Lyapunov exponent for quantum graphs that are elements of a subshift of finite type
Oleg Safronov
TL;DR
The paper studies a Schrödinger operator on quantum graphs whose edge multiplicities are given by a subshift of finite type, encoding spectral data through a locally constant SL(2,R) cocycle and its Lyapunov exponent. Under natural dynamical hypotheses (local product structure, full support, and a fixed point), the zero set of the Lyapunov exponent is finite; in a broad admissible regime the exponent is positive for almost all energies except possibly a few special values, and the zeros are precisely linked to the spectra of periodic operators. Techniques blend dynamical systems (holonomies, su-states, local product structure) with spectral theory and periodic approximations, yielding a concrete description ${\frak L}(A,\mu)=\bigcap_{p\in Per(T)}\{k:\ k^2\in\sigma(p)\}$ of the zero set. The results illuminate when quantum graph models exhibit localization-like behavior and demonstrate the power of periodic reduction in this setting.
Abstract
We consider the Schrödinger operator on the quantum graph whose edges connect the points of ${\Bbb Z}$. The numbers of the edges connecting two consecutive points $n$ and $n+1$ are read along the orbits of a shift of finite type. We prove that the Lyapunov exponent is potitive for energies $E$ that do not belong to a discrete subset of $[0,\infty)$. The number of points $E$ of this subset in $[(π(j-1))^2, (πj)^2]$ is the same for all $j\in {\Bbb N}$.