Group ring elements with large spectral density

Abstract

Given ${\delta }>0$ we construct a group $G$ and a group ring element $S\in \mathbb Z[G]$ such that the spectral measure $\mu$ of $S$ fulfils $\mu ((0,{\varepsilon })) > \frac{C}{|\log ({\varepsilon })|^{1+{\delta }}}$ for small ${\varepsilon }$. In particular the Novikov-Shubin invariant of any such $S$ is $0$. The constructed examples show that the best known upper bounds on $\mu ((0,{\varepsilon }))$ are not far from being optimal.

Appendix: A counterexample to Conjecture 1 from mathematical physics

In this appendix we present the following unpublished observation of the author and B. Virág: a counterexample to Conjecture 1 can be also deduced from the mathematical physics literature.

Let us specialize to the following situation. Let $\mathbf Z\curvearrowright X$ be an essentially free action (i.e. free on a subset of full measure). Let $t\in \mathbf Z$ be a fixed generator, let $F:X \rightarrow \mathbb R$ be a bounded measurable function and let $T\in {\Gamma }\ltimes L^\infty (X)$ be

$$\begin{aligned} T:= tF +Ft^{-1} = tF + t^{-1}(t.F). \end{aligned}$$

(10)

For $x\in X$ let $T(x):l^2(\mathbb Z)\rightarrow l^2(\mathbb Z)$ be defined on the standard basis vectors as

$$\begin{aligned} T(x)\zeta _k := F(t^k.x)\zeta _{k+1} + F(t^{k-1}.x)\zeta _{k-1}. \end{aligned}$$

(11)

Let us identify $l^2(\mathbf Z)$ with $l^2(\mathbb Z)$ in the natural way. As explained in Sect. 2, $T$ is an operator on $\int ^\oplus _X l^2(\mathbb Z) \, d\mu (x)$ which preserves the fibers $l^2(\mathbb Z)$. A direct check shows that $T$ decomposes as

$$\begin{aligned} \int ^\oplus _X T(x)\,d\mu (x). \end{aligned}$$

Since $\tau (T) = \int _X \langle T(x)\zeta _0,\zeta _0\rangle$, we have the following formula for the spectral measure of $\mu _T$ of $T$:

$$\begin{aligned} \mu _T (D) = \int _X \langle P(x,D)\zeta _0, \zeta _0\rangle \,d\mu (x), \end{aligned}$$

(12)

where $D\subset \mathbb R$ is a measurable set and $P(x,D)$ is the spectral projection of $T(x)$ corresponding to the set $D$ (see e.g. [13, Lemma 1.9]).

The measure $D\mapsto \langle P(x,D)\zeta _0, \zeta _0\rangle$ on $\mathbb R$ is called the rooted spectral measure of $T(x)$. Equation (12) can be concisely summarized as follows. The family $T(x)$, $x\in X$, is a random operator, and the spectral measure of $T$ is equal to the expected rooted spectral measure of $T(x)$.

We specialize further as follows. Let $(S,\nu )$ be a compact abelian group with the normalized Haar measure, let $A$ be its Pontryagin dual, and let $X=\prod _\mathbb ZS$ together with the product measure. Consider the action $\mathbf Z\curvearrowright X$ given by shifting the coordinates. Let $f:S \rightarrow \mathbb R$ be a function in the image of the Pontryagin duality map $\mathbb Q[A]\rightarrow L^\infty (S)$, and let $F:X \rightarrow \mathbb R$ be defined as $F((x_i)) := f(x_0)$.

For $x=(x_i)\in X$ the formula (11) becomes

$$\begin{aligned} T(x)\zeta _k = f(x_k)\zeta _{k+1} + f(x_{k-1})\zeta _{k-1}. \end{aligned}$$

(13)

Such families of operators have been studied in mathematical physics at least since [7]. We provide a small dictionary. First, usually there would be no reference to a concrete measure space $X$. The Eq. (13) would be written as

$$\begin{aligned} H\zeta _k = W(k)\zeta _{k+1} + W(k-1)\zeta _{k-1}, \end{aligned}$$

(14)

together with the assumption that the numbers $W(k)$ are random, independent, and distributed according to some probability measure ${\varphi }$ on $\mathbb R$. In the case of (13) the measure ${\varphi }$ is the push-forward of $\nu$ through $f$, i.e. ${\varphi }(D):=\nu (f^{-1}(D))$.

The operator $H$ is the Hamiltonian associated to the one-dimensional disordered chain whose disorder is i.i.d., distributed according to the law ${\varphi }$. The spectral measure of the original $T\in \mathbf Z\ltimes L^\infty (X)$ given by (10) is referred to as the expected spectral measure, or the expected density of states, in order to differentiate it from the rooted spectral measures of the operators $T(x)$, $x\in X$.

In his very impressive article, Dyson [7] considered a disorder distributed according to some specific measure ${\varphi }$. He showed that for that specific disorder the expected spectral measure $\mu _H$ has the property

$$\begin{aligned} \mu _H((0,{\varepsilon })) \approx \frac{1}{|\log ({\varepsilon })|^2}, \end{aligned}$$

(15)

where $\approx$ means that the ratio of both sides approaches a positive constant when ${\varepsilon }\rightarrow 0$. Since then, such behaviour is referred to as the Dyson’s singularity. It is the most eminent qualitative difference between the disordered chains and the situation without a disorder, i.e. when all $W(k)$ are equal to $1$.

We cannot use Dyson’s result to give a counterexample to Conjecture 1, because the measure ${\varphi }$ he considered is not supported in a bounded interval. Accordingly ${\varphi }$ is not a push-forward of $\nu$ through $F\in L^\infty (X)$ (since, by definition, such $F$ must be bounded, and so the push-forward of any measure through $F$ is supported in a bounded interval). In particular, the Hamiltonian considered by Dyson is not an element of the von Neumann algebra $\mathbf Z\ltimes L^\infty (X)$.

However, Dyson’s singularity (15) has been conjectured to appear for arbitrary measures ${\varphi }$ which have a non-atomic part. So far, the exact form (15) has been confirmed only via heuristic arguments (e.g. [9]). However, the following has been rigorously established. Recall that $G\in L^1(\mathbb R)$ is the density of a measure ${\varphi }$ if for every measurable $D\subset \mathbb R$ we have ${\varphi }(D) = \int _D G(x) \,dx$.

Theorem 11

(Campanino and Perez [4]) Let $W$ in (14) be distributed according to a measure ${\varphi }$ with a continuous density supported in an interval $(a, b)$, where $b>a>0$. Then the expected spectral measure $\mu _H$ of (14) has the property that

$$\begin{aligned} \mu _H((0,{\varepsilon })) \geqslant \frac{C}{{|\log ({\varepsilon })|^3}} \end{aligned}$$

for some $C>0$ and all sufficiently small ${\varepsilon }$.

As such, to provide a counterexample to Conjecture 1, it is enough to find a compact abelian group $(S,\nu )$ together with its Pontryagin dual $A$ and ${\widehat{f}}\in \mathbb Q[A]$ such that the push-forward of $\nu$ through $f:S\rightarrow \mathbb R$ has a continuous density supported in some interval $(a, b)$, where $b>a>0$. A simple exercise in the Fourier transform shows that we can take $A=\mathbf Z^3$ and ${\widehat{f}} = s_1+s_1^{-1}+s_2+s_2^{-1}+s_3+s_3^{-1}+7$.

Together with Lemma 5 we obtain the following.

Corollary 12

Let $T\in \mathbb Q[\mathbf Z^3\wr \mathbf Z]$ be given as

$$\begin{aligned} T&:= (s_1+s_1^{-1}+s_2+s_2^{-1}+s_3+s_3^{-1}+7)t\\&+\, t^{-1}(s_1+s_1^{-1}+s_2+s_2^{-1}+s_3+s_3^{-1}+7). \end{aligned}$$

Then the spectral measure $\mu _T$ of $T$ has the property that

$$\begin{aligned} \mu _T((0,{\varepsilon })) \geqslant \frac{C}{{|\log ({\varepsilon })|^3}} \end{aligned}$$

for some $C>0$ and all sufficiently small ${\varepsilon }$.

Remarks 13

At least one other example of a group ring element with an interesting spectral measure can be exhibited using mathematical physics literature. Let $a$ be the non-trivial element of $\mathbf Z_2$ and let $T\in \mathbb R[\mathbf Z_2\wr \mathbf Z]$ be given as $T:= t+t^{-1} + {\beta }\cdot a \in \mathbb R[\mathbf Z_2\wr \mathbf Z]$, where ${\beta }\in \mathbb R$. Then [23] implies, repeating the discussion above, that when ${\beta }$ is large enough, the spectral measure of $T$ has only singularly continuous part, i.e. it has no atoms and it is not possible to write it as $f{\lambda }$, where $f$ is a measurable function and ${\lambda }$ is the Lebesgue measure.

This is the only example of such behaviour known to the author. It has been conjectured that no singularly continuous part appears in the spectral measure of $T$ when $T$ is an element of the group ring of a torsion-free group ([29]). Unfortunately the mathematical physics literature does not seem to provide a counterexample to that conjecture.