{1}{t^{2}} log Ñ(t) = lim_{t↑∞} \frac{1}{t^{2}} log E[ exp (t\frac{g − 2}{4}∥B(0) − M∥_{1})],

where M = E[B(0)].

Proposition 2.2. For a D-dimensional Gaussian random variable 𝔹 = (B_{ω}^{(j,k)})_{1≤j<k≤d} with mean 0 and covariance β = (β_{(ℓ,m)}^{(j,k)})_{1≤j<k≤d,1≤ℓ<m≤d}, we have

(2.11)  lim_{t↑∞} \frac{1}{t^{2}} log E[exp(t∥B∥_{1})] = ∥\sqrt{β}∥_{(1←2)}^{2},

where B = (B_{ω,k}^{j})_{1≤j<k≤d} is the skew-symmetric d × d matrix valued random variable identified with 𝔹 by (2.3).

We prove Proposition 2.1 in Sections 3 and 4, and prove Proposition 2.2 in Section 5.

3. Proof of Proposition 2.1 (I): Upper estimate

In this section we prove

(3.1)  \overline{lim}_{t↑∞} \frac{1}{t^{2}} log Ñ(t) ≤ \overline{lim}_{t↑∞} \frac{1}{t^{2}} log E[ exp (t\frac{g − 2}{4}∥B(0) − M∥_{1})].

We first show the existence of N(λ) and Ñ(t), and fundamental estimates for them. For this purpose, we will use stochastic oscillatory integrals as in (1.6). Then we should change the gauge, since d^{∗}(∑_{j} A_{ω}^{j}(x)dx^{j}) may not be a usual function of x, where d^{∗} is the formal adjoint of the exterior derivative. Now we define a vector potential A^{R} = (A_{ω}^{R,j}(x))_{1≤j≤d} by

∑_{j=1}^{d} A_{ω}^{R,j}(x)dx^{j} = d^{∗}{ ∑_{j<k} C_{ω}^{R,(j,k)}(x)dx^{j} ∧ dx^{k}} (3.2)

in the sense of distributions, where C_{ω}^{R,(j,k)}(x) is a C^{1} function defined by

C_{ω}^{R,(j,k)}(x) := ∫ k_{d}(x − y){dρ(y/R)A_{ω}(y)}^{(j,k)}dy,

\{dρ(y/R)A_{ω}(y)\}^{(j,k)} = B_{ω}^{(j,k)}(y)ρ(\frac{y}{R}) + \frac{1}{R}((∂_{j}ρ)(\frac{y}{R})A_{ω}^{k}(y) − (∂_{k}ρ)(\frac{y}{R})A_{ω}^{j}(y)),

ρ ∈ C_{0}^{∞}(Λ(2) → [0, 1]) such that ρ = 1 on Λ(1), and k_{d}(x) is the fundamental solution of (−1) times the Laplacian −Δ = − ∑_{j=1}^{d} ∂_{j}^{2}. The explicit representation of k_{d}(x) is

k_{d}(x) = \begin{cases}\frac{Γ(d/2)}{(d − 2)2π^{d/2}|x|^{d−2}} if d > 2, \\ \frac{1}{2π} log \frac{1}{|x|} if d = 2, \end{cases}

where Γ(·) is the Gamma function. Thus the vector potential A^{R} is written more explicitly as

A_{ω}^{R,j}(x) = \frac{−Γ(d/2)}{2π^{d/2}} ∫ ∑_{k≠j} \frac{x^{k} − y^{k}}{|x − y|^{d}} \{dρ(y/R)A_{ω}(y)\}^{(j,k)}dy.

Since the de Rham-Hodge-Kodaira Laplacian □ := d^{∗}d + dd^{∗} acts as

□{ψ(x)dx^{j} ∧ dx^{k}} = −Δψ(x)dx^{j} ∧ dx^{k}

for any ψ ∈ C_{0}^{∞}(ℝ^{d}) and

d{ ∑_{j<k} C_{ω}^{R,(j,k)}(x)dx^{j} ∧ dx^{k}} = 0,

we have

d\{ ∑_{j=1}^{d} A_{ω}^{R,j}(x)dx^{j}\} = d\{ρ(\frac{x}{R}) ∑_{j=1}^{d} A_{ω}^{j}(x)dx^{j}\}

in the sense of distributions. Then there exists a C^{1} function g_{ω}(x) on Λ(R) such that

dg_{ω}(x) = ∑_{j=1}^{d} A_{ω}^{j}(x)dx^{j} − ∑_{j=1}^{d} A_{ω}^{R,j}(x)dx^{j}.

Since ρ(x/R) = 1 on Λ(R), the operator H_{ω,R} is unitarily equivalent with the operator { \atop H_{ω,R}} by the unitary operator defined by exp(ig_{ω}(x)), where { \atop H_{ω,R}} is the operator obtained by replacing A_{ω}^{j} by A_{ω}^{R,j} in the definition of H_{ω,R}. For the operator { \atop H_{ω,R}}, we can use stochastic oscillatory integrals since d^{∗}(∑_{j} A_{ω}^{R,j}dx^{j}) = 0.

Let w = (w^{1}(t), w^{2}(t), …, w^{d}(t)) be a d-dimensional Wiener process starting at 0 and M_{x}(t) be the solution of the ordinary differential equation

\begin{cases}\frac{d}{dt}M_{x}(t) = M_{x}(t)Ξ_{x}(t), \\ M_{x}(0) = I \end{cases} (3.3)

on the space End(V) of the endomorphisms on V, where

Ξ_{x}(t) := −\frac{g}{4} ∑_{j<k} iγ_{j}γ_{k}B_{ω}^{(j,k)}(x + w(t)).

Then the integral kernel exp(−t{ \atop H_{ω,R}}/2)(x, y), (t, x, y) ∈ (0, ∞) × Λ(R) × Λ(R), of the heat semigroup exp(−t{ \atop H_{ω,R}}/2):

exp(−t{ \atop H_{ω,R}}/2)(x, y)

= E^{w}[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{R,j}(x + w(s))dw^{j}(s))M_{x}(t)

×χ{x + w(s) ∈ Λ(R) for any 0 ≤ s ≤ t}|x + w(t) = y] (3.4)

×\frac{1}{(2πt)^{d/2}} exp (−\frac{|x − y|^{2}}{2t} ) ,

where E^{w} is the expectation with respect to the d-dimensional Wiener process w, dw^{j}(s) is the Itô stochastic differential, and, for each Borel set A in the Wiener space, χ(A) is its indicator function (cf. [5], [14], [25]). From (3.3), we have

∥M_{x}(t)^{∗}∥_{2} ≤ ∥I∥_{2} + ∫_{0}^{t} ∥Ξ_{x}(s)^{∗}∥_{op}∥M_{x}(s)^{∗}∥_{2}ds,

where ∥ · ∥_{op} is the operator norm. By Gronwall’s lemma, we have

∥M_{x}(t)^{∗}∥_{2} ≤ ∥I∥_{2} exp (∫_{0}^{t} ∥Ξ_{x}(s)^{∗}∥_{op}ds) .

Thus we have

∥M_{x}(t)∥_{2} ≤ c′_{1} exp (c′_{2} ∫_{0}^{t} g ∑_{j<k} |B_{ω}^{(j,k)}(x + w(s))|ds)

and

E^{ω}[∥M_{x}(t)∥_{2}^{p}] ≤ c_{1}^{p} exp (c_{2}(pgt)^{2} ∑_{j<k} β_{(j,k)}^{(j,k)}(0)) (3.5)

for any p ≥ 1, where c′_{j}, c_{j} > 0 depend only on the dimension d and E^{ω} is the expectation with respect to ω. Since the equality d(∑_{j} A_{ω}^{R,j}dx^{j}) = ∑_{j<k} B_{ω}^{(j,k)}dx^{j} ∧ dx^{k} holds only on Λ(R), we do not give an expression such as (1.6). Instead of this, we give the estimate (3.6) below:

Lemma 3.1. There exists the integrated density of states N(λ) defined in (2.6) of the random Pauli Hamiltonian H_{ω} formally defined in (2.4). Its Laplace-Stieltjes transform Ñ(t) exists for any t > 0. This is estimated as

|Ñ(\frac{t}{2}) − Ñ_{R}(\frac{t}{2})| ≤ \frac{c_{1}}{t^{d/2}} exp (c_{2}t^{2} − \frac{R^{2}}{2t} ), (3.6)

for some c_{1}, c_{2} > 0 independent of t and R, where

Ñ_{R}(\frac{t}{2}) := \frac{1}{R^{d}} ∫_{Λ(R)} E[ Tr [ exp ( − \frac{t}{2}{ \atop H_{ω,2R}})(x, x)]]dx.

Proof. To prove not only the existence of N(λ) but also that of Ñ(t), we apply the method by Kirsch-Martinelli [15] to the Laplace-Stieltjes transform Ñ(t). Let I be the class of sets [a, b) of the form [a, b) := {x ∈ ℝ^{d} : a^{j} ≤ x^{j} < b^{j}, j = 1, 2, …, d}, where a = (a^{j})_{1≤j≤d}, b = (b^{j})_{1≤j≤d}. For each [a, b) ⊂ I, let H_{ω,[a,b)} be the self-adjoint operator on L^{2}([a, b) → V), corresponding to the closure of the quadratic form q_{ω} in (2.5) with the domain C_{0}^{∞}((a, b) → V), N_{ω,[a,b)}(λ) be the number of eigenvalues of H_{ω,[a,b)} not exceeding λ and Ñ_{ω,[a,b)}(t) be its Laplace-Stieltjes transform, where (a, b) := {x ∈ ℝ^{d} : a^{j} < x^{j} < b^{j}, j = 1, 2, …, d} for a = (a^{j})_{1≤j≤d} and b = (b^{j})_{1≤j≤d}. By the gauge invariance, we can show that the eigenvalues of the operator H_{ω,[a,b)} are determined only by the magnetic field B_{ω}. Then we can show that, for each t > 0, I ∋ [a, b) ↦ Ñ_{ω,[a,b)}(t) is a superadditive process:

(i) Ñ_{T_{k}ω,I}(t) = Ñ_{ω,I+k}(t) for any I ∈ I and k ∈ ℝ^{d};

(ii) if I_{1}, …, I_{n} are disjoint sets in I and if I = ⋃_{j=1}^{n} I_{j} is also in I then Ñ_{ω,I}(t) ≥ ∑_{j=1}^{n} Ñ_{ω,I_{j}}(t);

(iii) sup{|I|^{−1}E[Ñ_{ω,I}(t)] : I ∈ I, |I| > 0} < ∞, where T_{k} : Ω → Ω such that 𝔹_{T_{k}ω}(x) = 𝔹_{ω}(x + k), and |I| = ∏_{j=1}^{d}(b_{j} − a_{j}) for I = [a, b), a = (a^{j})_{1≤j≤d}, b = (b^{j})_{1≤j≤d}. Thus by the Akcoglu-Krengel superadditive ergodic theorem, R^{−d}Ñ_{ω,R}(t) converges to a number Ñ(t) depending only on t almost surely and in L^{1} (cf. [1], [16]). Since Ñ(t) is monotone decreasing in t, we can modify Ñ(t) so that R^{−d}Ñ_{ω,R}(t) converges to Ñ(t) for any point of continuity of Ñ(t) and almost all ω as in [7], [15].

We next prove (3.6). By the L^{1} convergence to Ñ(t), we have

Ñ(\frac{t}{2}) = lim_{Q→∞} \frac{1}{Q^{d}} E[Ñ_{ω,Q}(\frac{t}{2})].

We use Mercer’s expansion theorem, the formula (3.4), the stationarity of the random magnetic field and the gauge invariance to rewrite this as

Ñ(\frac{t}{2}) = lim_{Q→∞} \frac{1}{Q^{d}} ∫_{Λ(Q)} E[ Tr [ exp ( − \frac{t}{2}{ \atop H_{ω,Q}})(x, x)]]dx

= lim_{Q→∞} ∫_{Λ(1)} E[ Tr [ exp ( − \frac{t}{2}{ \atop H_{ω,Q}})(Qx, Qx)]]dx

= lim_{Q→∞} ∫_{Λ(1)} E^{(ω,w)}[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{2Q,j}(w(s))dw^{j}(s)) Tr[M_{0}(t)]

×χ\{Qx + w(s) ∈ Λ(Q) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{d/2}} dx.

Similarly we have

Ñ_{R}(\frac{t}{2}) = ∫_{Λ(1)} E^{(ω,w)}[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{4R,j}(w(s))dw^{j}(s)) Tr[M_{0}(t)]

×χ\{Rx + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}|w(t) = 0]\frac{1}{(2πt)^{d/2}} dx.

In this expression we can replace A_{ω}^{4R,j} by A_{ω}^{2Q,j} since dA_{ω}^{4R,j} = dA_{ω}^{2Q,j} on Λ(4R) for large enough Q. Thus we have

|Ñ(\frac{t}{2}) − Ñ_{R}(\frac{t}{2})|

= lim_{Q→∞} | ∫_{Λ(1)} E^{(ω,w)}[ exp ( − i ∑_{j=1}^{d} ∫_{0}^{t} A_{ω}^{2Q,j}(w(s))dw^{j}(s)) Tr[M_{0}(t)]

×{χ{Qx + w(s) ∈ Λ(Q) for any 0 ≤ s ≤ t}

−χ\{Rx + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t\}\}|w(t) = 0]\frac{1}{(2πt)^{d/2}} dx|.

Using (3.5), we see that the right hand side is dominated by

\frac{c_{1}}{t^{d/2}} exp(c_{2}t^{2}) \overline{lim}_{Q→∞} ∫_{Λ(1)} E[|χ\{Qx + w(s) ∈ Λ(Q) for any 0 ≤ s ≤ t\}

−χ{Rx + w(s) ∈ Λ(2R) for any 0 ≤ s ≤ t}||w(t) = 0]dx.

Since

lim_{Q→∞} χ{Qx + w(s) ∈ Λ(Q) for any 0 ≤ s ≤ t} = 1

for each x ∈ Λ(1 − ε), we have

|Ñ(\frac{t}{2}) − Ñ_{R}(\frac{t}{2})|

≤ \frac{c_{1}}{t^{d/2}} exp(c_{2}t^{2})\{ ∫_{Λ(1)−Λ(1−ε)} dx

+ ∫_{Λ(1−ε)} P (Rx + w(s) ∉ Λ(2R) for some 0 ≤ s ≤ t|w(t) = 0)dx}

and

|Ñ(\frac{t}{2}) − Ñ_{R}(\frac{t}{2})|

≤ \frac{c_{1}}{t^{d/2}} exp(c_{2}t^{2}) ∫_{Λ(1)} P (Rx + w(s) ∉ Λ(2R) for some 0 ≤ s ≤ t|w(t) = 0)dx.

From this we can obtain (3.6) by a standard argument on the Wiener process. □

We now proceed to prove (3.1). We take small δ > 0 and use the semigroup property as follows:

Tr [ exp ( − \frac{t}{2}{ \atop H_{ω,2R}})(x, x)]

≤ ∫_{Λ(2R)} dy∥ exp ( − \frac{δ}{2}{ \atop H_{ω,2R}})(y, x)∥_{2}^{2} exp ( − (\frac{t}{2} − δ) inf spec H_{ω,2R}),

where, for any self-adjoint operator A, spec A is its spectral set. We take another small ε > 0 and use Hölder’s inequality as follows:

E^{ω}[ Tr [ exp ( − \frac{t}{2}{ \atop H_{ω,2R}})(x, x)]]

≤ E^{ω}[( ∫_{Λ(2R)} dy∥ exp ( − \frac{δ}{2}{ \atop H_{ω,2R}})(y, x)∥_{2}^{2})^{(1+ε)/ε}]^{ε/(1+ε)}

×E^{ω}[ exp ( − (1 + ε)(\frac{t}{2} − δ) inf spec H_{ω,2R})]^{1/(1+ε)}.

By (3.5), we have

E^{ω}[( ∫_{Λ(2R)} dy∥ exp ( − \frac{δ}{2}{ \atop H_{ω,2R}})(y, x)∥_{2}^{2})^{(1+ε)/ε}]^{ε/(1+ε)}

≤ E^{ω}[ ∫_{Λ(2R)} dy∥ exp ( − \frac{δ}{2}{ \atop H_{ω,2R}})(y, x)∥_{2}^{2(1+ε)/ε}]^{ε/(1+ε)}(2R)^{d/(1+ε)}

≤ E^{ω}[ ∫_{Λ(2R)} dyE^{w}[∥M_{y}(δ)∥_{2}^{2(1+ε)/ε}|y + w(δ) = x]

× exp ( − \frac{1 + ε}{ε} \frac{|x − y|^{2}}{δ} )]^{ε/(1+ε)} \frac{(2R)^{d/(1+ε)}}{(2πδ)^{d}}

≤ c_{ε,δ} exp (c\frac{1 + ε}{ε}δ^{2})R^{d/(1+ε)},

where c_{ε,δ} > 0 depends only on d, ε and δ. Therefore we obtain

Ñ_{R}(\frac{t}{2}) ≤ c_{ε,δ} exp (c\frac{1 + ε}{ε}δ^{2})R^{d/(1+ε)}Ñ_{R,0}((1 + ε)(\frac{t}{2} − δ))^{1/(1+ε)}, (3.7)

where

Ñ_{R,0}(t) := E[exp(−t inf spec H_{ω,2R})].

By the estimate

q_{ω}(Φ, Φ) = ||| ∑_{j=1}^{d} γ_{j}(i∂_{j} + A_{ω}^{j}(x))Φ(x)|||_{V}^{2}

+⟨Φ(x), (\frac{g}{2} − 1) ∑_{j<k} iγ_{j}γ_{k}B_{ω}^{(j,k)}(x)Φ(x)⟩_{V}

≥ −\frac{g − 2}{4}⟨Φ(x), ∥B_{ω}(x)∥_{1}Φ(x)⟩_{V} ,

we obtain

inf spec H_{ω,2R} ≥ −\frac{g − 2}{4} sup_{x∈Λ(2R)} ∥B_{ω}(x)∥_{1},

where ||| · |||_{V} is the L^{2}-norm on L^{2}(ℝ^{d} → V) (cf. Lemma 5.1 below). For arbitrarily taken small η > 0, since

sup_{x∈Λ(2R)} ∥B_{ω}(x)∥_{1} ≤ sup_{ι∈Λ(2R)∩((η/2)ℤ)^{d}} sup_{x∈Λ_{ι}(η)} ∥B_{ω}(x)∥_{1},

we have

Ñ_{R,0}(t) ≤ ∑_{ι∈Λ(2R)∩((η/2)ℤ)^{d}} E[ exp (t\frac{g − 2}{4} sup_{x∈Λ_{ι}(η)} ∥B_{ω}(x)∥_{1})]

≤ c_{η,d}R^{d}E[ exp (t\frac{g − 2}{4} sup_{x∈Λ(η)} ∥B_{ω}(x)∥_{1})],

where Λ_{ι}(η) = Λ(η) + ι and c_{η,d} is a constant depending only on η and d. Now, setting B̊_{ω}(x) = E[B_{ω}(x)|B_{ω}(0)] − M and \overline{B}_{ω}(x) = B_{ω}(x) − B̊_{ω}(x) − M, we write B_{ω}(x) = M + B̊_{ω}(x) + \overline{B}_{ω}(x). (3.8) We note that B̊_{ω}(x) and \overline{B}_{ω}(x) are independent and that

B̊_{ω}^{(j,k)}(x) = ∑_{ℓ<m} ∑_{p<q} β_{(ℓ,m)}^{(j,k)}(x)(β^{−1})_{(p,q)}^{(ℓ,m)}(0)(B_{ω}^{(p,q)}(0) − M^{(p,q)}). (3.9)

By the independence we have

Ñ_{R,0}(t) ≤ c_{η,d}R^{d} exp (t\frac{g − 2}{4}∥M∥_{1})Ñ_{R,1}(\frac{g − 2}{4}t)Ñ_{R,2}(\frac{g − 2}{4}t), (3.10)

where

Ñ_{R,1}(t) := E[ exp (t sup_{x∈Λ(η)} ∥B̊_{ω}(x)∥_{1})]

and

Ñ_{R,2}(t) := E[ exp (t sup_{x∈Λ(η)} ∥\overline{B}_{ω}(x)∥_{1})].

By the expression (3.9), we can take a function ℓ(s) of s > 0, independent of ω, such that lim_{s↓0} ℓ(s) = 0, |B̊_{ω}^{(j,k)}(x) − (B_{ω}^{(j,k)}(0) − M^{(j,k)})| ≤ ℓ(|x|)∥B_{ω}(0) − M∥_{1} (3.11) and

sup_{x∈Λ(η)} ∥B̊_{ω}(x)∥_{1} ≤ ∥B_{ω}(0) − M∥_{1}(1 + ℓ(η)).

Thus we have Ñ_{R,1}(t) ≤ E[exp{(1 + ℓ(η))t∥B_{ω}(0) − M∥_{1}}]. (3.12)

On the other hand, by the continuity of the sample path x ↦ B(x), we have

lim_{η↓0} P ( sup_{x∈Λ(η)} ∥\overline{B}_{ω}(x)∥_{1} ≤ 1) = 1

Thus, by the theory of Fernique [12], we can take a function k(η) such that

P ( sup_{x∈Λ(η)} ∥\overline{B}_{ω}(x)∥_{1} ≥ μ) ≤ exp(−μ^{2}k(η))

for any μ ≥ 1 and lim_{η↓0} k(η) = ∞. Thus we have

Ñ_{R,2}(t) = 1 + ∫_{0}^{∞} te^{tμ}P ( sup_{x∈B(η)} ∥\overline{B}_{ω}(x)∥_{1} ≥ μ)dμ

≤ e^{t} + \sqrt{\frac{π}{k(η)}}t exp (\frac{t^{2}}{4k(η)}). (3.13)

We now combine (3.6), (3.7), (3.10), (3.12) and (3.13), and put R = t^{2}. Then we obtain

\overline{lim}_{t↑∞} \frac{1}{(t/2)^{2}} log Ñ(\frac{t}{2})

≤ \overline{lim}_{t↑∞} \frac{1}{t^{2}} log E[ exp (t\frac{g − 2}{4}∥B_{ω}(0) − M∥_{1})](1 + ε)(1 + r(η))^{2}

+\frac{1 + ε}{4k(η)}(\frac{g − 2}{4})^{2}.

Since η and ε are arbitrary, we obtain (3.1).

4. Proof of Proposition 2.1 (II): Lower estimate

In this section we prove

(4.1)  \underline{lim}_{t↑∞} \frac{1}{t^{2}} log Ñ(t) ≥ \underline{lim}_{t↑∞} \frac{1}{t^{2}} log E[ exp (t\frac{g − 2}{4}∥B(0) − M∥_{1})].

By the Akcoglu-Krengel superadditive ergodic theorem (cf. [1], [7], [15], [16]), we have

Ñ(t) ≥ E[Ñ_{ω,1}(t)] = E[Tr[exp(−tH_{ω,1})]] ≥ E[exp(−t inf spec H_{ω,1})]
≥ E[exp(−t inf spec H_{ω,η})]

for any 0 < η < 1. According to the expression (3.8), we write the vector potential as A_{ω} = Å_{ω}(x) + \overline{A}_{ω}(x), where Å_{ω}(x) and \overline{A}_{ω}(x) are vector potentials obtained by replacing B_{ω} by B̊_{ω} and M + \overline{B}_{ω}, respectively, in (2.2). We denote the Dirac operator with the magnetic field B̊_{ω} by D̊_{ω}:

D̊_{ω} = ∑_{j=1}^{d} γ_{j}(i∂_{j} + Å_{ω}^{j}).

Let (D̊_{ω}^{2})_{η} be the self-adjoint operator corresponding to the closure of the quadratic form

r_{ω}(Φ, Ψ) := ⟨D̊_{ω}Φ(x), D̊_{ω}Ψ(x)⟩_{V}

with the domain C_{0}^{∞}(Λ(η) → V). For arbitrarily fixed ε > 0, we take Φ^{∘}_{ω} ∈ C_{0}^{∞}(Λ(η) → V) so that |||Φ^{∘}_{ω}|||_{V} = 1 and r_{ω}(Φ^{∘}_{ω}, Φ^{∘}_{ω}) ≤ inf spec((D̊_{ω}^{2})_{η}) + ε. Then, for arbitrarily fixed δ > 0, we have

inf spec H_{ω,η}

≤ ∑_{j=1}^{d} |||(i∂_{j} + Å_{ω}^{j} + \overline{A}_{ω}^{j})Φ^{∘}_{ω}|||_{V}^{2} + ⟨Φ^{∘}_{ω}, \frac{g}{2} ∑_{j<k} iγ_{j}γ_{k}B_{ω}^{(j,k)}Φ^{∘}_{ω}⟩_{V}

≤ (1 + δ)r_{ω}(Φ^{∘}_{ω}, Φ^{∘}_{ω}) + (1 + \frac{1}{δ})||| ∑_{j=1}^{d} γ_{j}\overline{A}_{ω}^{j}Φ^{∘}_{ω}|||_{V}^{2} + \frac{g − 2}{2}⟨Φ^{∘}_{ω}, ∑_{j<k} iγ_{j}γ_{k}B̊^{(j,k)}Φ^{∘}_{ω}⟩_{V}

+\frac{g − 2}{2}⟨Φ^{∘}_{ω}, ∑_{j<k} iγ_{j}γ_{k}(M^{(j,k)} + \overline{B}_{ω}^{(j,k)}(x))Φ^{∘}_{ω}⟩_{V}

≤ (1 + δ) inf spec((D̊_{ω}^{2})_{η}) + (1 + δ)ε + c(1 + \frac{1}{δ}) sup_{x∈Λ(η)} |\overline{A}_{ω}(x)|^{2}

+\frac{g − 2}{2}⟨Φ^{∘}_{ω}, ∑_{j<k} iγ_{j}γ_{k}B̊^{(j,k)}Φ^{∘}_{ω}⟩_{V} + c\frac{g − 2}{2} sup_{x∈Λ(η)} |𝕄 + \overline{𝔹}_{ω}(x)|

for some c > 0. By the independence, we have

Ñ(t) ≥ Ñ^{0}(t)Ñ^{1}(t),

where

Ñ^{0}(t) := E[ exp { − t((1 + δ) inf spec((D̊_{ω}^{2})_{η})

+\frac{g − 2}{2}⟨Φ^{∘}_{ω}, ∑_{j<k} iγ_{j}γ_{k}B̊_{ω}^{(j,k)}Φ^{∘}_{ω}⟩_{V} )\}]

and

Ñ^{1}(t) := exp(−t(1 + δ)ε)E[ exp \{ − tc((1 + \frac{1}{δ}) sup_{x∈Λ(η)} |\overline{A}_{ω}(x)|^{2}

+\frac{g − 2}{2} sup_{x∈Λ(η)} |𝕄 + \overline{𝔹}_{ω}(x)|)\}].

By Jensen’s inequality, we have

\underline{lim}_{t↑∞} \frac{1}{t^{2}} log Ñ^{1}(t)

≥ − \overline{lim}_{t↑∞} \frac{c}{t}E[(1 + \frac{1}{δ}) sup_{x∈Λ(η)} |\overline{A}_{ω}(x)|^{2} + \frac{g − 2}{2} sup_{x∈Λ(η)} |𝕄 + \overline{𝔹}_{ω}(x)|].

Since

E[ sup_{x∈Λ(η)} |\overline{A}_{ω}(x)|^{2}] and E[ sup_{x∈Λ(η)} |𝕄 + \overline{𝔹}_{ω}(x)|]

are finite, we have

\underline{lim}_{t↑∞} \frac{1}{t^{2}} log Ñ^{1}(t) ≥ 0.

By the continuity (3.11), we can choose Φ^{∘}_{ω} so that

⟨Φ^{∘}_{ω}, ∑_{j<k} iγ_{j}γ_{k}B̊^{(j,k)}Φ^{∘}_{ω}⟩_{V} ≤ −\frac{1}{2}(1 − 2Dℓ(η))∥B_{ω}(0) − M∥_{1}.

By using also Lemma 4.1 below, we have

Ñ^{0}(t) ≥ exp(−t(1 + δ)cη^{−2})E[ exp \{t∥B_{ω}(0) − M∥_{1}(\frac{g − 2}{4}(1 − 2Dℓ(η)) − cr(η))\}].

Since η is arbitrary, we obtain (4.1).

Lemma 4.1. We assume M = 0. Then there exist a constant c > 0 such that (4.2) inf spec((D̊_{ω}^{2})_{η}) ≤ c(η^{−2} + ∥B_{ω}(0)∥_{1}ℓ(η)) for small enough η > 0, where ℓ(η) is the function taken in (3.11).

Proof. By (3.11) we have

|Å_{ω}(x) − Â_{ω}(x)| ≤ c ℓ(η)∥B_{ω}(0)∥_{1}|x|

on x ∈ Λ(η), where Â_{ω} = (Â_{ω}^{j})_{1≤j≤d} is

Â_{ω}^{j} = \frac{1}{2} ∑_{k=1}^{d} B_{ω,j}^{k}(0)x^{k}.

Then the proof is reduced to an estimate in the case of the uniform magnetic field as follows:

inf spec((D̊_{ω}^{2})_{η})

≤ c inf \{||| ∑_{j=1}^{d} γ_{j}(i∂_{j} + \frac{1}{2} ∑_{k=1}^{d} B_{ω,j}^{k}(0)x^{k})Φ(x)|||_{V}^{2} + ℓ(η)^{2}∥B_{ω}(0)∥_{1}^{2}||||x|Φ|||_{V}^{2}

: Φ ∈ C_{0}^{∞}(Λ(η) → V), |||Φ|||_{V} = 1}.

Changing the variable x → x/\sqrt{∥B_{ω}(0)∥_{1}}, we rewrite

inf spec((D̊_{ω}^{2})_{η})

≤ c∥B_{ω}(0)∥_{1} inf \{||| ∑_{j=1}^{d} γ_{j}(i∂_{j} + \frac{1}{2} ∑_{k=1}^{d} \frac{B_{ω,j}^{k}(0)}{\sqrt{∥B_{ω}(0)∥_{1}}} x^{k})Φ(x)|||_{V}^{2} + ℓ(η)^{2}||||x|Φ|||_{V}^{2}

: Φ ∈ C_{0}^{∞}(Λ(η\sqrt{∥B_{ω}(0)∥_{1}}) → V), |||Φ|||_{V} = 1\}.

Since

U_{ω}^{∗}B_{ω}(0)U_{ω} = ( \begin{matrix}B_{ω}^{⟨1⟩} \\ & . & . & . \\ & & & & B_{ω}^{⟨[d/2]⟩} \\ & & & & & (0) \end{matrix} ) ,  B_{ω}^{⟨j⟩} = ( \begin{matrix}0 & B_{ω}^{(j)} \\ −B_{ω}^{(j)} & 0 \end{matrix} ) ,  B_{ω}^{(j)} ≥ 0,

for some orthogonal matrix U_{ω}, we have

inf spec((D̊_{ω}^{2})_{η}) ≤ c ∑_{h=0}^{[d/2]} I_{ω}^{h},

where

I_{ω}^{0} := ∥B_{ω}(0)∥_{1} inf {|||ϕ′(x)|||_{1}^{2} + ℓ(η)^{2}|||xϕ(x)|||_{1}^{2}
: ϕ ∈ C_{0}^{∞}(Λ_{1}(η\sqrt{∥B_{ω}(0)∥_{1}/d}) → ℝ), |||ϕ|||_{1} = 1\}

and

I_{ω}^{h} := ∥B_{ω}(0)∥_{1} inf \{|||\{γ_{1}(i∂_{1} − \frac{B_{ω}^{(h)}x^{2}}{2∥B_{ω}(0)∥_{1}} )

+γ_{2}(i∂_{2} + \frac{B_{ω}^{(h)}x^{1}}{2∥B_{ω}(0)∥_{1}} )\}Φ(x)|||_{2}^{2} + ℓ(η)^{2}||||x|Φ|||_{2}^{2}

: Φ ∈ C_{0}^{∞}(Λ_{2}(η\sqrt{∥B_{ω}(0)∥_{1}/d}) → ℂ^{2}), |||Φ|||_{2} = 1\}

for 1 ≤ h ≤ [d/2]. In these expressions, ||| · |||_{j} is the norm of L^{2}(ℝ^{j} → ℂ^{j}) and Λ_{j}(R) := (−R/2, R/2)^{j} for j = 1, 2 and R > 0, and γ_{1}, γ_{2} are the matrices defined by

γ_{1} = ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ) , γ_{2} = ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) .

By changing the variable x → x/\sqrt{ℓ(η)}, we have

I_{ω}^{0} = ∥B_{ω}(0)∥_{1}ℓ(η) inf{|||ϕ′|||_{1}^{2} + |||xϕ|||_{1}^{2}

: ϕ ∈ C_{0}^{∞}(Λ_{1}(η\sqrt{ℓ(η)∥B_{ω}(0)∥_{1}/d}) → ℝ), |||ϕ|||_{1} = 1\}. (4.3)

By choosing ϕ(x) = ζ(x/(η\sqrt{ℓ(η)∥B_{ω}(0)∥_{1}/d})) with ζ ∈ C_{0}^{∞}(Λ_{1}(1) → [0, 1]) such that ζ = 1 on Λ_{1}(1/2), we have

I_{ω}^{0} ≤ c(η^{−2} + η^{2}ℓ(η)^{2}∥B_{ω}(0)∥_{1}^{2}). (4.4)

This estimate is efficient only when ∥B_{ω}(0)∥_{1} is small so that η\sqrt{ℓ(η)∥B_{ω}(0)∥_{1}/d} ≤ 1. When η\sqrt{ℓ(η)∥B_{ω}(0)∥_{1}/d} ≥ 1, we replace η\sqrt{ℓ(η)∥B_{ω}(0)∥_{1}/d} by 1 in (4.3). Then we have

I_{ω}^{0} ≤ ∥B_{ω}(0)∥_{1}ℓ(η) inf{|||ϕ′|||_{1}^{2} + |||xϕ|||_{1}^{2}
: ϕ ∈ C_{0}^{∞}(Λ_{1}(1) → ℝ), |||ϕ|||_{1} = 1}
≤ c∥B_{ω}(0)∥_{1}ℓ(η).

Thus we always have

I_{ω}^{0} ≤ c(η^{−2} + ∥B_{ω}(0)∥_{1}ℓ(η)).

For I_{ω}^{h}, we take Φ(x) = (ϕ(x^{1})ϕ(x^{2}), 0)^{∗} with a real valued function ϕ of one variable. Then we have

I_{ω}^{h} ≤ 2∥B_{ω}(0)∥_{1} inf \{|||ϕ′(x) + \frac{B_{ω}^{(h)}x}{2∥B_{ω}(0)∥_{1}} ϕ(x)|||_{1}^{2} + ℓ(η)^{2}|||xϕ|||_{1}^{2}

: ϕ ∈ C_{0}^{∞}(Λ_{1}(η\sqrt{∥B_{ω}(0)∥_{1}/d}) → ℝ), |||ϕ|||_{1} = 1\}. (4.5) By

|||ϕ′(x) + \frac{B_{ω}^{(h)}x}{2∥B_{ω}(0)∥_{1}} ϕ(x)|||_{1}^{2} ≤ 2(|||ϕ′(x)|||_{1}^{2} + |||\frac{B_{ω}^{(h)}x}{2∥B_{ω}(0)∥_{1}} ϕ(x)|||_{1}^{2}),

we can estimate I_{ω}^{h} as in our estimate of I_{ω}^{0}:

I_{ω}^{h} ≤ c∥B_{ω}(0)∥_{1} inf \{|||ϕ′|||_{1}^{2} + (\frac{(B_{ω}^{(h)})^{2}}{∥B_{ω}(0)∥_{1}^{2}} + ℓ(η)^{2})|||xϕ|||_{1}^{2}

: ϕ ∈ C_{0}^{∞}(Λ_{1}(η\sqrt{∥B_{ω}(0)∥_{1}/d}) → ℝ), |||ϕ|||_{1} = 1\}

= c\sqrt{(B_{ω}^{(h)})^{2} + ∥B_{ω}(0)∥_{1}^{2}ℓ(η)^{2}} inf \{|||ϕ′|||_{1}^{2} + |||xϕ|||_{1}^{2}

: ϕ ∈ C_{0}^{∞}(Λ_{1}(η((B_{ω}^{(h)})^{2} + ∥B_{ω}(0)∥_{1}^{2}ℓ(η)^{2})^{1/4}/\sqrt{d}) → ℝ), |||ϕ|||_{1} = 1\},

from which we obtain I_{ω}^{h} ≤ c(η^{−2} + \sqrt{(B_{ω}^{(h)})^{2} + ∥B_{ω}(0)∥_{1}^{2}ℓ(η)^{2}}). (4.6) On the other hand we choose ϕ in (4.5) as

ϕ(x) = ζ(\frac{x}{η}\sqrt{d/∥B_{ω}(0)∥_{1}}) exp ( − \frac{B_{ω}^{(h)}x^{2}}{4∥B_{ω}(0)∥_{1}} ),

where ζ is the function used in (4.4). Then we have

ϕ′(x) + \frac{B_{ω}^{(h)}x}{2∥B_{ω}(0)∥_{1}} ϕ(x) = \frac{\sqrt{d/∥B_{ω}(0)∥_{1}}}{η} ζ′(\frac{x}{η}\sqrt{d/∥B_{ω}(0)∥_{1}}) exp ( − \frac{B_{ω}^{(h)}x^{2}}{4∥B_{ω}(0)∥_{1}} ).

By a simple estimate we have ∥ϕ∥_{1}^{2} ≥ c\{(η\sqrt{∥B_{ω}(0)∥_{1}}) ∧ \sqrt{∥B_{ω}(0)∥_{1}/B_{ω}^{(h)}}\},

|||ϕ′(x) + \frac{B_{ω}^{(h)}x}{2∥B_{ω}(0)∥_{1}} ϕ(x)|||_{1}^{2} ≤ \frac{c}{η^{2}∥B_{ω}(0)∥_{1}} exp ( − c′B_{ω}^{(h)}η^{2})\{(η\sqrt{∥B_{ω}(0)∥_{1}}) ∧ \sqrt{∥B_{ω}(0)∥_{1}/B_{ω}^{(h)}}\}

and

∥xϕ∥_{1}^{2} ≤ c\{(η\sqrt{∥B_{ω}(0)∥_{1}}) ∧ (\sqrt{∥B_{ω}(0)∥_{1}/B_{ω}^{(h)}})\}^{3},

from which we obtain

I_{ω}^{h} ≤ c(\frac{1}{η^{2}} + ℓ(η)^{2}∥B_{ω}(0)∥_{1}(\frac{∥B_{ω}(0)∥_{1}}{B_{ω}^{(h)}} )^{2}). (4.7)

We use (4.7) when B_{ω}^{(h)} ≥ ∥B_{ω}(0)∥_{1}ℓ(η), and use (4.6) when B_{ω}^{(h)} ≤ ∥B_{ω}(0)∥_{1}ℓ(η). Then we always obtain (4.2). □

5. Skew-symmetric matrix valued Gaussian random variables

In this section we first prove Proposition 2.2 and discuss the representation of the norm ∥ · ∥_{(1←2)}.

Proof of Proposition 2.2. Since 𝔹 can be realized as

B_{ω}^{(j,k)} = ∑_{1≤ℓ<m≤d} \sqrt{β}_{(ℓ,m)}^{(j,k)}X_{ω}^{(ℓ,m)},

where X_{ω}^{(ℓ,m)}, 1 ≤ ℓ < m ≤ d, are independent unit normal random variables, we have

I(t) := E[exp(t∥B∥_{1})]

= ∫_{ℝ^{D}} exp \{t∥( ∑_{1≤ℓ<m≤d} \sqrt{β}_{(ℓ,m)}^{(·,·)}x^{(ℓ,m)})∥_{(1)} − \frac{|x^{(·,·)}|^{2}}{2} \} ∏_{1≤ℓ<m≤d} \frac{dx^{(ℓ,m)}}{\sqrt{2π}} .

By changing the variable x^{(·,·)} → tx^{(·,·)}, we have

I(t) := E[exp(t∥B∥_{1})]

= t^{D} ∫_{ℝ^{D}} exp \{t^{2}(∥( ∑_{1≤ℓ<m≤d} \sqrt{β}_{(ℓ,m)}^{(·,·)}x^{(ℓ,m)})∥_{(1)} − \frac{|x^{(·,·)}|^{2}}{2} )\} ∏_{1≤ℓ<m≤d} \frac{dx^{(ℓ,m)}}{\sqrt{2π}} .

Thus we obtain

lim_{t↑∞} \frac{1}{t^{2}} log I(t) = sup \{∥( ∑_{1≤ℓ<m≤d} \sqrt{β}_{(ℓ,m)}^{(·,·)}x^{(ℓ,m)})∥_{(1)} − \frac{|x^{(·,·)}|^{2}}{2} : (x^{(·,·)}) ∈ ℝ^{D}\}.

It is easy to see that the right hand side coincides with the right hand side of (2.11). □

In the case of d = 2 or 3, the trace norm ∥X∥_{1} and the Hilbert-Schmidt norm ∥X∥_{2} are same up to constants and we can represent the norm ∥α∥_{(1←2)} explicitly as follows:

Example 1. In the case of d = 2, 𝕏 and α are scalars. Then it is easy to see that ∥𝕏∥_{(1)} = \sqrt{2}∥𝕏∥_{(2)} = 2|𝕏| and ∥α∥_{(1←2)} = \sqrt{2}|α|. Therefore we obtain Theorem 1.1 from Theorem 2.1.

Example 2. In the case of d = 3, the eigenvalues of X are i|𝕏|, −i|𝕏|, 0. Thus we have ∥𝕏∥_{(1)} = \sqrt{2}∥𝕏∥_{(2)} = 2|𝕏| and

∥α∥_{(1←2)} = \sqrt{2} max\{|α(𝕏)| : 𝕏 ∈ ℝ^{3}, |𝕏| = 1\}
= \sqrt{2}μ(α),

where μ(α) is the largest eigenvalue of α. Therefore the result of Proposition 2.2 is

lim_{t↑∞} \frac{1}{t^{2}} log E[exp(t∥B∥_{1})] = 2μ(β)

and the result of Theorem 2.1 is

lim_{λ↓−∞} \frac{1}{λ^{2}} log N(λ) = \frac{−2}{(g − 2)^{2}μ(β(0))}.

For higher dimensional cases, we give the following:

Lemma 5.1. (i) For any 𝕏 = (X^{(j,k)})_{1≤j<k≤d} ∈ ℝ^{D}, we have

∥𝕏∥_{(1)} = 2∥𝕏̂∥_{op},

where ∥𝕏̂∥_{op} is the operator norm of

𝕏̂ = ∑_{1≤j<k≤d} X^{(j,k)}iγ_{j}γ_{k}

in End(V).

(ii) For any D × D matrix α = (α_{(ℓ,m)}^{(j,k)})_{1≤j<k≤d,1≤ℓ<m≤d}, we have

∥α∥_{(1←2)}^{2} = 2 max { ∑_{1≤j<k≤d} α^{(j,k)}(𝕏)^{2} + 2 ∑_{1≤j<k<ℓ<m≤d}(−α^{(j,k)}(𝕏)α^{(ℓ,m)}(𝕏)

+α^{(j,ℓ)}(𝕏)α^{(k,m)}(𝕏) − α^{(j,m)}(𝕏)α^{(k,ℓ)}(𝕏))(v, γ_{j}γ_{k}γ_{ℓ}γ_{m}v)_{V}

: 𝕏 ∈ ℝ^{D}, |𝕏| = 1, v ∈ V, |v|_{V} = 1},

where (·, ·)_{V} and | · |_{V} are the inner product and the norm of V, respectively.

We here note that (v, γ_{j}γ_{k}γ_{ℓ}γ_{m}v)_{V} is real since γ_{j}γ_{k}γ_{ℓ}γ_{m} is an Hermitian matrix. By this lemma we have the following:

Example 3. In the case of d = 4, since

{(v, γ_{1}γ_{2}γ_{3}γ_{4}v)_{V} : v ∈ V, |v|_{V} = 1} = [−1, 1],

we have

∥α∥_{(1←2)}^{2}

= 2 max { ∑_{j<k} α^{(j,k)}(𝕏)^{2} + 2τ(α^{(1,2)}(𝕏)α^{(3,4)}(𝕏) − α^{(1,3)}(𝕏)α^{(2,4)}(𝕏) + α^{(1,4)}(𝕏)α^{(2,3)}(𝕏))

: 𝕏 ∈ ℝ^{6}, |𝕏| = 1, −1 ≤ τ ≤ 1}

= 2( max{(α^{(1,2)}(𝕏) + α^{(3,4)}(𝕏))^{2} + (α^{(1,3)}(𝕏) − α^{(2,4)}(𝕏))^{2} + (α^{(1,4)}(𝕏) + α^{(2,3)}(𝕏))^{2}

: 𝕏 ∈ ℝ^{6}, |𝕏| = 1}

∨ max{(α^{(1,2)}(𝕏) − α^{(3,4)}(𝕏))^{2} + (α^{(1,3)}(𝕏) + α^{(2,4)}(𝕏))^{2} + (α^{(1,4)}(𝕏) − α^{(2,3)}(𝕏))^{2}

: 𝕏 ∈ ℝ^{6}, |𝕏| = 1}).

In particular if α_{(ℓ,m)}^{(j,k)} = δ_{jℓ}δ_{km}α^{(j,k)}, which corresponds to the case where the random variables B^{(j,k)}(0), 1 ≤ j < k ≤ 4, are mutually independent, then we have

∥α∥_{(1←2)}^{2} = 2{((α^{(1,2)})^{2} + (α^{(3,4)})^{2}) ∨ ((α^{(1,3)})^{2} + (α^{(2,4)})^{2}) ∨ ((α^{(1,4)})^{2} + (α^{(2,3)})^{2}}.

At present we do not have other simple representations for general higher dimensional cases, except for the following:

Example 4. Consider the case where the components B_{ω}^{(j,k)}(0), 1 ≤ j < k ≤ d, of the value of the magnetic field at a point are independently and identically distributed with the variance β(0). This corresponds to the case where the D × D matrix α is a constant times the identity matrix I, for which

∥I∥_{(1←2)} = max{∥𝕏∥_{(1)} : 𝕏 ∈ ℝ^{D}, ∥𝕏∥_{(2)} = 1}.

We note that, if the nonzero eigenvalues of X are ±iλ_{1}, ±iλ_{2}, …, ±iλ_{h}, 0 ≤ h ≤ [d/2], then ∥X∥_{1} = 2(|λ_{1}| + |λ_{2}| + ⋯ + |λ_{h}|) and ∥X∥_{2} = \sqrt{2(|λ_{1}|^{2} + |λ_{2}|^{2} + ⋯ + |λ_{h}|^{2})}, and that h varies over {0, 1, …, [d/2]} and (λ_{1}, λ_{2}, …, λ_{h}) varies over (ℝ − 0)^{h} as 𝕏 varies over ℝ^{D}. Thus we obtain

∥I∥_{(1←2)} = \sqrt{2[\frac{d}{2}]}.

Thus the result of Theorem 2.1 is

lim_{λ↓−∞} \frac{1}{λ^{2}} log N(λ) = \frac{−2}{(g − 2)^{2}[d/2]β(0)}.

A statement corresponding to Proposition 2.2 is the following: if B_{ω}^{(j,k)}, 1 ≤ j < k ≤ d are independently and identically distributed Gaussian random variable with the mean 0 and the variance β, then the corresponding random skew-symmetric matrix B = (B_{ω,k}^{j}) satisfies

lim_{t↑∞} \frac{1}{t^{2}} log E[exp(t∥B∥_{1})] = 2[\frac{d}{2}]β.

In this case we also know the explicit form of the joint probability distribution of the eigenvalues (cf. Mehta [19] §3.4).

References

[1] M. A. Akcoglu and U. Krengel: Ergodic theorems for superadditive processes, J. Reine Angew. Math. 323 (1981), 53–67.

[2] F. Bentosela, R. M. Cavalcanti, P. Exner and V. A. Zagrebnov: Anomalous electron trapping by localized magnetic fields, J. Phys. A 32 (1999), no. 16, 3029–3039.

[3] K. Broderix, N. Heldt and H. Leschke: Partition function and the density of states for an electron in the plane subjected to a random potential and a magnetic field, Phys. Rev. B 40 (1989), no.11, 7479–7486.

[4] K. Broderix, D. Hundertmark and H. Leschke: Self-averaging, decomposition and asymptotic properties of the density of states for random Schrödinger operators with constant magnetic field, Path integrals from meV to MeV: Tutzing ’92, 98–107, World Sci. Publishing, River Edge, NJ, 1993.

[5] K. Broderix, D. Hundertmark and H. Leschke: Continuity properties of Schrödinger semigroups with magnetic fields, Rev. Math. Phys. 12 (2000) 181–225.

[6] H. Cartan: Differential forms, Hermann, Paris, 1970.

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

[8] A. Casher and H. Neuberger, The density of states of a two-dimensional electron gas in a random magnetic field, Phys. Lett. B 139 (1984) 67–69.

[9] A. Comtet, A. Georges and P. Le Doussal, Exact density of states of a two-dimensional Dirac operator in a random magnetic field, XVIIth International Colloquium on Group Theoretical Methods in Physics, Sainte-Adlè, PQ, 1988, World Sci. Publishing, Teaneck, NJ, 1989, 226–229.

[10] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon: Schrödinger operators, Springer, Berlin, Heidelberg, New York, 1987.

[11] S. Doi, A. Iwatsuka and T. Mine: The uniqueness of the integrated density of states for the Schrödinger operators with magnetic fields, Math. Z. 237 (2001), 335-371.

[12] X. Fernique: Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris Série A 270 (1970), 1698– 1699.

[13] T. Hupfer, H. Leschke, P. Müller and S. Warzel: Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, preprint.

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

[15] W. Kirsch and F. Martinelli: On the density of states of Schrödinger operators with a random potential, J. Phys. A 15 (1982), 2139–2156.

[16] U. Krengel: Ergodic theorems, With a supplement by Antoine Brunel, Walter de Gruyter & Co., Berlin-New York, 1985.

[17] H. Leinfelder and C. G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), 1–19.

[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] M. L. Mehta: Random matrices, 2nd. ed., Academic Press, Boston, 1991.

[20] S. Nakamura: A remark on the Dirichlet-Neumann decoupling and the integrated density of states, J. Funct. Anal. 179 (2001), 136–152.

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

[22] L. A. Pastur: Spectra of random self adjoint operators, Russ. Math. Surv. 28 (1973), 1-67.

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

[24] M. Reed and B. Simon: Methods of modern mathematical physics. I. Functional analysis, 2nd ed., Academic Press, New York, 1980.

[25] B. Simon: Functional integration and quantum physics, Academic Press, New York, 1979.

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

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