--- title: 'Ergodic Hankel Operators: Dynamics & Spectra' url: https://www.emergentmind.com/topics/ergodic-hankel-operators type: topic --- # Ergodic Hankel Operators: Dynamics & Spectra Ergodic Hankel operators arise in two distinct but related senses in recent research. In one sense, Hankel operators are studied between Köthe and power series spaces, where ergodic phenomena enter through the backward and forward shift operators and through the Cesàro behavior of the columns of a Hankel matrix [2404.00574]. In a second, more directly spectral sense, ergodic Hankel operators are introduced as ergodic families of bounded self-adjoint Hankel integral operators on $L^2(0,\infty)$, realized after transplantation as covariant operators on $L^2(\mathbb{R})$ and analyzed via the Integrated Density of States (IDS), Floquet theory, and random-operator methods [2509.25010]. The term therefore does not refer to a single universal formalism; rather, it currently designates both a dynamical viewpoint on Hankel-generated shift behavior and a spectral theory of ergodic operator families. ## 1. Terminological scope and basic operator models A Hankel operator is characterized by dependence on the sum of indices or variables. In the sequence-space setting, a Hankel matrix $(a_{m,n})$ is one whose entries are constant along skew diagonals, and for a sequence $\theta=(\theta_n)_{n\in\mathbb{N}_0}$ the associated infinite Hankel matrix is $$ H_\theta=[\theta_{i+j}]_{i,j\ge 0}. $$ When defined between Köthe spaces, its action on the canonical basis is $$ H_\theta e_n=(\theta_{n-1},\theta_n,\theta_{n+1},\ldots)=\sum_{j=1}^\infty \theta_{j+n-2}e_j, $$ and for $x=\sum_{n=1}^\infty x_n e_n$ one sets $$ H_\theta x=\sum_{n=1}^\infty x_n H_\theta e_n $$ provided the series is well defined [2404.00574]. In the half-line integral setting, a Hankel operator $H$ on $L^2(0,\infty)$ is realized by $$ (Hf)(x)=\int_0^\infty h(x+y)f(y)\,dy,\qquad x>0, $$ with kernel $K(x,y)=h(x+y)$. If $h$ is real-valued, the operator is self-adjoint. A sufficient boundedness condition used throughout is $$ h(t)\le \frac{C_h}{t},\qquad t>0, $$ which, via the Carleman estimate, $$ \int_0^\infty\int_0^\infty f(t)\overline{f(s)}(t+s)^{-1}\,dt\,ds\le \pi\|f\|^2_{L^2(0,\infty)}, $$ implies $\|H\|\le \pi C_h$ [2509.25010]. A central structural device in the half-line theory is the unitary transplantation $$ (Ef)(\xi)=e^{\xi/2}f(e^\xi),\qquad \xi\in\mathbb{R}, $$ from $L^2(0,\infty)$ to $L^2(\mathbb{R})$. If $K:=EHE^*$, then $$ K(x,y)=e^{(x+y)/2}h(e^x+e^y). $$ For kernels of the form $h(t)=P(\log t)/t$, this becomes $$ K(x,y)=\frac{P(\log(e^x+e^y))}{2\cosh((y-x)/2)} =\frac{P(x+\log(1+e^{y-x}))}{2\cosh((y-x)/2)}, $$ which makes the covariance structure of ergodic families explicit after passing to logarithmic coordinates [2509.25010]. This bifurcation of models is crucial. In the Köthe-space literature, ergodicity is not attributed to the powers $H_\theta^n$ themselves. In the half-line theory, by contrast, ergodicity is built into a covariant family $\{H_\omega\}$ indexed by an ergodic probability space [2404.00574; 2509.25010]. ## 2. Hankel operators between Köthe and power series spaces The sequence-space framework begins with Köthe spaces $$ K(a_{n,k})=\left\{x=(x_n)_{n\in\mathbb{N}}:\|x\|_k:=\sum_{n=1}^\infty |x_n|a_{n,k}<\infty\ \text{for all }k\in\mathbb{N}\right\}, $$ where $(a_{n,k})_{n,k\in\mathbb{N}}$ is a Köthe matrix. Every Köthe space is a Fréchet space, and its dual is $$ (K(a_{n,k}))'=\left\{y=(y_n)_{n\in\mathbb{N}}:\sup_{n\in\mathbb{N}} |y_n|a_{n,k}^{-1}<+\infty\ \text{for some }k\in\mathbb{N}\right\} $$ [2404.00574]. The principal concrete subclasses are the power series spaces associated with a non-negative increasing sequence $\alpha=(\alpha_n)_{n\in\mathbb{N}}$ satisfying $\lim_{n\to\infty}\alpha_n=\infty$: $$ \Lambda_1(\alpha):=\left\{x=(x_n)_{n\in\mathbb{N}}:\|x\|_k:=\sum_{n=1}^\infty |x_n|e^{-\alpha_n/k}<\infty\ \text{for all }k\in\mathbb{N}\right\}, $$ $$ \Lambda_\infty(\alpha):=\left\{x=(x_n)_{n\in\mathbb{N}}:\|x\|_k:=\sum_{n=1}^\infty |x_n|e^{k\alpha_n}<\infty\ \text{for all }k\in\mathbb{N}\right\}. $$ These are Köthe spaces, and every power series space is Montel. The examples $$ O(\mathbb{C}^d)\cong \Lambda_\infty(n^{1/d}),\qquad O(\mathbb{D}^d)\cong \Lambda_1(n^{1/d}) $$ place the theory in a standard holomorphic-function context [2404.00574]. For Hankel operators between such spaces, the basic continuity criterion is expressed on the canonical basis. The operator $H_\theta:K(a_{n,k})\to K(b_{n,k})$ is well defined and continuous if and only if 1. $H_\theta e_n\in K(b_{n,k})$ for every $n\in\mathbb{N}$, and 2. for every $k\in\mathbb{N}$ there exists $m\in\mathbb{N}$ such that $$ \sup_{n\in\mathbb{N}} \frac{\|H_\theta e_n\|_k}{\|e_n\|_m}<\infty. $$ A necessary consequence is that if $H_\theta$ is continuous, then $\theta=H_\theta e_1\in K(b_{n,k})$ and also $\theta\in (K(a_{n,k}))'$ [2404.00574]. The paper develops several sufficient conditions for continuity and compactness, especially for mappings between power series spaces. Among the principal statements are the following. - For every $\theta\in\Lambda_\infty(\beta)$, the Hankel operator $$ H_\theta:\Lambda_\infty(\alpha)\to \Lambda_\infty(\beta) $$ is continuous and compact [2404.00574]. - If $\alpha_n\le A\beta_n+B$ for all $n\in\mathbb{N}$ and some $A,B>0$, then for every $\theta\in\Lambda_\infty(\beta)$, $$ H_\theta:\Lambda_1(\alpha)\to \Lambda_\infty(\beta) $$ is well defined, continuous, and compact [2404.00574]. - If $\beta$ is stable and $\beta_n\le A\alpha_n+B$ for all $n\in\mathbb{N}$, then for every $\theta\in\Lambda_1(\beta)$, $$ H_\theta:\Lambda_\infty(\alpha)\to \Lambda_1(\beta) $$ is well defined, continuous, and compact [2404.00574]. - If $\Lambda_1(\beta)$ is a nuclear finite-type power series space, then for every $\theta\in(\Lambda_1(\alpha))'$, $$ H_\theta:\Lambda_1(\alpha)\to \Lambda_1(\beta) $$ is well defined, continuous, and compact [2404.00574]. These results show that, in the Köthe-space setting, compactness is pervasive under natural growth assumptions. This suggests that the ergodic questions relevant there are less about long-time iteration of a fixed Hankel operator and more about how Hankel structure encodes the dynamics of related operators, especially shifts. ## 3. Shift dynamics and the first ergodic interpretation The ergodic content of the Köthe-space theory is mediated by the backward and forward shift operators on $\Lambda_r(\alpha)$, $r\in\{1,\infty\}$: $$ B(\theta)=(\theta_{n+1})_{n\in\mathbb{N}}, $$ $$ F(\theta)=(\theta_{n-1})_{n\in\mathbb{N}},\qquad \theta_{-n}=0\ \text{for all }n\in\mathbb{N}. $$ These operators are well defined and continuous when $\alpha$ is weakly-stable, meaning $$ \limsup_{n\in\mathbb{N}}\frac{\alpha_{n+1}}{\alpha_n}<\infty, $$ and the paper works under the stronger assumption that $\alpha$ is stable [2404.00574]. A Toeplitz operator associated with $\theta=(\theta_n)_{n\in\mathbb{N}_0}$ is also introduced: $$ \widehat{T}_\theta e_n=(0,\ldots,0,\theta_0,\theta_1,\theta_2,\ldots)=\sum_{j=n}^\infty \theta_{j-n}e_j. $$ The decisive identities are $$ F^n(\theta)=\widehat{T}_\theta(e_{n+1}),\qquad B^n(\theta)=H_\theta(e_{n+1}). $$ They identify the iterates of the forward shift with Toeplitz columns and the iterates of the backward shift with Hankel columns [2404.00574]. For a continuous operator $T$ on a Fréchet space $E$, the $n$-th Cesàro mean is $$ T^{[n]}:=\frac{1}{n}\sum_{m=1}^n T^m. $$ The operator is mean ergodic if $\lim_{n\to\infty}T^{[n]}x$ exists in $E$ for every $x\in E$, and it is Cesàro bounded if $\{T^{[n]}:n\in\mathbb{N}\}$ is equicontinuous in $L(E)$. On a Montel Fréchet space, Kalmes–Santacreu’s criterion gives that $T$ is mean ergodic if and only if it is Cesàro bounded and $$ \lim_{n\to\infty}\frac{1}{n}T^n x=0 $$ for every $x\in E$ [2404.00574]. Using the identities above and continuity of the Toeplitz and Hankel actions, the paper proves $$ \lim_{n\to\infty}F^{[n]}(\theta)=0,\qquad \lim_{n\to\infty}B^{[n]}(\theta)=0 $$ for all $\theta\in\Lambda_r(\alpha)$, $r=1,\infty$. Concretely, $$ \lim_{n\to\infty}\frac{1}{n}\sum_{m=1}^n H_\theta(e_{m+1})=0. $$ Accordingly, the forward and backward shifts on $\Lambda_r(\alpha)$, for stable $\alpha$, are mean ergodic and Cesàro bounded [2404.00574]. This gives the first modern meaning of “ergodic Hankel operators”: not that $H_\theta$ itself is shown to be mean ergodic under iteration, but that Hankel columns realize the iterates of an ergodic shift. A common misconception is therefore to equate the result with a theorem about the Cesàro means of $H_\theta^n$. The paper explicitly does not analyze mean ergodicity or Cesàro boundedness of the powers $H_\theta^n$ themselves [2404.00574]. ## 4. Ergodic families on the half-line A distinct theory is developed for bounded self-adjoint Hankel operators on $L^2(0,\infty)$ that form ergodic families under dilations [2509.25010]. In logarithmic coordinates, these become covariant families on $L^2(\mathbb{R})$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with an ergodic group $\{T_a\}$, where $a\in\mathbb{R}$ in the continuous case or $a\in\mathbb{Z}$ in the discrete case. Let $$ (U_\tau f)(\xi)=f(\xi+\tau),\qquad \xi\in\mathbb{R}, $$ for a fixed period $\tau>0$. A measurable family of bounded operators $K_\omega$ on $L^2(\mathbb{R})$ is ergodic with period $\tau$ if $$ K_{T_a\omega}U_{\tau a}=U_{\tau a}K_\omega,\qquad \forall a. $$ In $L^2(0,\infty)$ this is equivalent to $$ H_{T_a\omega}D_{\tau a}=D_{\tau a}H_\omega, $$ where $$ (D_\tau f)(t)=e^{\tau/2}f(e^\tau t). $$ Thus the natural ergodic symmetry for half-line Hankel operators is multiplicative in the original variable and translational after logarithmic transplantation [2509.25010]. If one writes $$ h_\omega(t)=\frac{P_\omega(\log t)}{t}, $$ then ergodicity is equivalent to $$ P_{T_a\omega}(\xi)=P_\omega(\xi+\tau a),\qquad \xi\in\mathbb{R}. $$ Under the standing bound $h_\omega(t)\le C_h/t$ almost surely, one has $$ \operatorname*{ess\,sup}_\xi |P_\omega(\xi)|\le C_h $$ almost surely [2509.25010]. The positive case has an especially rigid structure. A bounded Hankel operator is positive if and only if $$ h(t)=\int_0^\infty e^{-tx}\,d\sigma(x),\qquad t>0, $$ with $\sigma(\{0\})=0$ and the Carleson condition $$ \sigma((0,a))\le C_\sigma a,\qquad \forall a>0. $$ For positive ergodic Hankel operators there exists an ergodic family of Borel measures $\Sigma_\omega$ on $\mathbb{R}$ such that $$ h_\omega(t)=\int_{\mathbb{R}} e^{-t e^{-\xi}}e^{-\xi}\,d\Sigma_\omega(\xi),\qquad t>0, $$ and $$ K_\omega(x,y)=\int_{\mathbb{R}} \beta(x-\xi)\beta(y-\xi)\,d\Sigma_\omega(\xi), $$ with $$ \beta(\xi)=e^{-e^\xi}e^{\xi/2}. $$ The covariance condition becomes $$ \Sigma_{T_a\omega}(\Delta)=\Sigma_\omega(\Delta+\tau a) $$ for Borel $\Delta\subset\mathbb{R}$, together with the uniform local boundedness estimate $$ \Sigma_\omega((x-1,x+1))\le C_\Sigma $$ uniformly in $x$ and $\omega$ [2509.25010]. This framework produces a spectral theory analogous in several respects to that of ergodic Schrödinger operators, but with genuinely Hankel-specific geometry arising from the $x+y$ kernel dependence and the dilation covariance. ## 5. Integrated Density of States and structural spectral results The Integrated Density of States is the central invariant of the ergodic half-line theory. For a Borel set $\Delta\subset\mathbb{R}$ separated from $0$, the IDS measure $\nu$ is defined by $$ \nu(\Delta)=\mathbb{E}\{\chi_\Delta(K_\omega)(0,0)\} $$ in the continuous case, and by $$ \nu(\Delta)=\mathbb{E}\left\{\frac{1}{\tau}\int_0^\tau \chi_\Delta(K_\omega)(x,x)\,dx\right\} $$ in the discrete case, with $\nu(\{0\})=0$ by convention [2509.25010]. Finite-volume approximations yield self-averaging. Almost surely, $$ \nu(\Delta)=\lim_{M\to\infty}\frac{1}{2M}\operatorname{Tr}\bigl(\chi_M\chi_\Delta(K_\omega)\chi_M\bigr), $$ where $\chi_M=\chi_{(-M,M)}$. Equivalently, for continuous $\phi$ compactly supported in $\mathbb{R}\setminus\{0\}$, $$ \lim_{M\to\infty}\frac{1}{2M}\operatorname{Tr}\bigl(\chi_M\phi(K_\omega)\chi_M\bigr)=\int \phi(\lambda)\,d\nu(\lambda). $$ The support of $\nu$ coincides with the almost-sure spectrum of $K_\omega$ [2509.25010]. Two Szegő-type formulations are established. First, $$ \lim_{M\to\infty}\frac{1}{2M}\operatorname{Tr}\phi(\chi_MK_\omega\chi_M)=\int \phi(\lambda)\,d\nu(\lambda). $$ Second, in the positive case, if $K_\omega^{(M)}$ is obtained by restricting $\Sigma_\omega$ to $(-M,M)$, then $$ \lim_{M\to\infty}\frac{1}{2M}\operatorname{Tr}\phi(K_\omega^{(M)})=\int \phi(\lambda)\,d\nu(\lambda) $$ for continuous $\phi$ supported away from $0$ [2509.25010]. Several general properties follow. - The spectrum and its multiplicity are non-random [2509.25010]. - The second moment of the IDS is finite in general, and the first moment is finite if $K_\omega$ is positive [2509.25010]. - If $K_\omega$ is positive, then $\nu$ has no atoms [2509.25010]. A particularly notable theorem identifies the total mass of the positive IDS. If $K_\omega$ is positive and bounded with associated $\Sigma_\omega$, and $$ D(\operatorname{supp}\Sigma_\omega)=\frac{1}{\tau}\mathbb{E}\{\#(\operatorname{supp}\Sigma_\omega\cap[0,\tau))\}, $$ then $$ \nu(\mathbb{R}_+)=D(\operatorname{supp}\Sigma_\omega). $$ In the terminology of the paper, $\nu(\mathbb{R}_+)$ is the mean number of spectral bands per unit period, counting flat and non-flat bands but ignoring $0$; it equals the mean number of atoms of $\Sigma_\omega$ per period [2509.25010]. The Carleman operator $h(t)=1/t$ provides an explicit benchmark. In that case, the IDS satisfies $$ \int_\lambda^\infty d\nu_C(x)=\frac{2}{\pi}\operatorname{arcsech}(\lambda/\pi) =\frac{2}{\pi}\log\left[\frac{\pi}{\lambda}+\sqrt{\left(\frac{\pi}{\lambda}\right)^2-1}\right],\qquad 0<\lambda<\pi, $$ with density $$ \nu_C(\lambda)=\frac{\chi_{(0,\pi)}(\lambda)}{\lambda\sqrt{\pi^2-\lambda^2}}. $$ In particular, $$ \nu_C((\lambda,\infty))=\frac{2}{\pi}\log(1/\lambda)+O(1),\qquad \lambda\to 0^+. $$ This explicit asymptotic makes precise the logarithmic singularity at zero in the model case [2509.25010]. ## 6. Periodic and random ergodic Hankel operators The periodic theory begins with $\tau$-periodic kernels of the form $$ h(t)=\frac{P(\log t)}{t}, $$ where $P$ is $\tau$-periodic. In $L^2(\mathbb{R})$, the corresponding operator commutes with $U_\tau$, and this admits a Floquet–Bloch decomposition $$ UAU^*=\int_{(-\pi/\tau,\pi/\tau)}^\oplus A(k)\,dk. $$ For smooth periodic Hankel operators satisfying $$ \sum |\widetilde{P}_m|(1+|m|)^{1/2}<\infty, $$ the fibers $K(k)$ are trace class and have matrix entries $$ [K(k)]_{n,m}=B\!\left(\frac12-i\frac{2\pi}{\tau}n-ik,\frac12+i\frac{2\pi}{\tau}m+ik\right)\widetilde{P}_{n-m} =\overline{\gamma_n(k)}\,\widetilde{\Sigma}_{n-m}\,\gamma_m(k), $$ where $\gamma_n(k)=\Gamma(1/2+(2\pi/\tau)n+k)$ [2509.25010]. The nonzero spectrum is organized by analytic band functions $\{E_n(k)\}$. The IDS in the periodic case has no singular continuous part. More precisely, $$ \nu^{ac}=\frac1\tau\sum_n \nu_n, $$ where each $\nu_n$ is a purely absolutely continuous probability measure supported on a closed interval $\sigma_n=E_n((0,\pi/\tau))$, and $$ \nu^{pp}=\frac1\tau\sum_n (\delta_{\lambda_n}+\delta_{-\lambda_n}), $$ where $\lambda_n>0$ are flat band energies. For positive periodic Hankel operators, there are no flat bands, so $\nu^{pp}=0$ [2509.25010]. Two explicit periodic examples illustrate the range of behaviors. - If $\Sigma=\sum_{n\in\mathbb{Z}}\delta_{\tau n}$, then $K(k)$ is rank one and its nonzero eigenvalue is $$ E_0(k)=\frac1\tau\sum_{n\in\mathbb{Z}} \pi\,\operatorname{sech}\!\left(\pi\left(\frac{2\pi}{\tau}n+k\right)\right). $$ The resulting single band is absolutely continuous and has square-root edge singularities [2509.25010]. - If $$ \Sigma=\sum_{n\in\mathbb{Z}}(\delta_{\tau n}-\delta_{\tau/2+\tau n}), $$ then $K(k)$ is rank two with constant eigenvalues $\pm E_*$, and the IDS is pure point with two atoms at $\pm E_*$ [2509.25010]. The random Kronig–Penney–Hankel (rKPH) model supplies the random counterpart. It is defined on $L^2(\mathbb{R})$ by $$ K_\omega=\sum_{n\in\mathbb{Z}}\kappa_\omega(n)\,|\psi_n\rangle\langle\psi_n|, $$ where $\psi_n(\xi)=\beta(\xi-\tau n)$, $\|\psi_n\|^2=1/2$, and $\{\kappa_\omega(n)\}$ are i.i.d. random variables with compact support $$ \operatorname{supp}\mu_0\subset [\kappa_{\min},\kappa_{\max}]\subset(0,\infty). $$ Its associated ergodic measure is $$ \Sigma_\omega=\sum_{n\in\mathbb{Z}}\kappa_\omega(n)\delta_{\tau n} $$ [2509.25010]. Let $$ \phi(k)=\sum_{n\in\mathbb{Z}}\langle \psi_n,\psi_0\rangle e^{ink} =\frac{\pi}{\tau}\sum_{m\in\mathbb{Z}}\operatorname{sech}\!\left(\frac{\pi(k-2\pi m)}{\tau}\right) =E_0(k/\tau), $$ and let $E_{\min}0$, then $$ \lim_{\delta\to 0+}\frac{\log(-\log \nu((\sigma_{\max}-\delta,\sigma_{\max}]))}{\log\delta}=-\frac12, $$ $$ \lim_{\delta\to 0+}\frac{\log(-\log \nu([\sigma_{\min},\sigma_{\min}+\delta)))}{\log\delta}=-\frac12. $$ The exponent is the one-dimensional Lifshitz exponent $1/2$ [2509.25010]. - **Wegner bound**: if $\mu_0$ has bounded density $\rho\le \rho_{\max}$, then $\nu$ is absolutely continuous with bounded density and for every interval $I$, $$ \nu(I)\le C_W |I|. $$ In finite volume, this corresponds to $$ \mathbb{E}[\operatorname{Tr}\chi_I(K_{\omega,M})]\le \text{const}\cdot |I|\cdot (2M) $$ [2509.25010]. - **Anderson localization**: if $\mu_0$ is uniformly Hölder continuous of order $\gamma$ on $[\kappa_{\min},\kappa_{\max}]$, then there exists $\tau_0>0$ such that for $\tau\ge \tau_0$, the operator $K_\omega$ has pure point spectrum almost surely, with exponentially decaying eigenfunctions [2509.25010]. These results establish an operator-theoretic analogue of periodic and random Schrödinger theory in a genuinely Hankel setting. ## 7. Conceptual distinctions, limitations, and open directions The recent literature makes clear that “ergodic Hankel operators” has two non-equivalent uses. The sequence-space literature analyzes Hankel operators as maps between Köthe or power series spaces and derives ergodic conclusions for the backward and forward shifts through the identities $$ B^n(\theta)=H_\theta(e_{n+1}),\qquad F^n(\theta)=\widehat{T}_\theta(e_{n+1}). $$ Its main ergodic conclusion is the vanishing of Cesàro averages of shift iterates, equivalently the Cesàro averages of Hankel columns, not the mean ergodicity of the operator powers $H_\theta^n$ [2404.00574]. By contrast, the half-line theory introduces ergodic families of Hankel operators in the same structural sense in which one speaks of ergodic Schrödinger operators: a probability space, a covariant group action, self-averaging spectral data, an IDS, and periodic/random models [2509.25010]. A plausible implication is that the latter framework will likely become the standard meaning of the term in spectral theory, while the former remains indispensable for locally convex and function-space operator theory. Each framework also has explicit limitations. In the Köthe-space setting, the paper does not provide criteria for mean ergodicity or Cesàro boundedness of $H_\theta$ itself under iteration, and the sufficient continuity and compactness conditions depend on stability assumptions and inequalities relating Köthe weights to exponential weights [2404.00574]. In the half-line setting, the theory is currently formulated for bounded operators, with positivity playing a particularly strong role through the Laplace-transform representation, the Carleson condition, and continuity of the IDS [2509.25010]. The open problems stated in the spectral theory of ergodic Hankel operators are correspondingly broad. They include quasi- or almost-periodic Hankel operators, unbounded ergodic Hankel operators, detailed IDS asymptotics at zero, coexistence and interaction of flat and non-flat bands, and the development of Hankel analogues of transfer matrices, Lyapunov exponents, and $m$-functions [2509.25010]. From the perspective of the Köthe-space theory, a natural unresolved direction is a direct study of ergodicity for the powers or Cesàro means of Hankel operators themselves rather than only for the shifts they encode [2404.00574]. Taken together, these two strands show that ergodic Hankel operators now occupy a dual position in operator theory: as structured maps between Köthe-type spaces whose columns exhibit ergodic averaging through shift dynamics, and as a new class of ergodic self-adjoint Hankel families with a developed IDS theory, periodic band structure, and random localization phenomena [2404.00574; 2509.25010]. Source: https://www.emergentmind.com/topics/ergodic-hankel-operators