Analyticity of density of states for the Cauchy distribution

In this short note we show that many random operators with the Cauchy distribution have analytic density of states, independent of disorder. This theorem applied to the Anderson model shows analyticity through the mobility edge (or the mobility band) on the Bethe lattice for small disorder and some models of Random Schrödinger operators.

The density of states of random operators is of interest for many random models. We are concerned in particular with random models coming from i.i.d random variables with the Cauchy distribution. For these models there is an exact formula for the density of states.

One of the earliest models with the Cauchy distribution was the Lloyd model studied by Lloyd [8] on $\ell^{2}(\mathbb{Z}^{3})$, in which he obtained an exact formula for the density of states. In the book of Carmona-Lacroix [4], page 329 and in problem VI.5.5 one finds calculations giving the exact value for the integrated density of states (IDS) for the Anderson model on the lattice $\ell^{2}(\mathbb{Z}^{d})$ with the Cauchy distribution. On the Bethe lattice Accosta-Klein [1], proved that the density of states is analytic in a strip about the real axis for any distribution close to and including the Cauchy distribution, in a function space. We note that Accosta-Klein [1] result is the first known result where smoothness of DOS is shown in the region where absolutely continuous spectrum is present for random operators.

In this note we show that the exact expressions for the IDS are valid in a much larger context both for the discrete and some continuous models, whenever i.i.d random variables with the Cauchy distribution are involved. We use the Trotter product formula for proving our result.

We consider a separable Hilbert space $\mathcal{H}$ and a self-adjoint operator $H_{0}$ with domain $D(H_{0})\subset\mathcal{H}$, a sequence $\{P_{n}\}$ of mutually orthogonal projections such that $\sum_{n\in\mathbb{N}}P_{n}=Id$ and consider the operator $\displaystyle{N^{\alpha}=\sum_{n\in\mathbb{N}}n^{\alpha}P_{n},~{}~{}\alpha>1,}$ with domain $D(N^{\alpha})\subset\mathcal{H}$. We consider i.i.d random variables $\{\omega_{n},~{}n\in\mathbb{N}\}$ with the Cauchy distribution whose density is given by $\psi_{\lambda}(x)=\frac{1}{\pi}\frac{\lambda}{\lambda^{2}+x^{2}},~{}\lambda>0$. If we consider an $\alpha>1$, then under the assumption on the random variables $\{\omega_{n}\}$, an application of the Borel-Cantelli Lemma implies that the set

$\displaystyle\Omega_{0}=\{(\omega_{n}):|\omega_{n}|\leq|n|^{\alpha},~{}\mathrm{for~{}all~{}but~{}finitely~{}many}~{}n\}$

(1)

has probability $1$. Therefore the domain of the self-adjoint operator $V^{\omega}=\sum_{n\in\mathbb{N}}\omega_{n}P_{n}$ contains $D(N^{\alpha})$ for every $\omega\in\Omega_{0}$. We assume that $D(H_{0})\cap D(N^{\alpha})$ is dense in $\mathcal{H}$ and that the random operators

$\displaystyle H^{\omega}=H_{0}+V^{\omega},~{}~{}V^{\omega}=\sum_{n\in\mathbb{N}}\omega_{n}P_{n}$

(2)

are essentially self-adjoint on $D(H_{0})\cap D(N^{\alpha})$ for every $\omega\in\Omega_{0}$. In the following we denote by $E_{A}()$, the spectral projection of a self-adjoint operator $A$.

Then we have, with $\mathbb{E}$ denotes integration with respect to $\omega$ and $*$ denotes convolution :

We have a few corollaries of this theorem applied to the Anderson model on $\mathbb{Z}^{d}$ and the Bethe lattice, where we denote by $\delta_{0}$ the unit vector supported at $0$ in $\mathbb{Z}^{d}$ or a chosen and fixed root in the Bethe lattice. In both these cases, the operator $H_{0}$, being the adjascency matrix, is bounded. Therefore the conditions of the Theoremm 1.1 are satisfied and the following corollary results by setting $\phi=\psi=\delta_{0}$ in the Theorem 1.1.

The second corollary is for Random Schrödinger Operators on $L^{2}(\mathbb{R})$. Consider a collection of smooth bump functions $\{u_{n},n\in\mathbb{Z}\}$ supported in a neighbourhood of unit cubes centered at $n\in\mathbb{Z}$. Let,

$\displaystyle H^{\omega}=-\Delta+V^{\omega},~{}~{}V^{\omega}(x)=\sum_{n\in\mathbb{Z}}\omega_{n}u_{n}(x),~{}~{}\sum_{n}u_{n}(x)=1,~{}~{}\forall x\in\mathbb{R},$

(4)

where $-\Delta$ is the Laplacian defined on its maximal domain $D(-\Delta)$, $\{\omega_{n}\}$ are i.i.d random variables with Cauchy distribution $\psi_{\lambda}(x)dx$.

Then we have the corollary, where as before $*$ denotes convolution. The only statement to verify to prove the corrolary is the essential self-adjointness of the random Schrödinger operators involved, which follows from a theorem of Kirsch-Martinelli [6]. We note that if the essential self-adjointness holds for the operators on $L^{2}(\mathbb{R}^{d})$, then the same result extends to that case also.

The proof of this is by the use of Trotter product formula.

We start by choosing a sequence $Q_{M}$ of finite rank orthogonal projections such that

		$\displaystyle\lim_{M\rightarrow\infty}Q_{M}=Id,~{}~{}[Q_{M},P_{n}]=0,~{}~{}Q_{M,n}=Q_{M}P_{n}~{}~{}\forall~{}n\in\mathbb{N}$		(6)
		$\displaystyle\forall~{}M\exists~{}N(M)~{}s.t.~{}Rank(Q_{M,n})\neq 0,\forall~{}n\leq N(M),~{}N(M)\rightarrow\infty~{}\mathrm{as}~{}M\rightarrow\infty,$		(7)

where the limits are in the stong sense. The existence of such $Q_{M}$ is not hard to see, since $P_{n}$s are mutually orthogonal and add to the identity.

We will prove the theorem by showing that the Fourier transform of $\mathcal{N}$ is the product of Fourier transforms of $\mu_{\phi,\psi}$ and $\psi_{\lambda}$. By the spectral theorem for self-adjoint operators, the Fourier transform of $\mathcal{N}$ is given by

$\displaystyle\int e^{itx}~{}d\mathcal{N}(x)=\mathbb{E}\big{(}\langle\phi,e^{itH^{\omega}}\psi\rangle\big{)}.$

(8)

We set $V^{\omega}=\sum\omega_{n}P_{n}$ and use the Trotter product formula [9, Theorem VIII.31], to write the right hand side of equation (8) as

		$\displaystyle\mathbb{E}\big{(}\langle\phi,e^{itH^{\omega}}\psi\rangle\big{)}=\lim_{k\rightarrow\infty}\mathbb{E}\big{(}\langle\phi,\big{(}e^{i\frac{t}{k}H_{0}}e^{i\frac{t}{k}V^{\omega}}\big{)}^{k}\psi\big{)}$
		$\displaystyle=\lim_{k\rightarrow\infty}\lim_{M\rightarrow\infty}\mathbb{E}\big{(}\langle\phi,\big{(}e^{i\frac{t}{k}H_{0}}e^{i\frac{t}{k}V^{\omega}}Q_{M}\big{)}^{k}\psi\rangle\big{)}$
		$\displaystyle=\lim_{k\rightarrow\infty}\lim_{M\rightarrow\infty}\sum_{n_{1},\dots,n_{k}=1}^{N(M)}\mathbb{E}\bigg{(}\langle\phi,\big{(}\prod_{j=1}^{\begin{subarray}{c}k\\ \rightarrow\end{subarray}}e^{i\frac{t}{k}H_{0}}e^{i\frac{t}{k}V^{\omega}}Q_{M,n_{j}}\big{)}\psi\rangle\bigg{)},$		(9)

where in the second equality we used the Lebesgue dominated convergence theorem to interchange the limit and expectation and in the next equality the arrow on the product denotes an ordered product with increasing index $j$ and finally the sum and the expectation are interchanged using Fubini’s theorem. Since we have from definitions of $V^{\omega},Q_{M,n}$ that $e^{i\frac{t}{k}V^{\omega}}Q_{M,n_{j}}=e^{i\frac{t}{k}\omega_{n_{j}}}$, the above equality becomes

		$\displaystyle\mathbb{E}\big{(}\langle\phi,e^{itH^{\omega}}\psi\rangle\big{)}$
		$\displaystyle=\lim_{k\rightarrow\infty}\lim_{M\rightarrow\infty}\sum_{n_{1},\dots,n_{k}=1}^{N(M)}\mathbb{E}\bigg{(}\langle\phi,\big{(}\prod_{j=1}^{\begin{subarray}{c}k\\ \rightarrow\end{subarray}}e^{i\frac{t}{k}H_{0}}e^{i\frac{t}{k}\omega_{n_{j}}}Q_{M,n_{j}}\big{)}\psi\rangle\bigg{)}$
		$\displaystyle=\lim_{k\rightarrow\infty}\lim_{M\rightarrow\infty}\sum_{n_{1},\dots,n_{k}=1}^{N(M)}e^{-\lambda|t|}\langle\phi,\big{(}\prod_{j=1}^{\begin{subarray}{c}k\\ \rightarrow\end{subarray}}e^{i\frac{t}{k}H_{0}}Q_{M,n_{j}}\big{)}\psi\rangle\bigg{)}$
		$\displaystyle=\lim_{k\rightarrow\infty}\lim_{M\rightarrow\infty}e^{-\lambda|t|}\langle\phi,\big{(}\prod_{j=1}^{\begin{subarray}{c}k\\ \rightarrow\end{subarray}}e^{i\frac{t}{k}H_{0}}Q_{M}\big{)}\psi\rangle\bigg{)}$
		$\displaystyle=e^{-\lambda|t|}\langle\phi,e^{itH^{\omega}}\psi\rangle,$		(10)

where in the second equality we used the fact that when the product is expanded for a fixed configuration of $(n_{1},\dots,n_{k})$, we will get an expression of the form

$$e^{i\frac{t}{k}(\alpha_{1}\omega_{n_{1}}+\alpha_{2}\omega_{n_{2}}+\dots+\alpha_{k}\omega_{n_{k}})},$$

with the $\alpha_{j}$’s denoting the number of times the $\omega_{n_{j}}$ occurs in the product and for any configuration of $(n_{1},\dots,n_{k})$ we have $\sum\alpha_{j}=k$. This fact together with the fact that $\omega_{n_{j}}$s are independent random variables for distint $n_{j}$ and the expectation with respect to the Cauchy distribution satisfies

$$\int e^{isx}\psi_{\lambda}(x)dx=e^{-\lambda|s|},~{}~{}\mathbb{E}e^{is\omega_{n}}e^{iw\omega_{m}}=e^{-\lambda|s+w|},~{}n\neq m$$

shows that the expectation always gives the value $e^{-\lambda|t|}$ for any configuration of $(n_{1},\dots,n_{k})$ in the sum, so we could take that factor out and resum the expression.

The equation (2) shows that $\mathbb{E}\langle\phi,e^{itH^{\omega}}\psi\rangle$ is integrable as a function of $t$ and so by Fourier inversion, the Theorem follows. ∎