arXiv subjects

Lian Haeming

Publications and source records attributed to Lian Haeming.

3 recordsLinked to original sources

arXiv2024

Quasiballistic Transport for Discrete One-Dimensional Quasiperidic Schr\"odinger Operators

We obtain (up to logarithmic scaling) the power-law lower bound $M_{p}(T_{k})\gtrsim T_{k}^{(1-\delta)p}$ on a subsequence $T_{k}\rightarrow\infty$, uniformly across $p>0$, for discrete one-dimensional quasiperiodic Schr\"odinger operators with frequencies satisfying $\beta(\alpha)>\frac{3}{\delta}\min_{\sigma}\gamma$. We achieve this by obtaining a quantitative ballistic lower bound for the Abel-averaged time evolution of general periodic Schr\"odinger operators in terms of the bandwidths. A similar result without uniformity, which assumes $\beta(\alpha)>\frac{C}{\delta}\min_{\sigma}\gamma$, was obtained earlier by Jitomirskaya and Zhang, for an implicit constant $C<\infty$.

math.SP↗

arXiv2024

On the Real Eigenvalues of the Non-Hermitian Anderson Model

We study the non-Hermitian Anderson model on the ring. We provide the exact rate of decay of the sensitivity of the eigenvalues to the non-Hermiticity parameter $g$, on the logarithmic scale, as the Lyapunov exponent minus the non-Hermiticity parameter. Namely, for $0 < g < \gamma(\lambda_{0})$ we show that $-\frac{1}{n}\log|\lambda_{g}-\lambda_{0}|\sim \gamma(\lambda_{0})-g$ and that the eigenvalue remains real for all such $g$. This provides an alternative proof to that of Goldsheid and Sodin that the perturbed eigenvalue remains real and specifies the exact rate at which the eigenvalue is exponentially close to the unperturbed eigenvalue.

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arXiv2022

On the Bandwidths of Periodic Approximations to Discrete Schr\"odinger Operators

We study how the spectral properties of ergodic Schr\"odinger operators are reflected in the asymptotic properties of its periodic approximation as the period tends to infinity. The first property we address is the asymptotics of the bandwidths on the logarithmic scale, which quantifies the sensitivity of the finite volume restriction to the boundary conditions. We show that the bandwidths can always be bounded from below in terms of the Lyapunov exponent. Under an additional assumption satisfied by i.i.d potentials, we also prove a matching upper bound. Finally, we provide an additional assumption which is also satisfied in the i.i.d case, under which the corresponding eigenvectors are exponentially localised with a localisation centre independent of the Floquet number.

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