arXiv subjects

Zhiyan Zhao

Publications and source records attributed to Zhiyan Zhao.

15 recordsLinked to original sources

arXiv2026

Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential

We consider the discrete Schrödinger operator $H = -Δ+ V$ on $\ell^2(\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in\mathbb{N}^*$, where $Δ$ is the standard discrete Laplacian and $V_n = o(|n|^{-1})$ as $|n| \to \infty$. %We prove the absence of singular continuous spectrum for $H$. For the unitary evolution $e^{-i tH}$, we prove that it exhibits ballistic transport in the sense that, for any $r > 0$, the weighted $\ell^2-$norm $$\|e^{-i tH}u\|_r:=\left(\sum_{n\in\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\right)^\frac12 $$ grows at rate $\simeq t^r$ as $t\to \infty$, provided that the initial state $u$ is in the absolutely continuous subspace and satisfies $\|u\|_r<\infty$. The proof relies on commutator methods and Mourre estimate, which yields quantitative lower bounds on transport for operators with purely absolutely continuous spectrum over appropriate spectral intervals.

math-ph↗

arXiv2026

Long-time stability for nonlinear Maryland models

For the $d-$dimensional nonlinear Maryland model \begin{equation}\label{eq-abs} \ri\partial_t q_n=\tanπ(n\cdot\varpi+x)q_n+ε(Δq)_n+|q_n|^2q_n,\quad n\in{\Z^d}, \end{equation} with $d\in\N^*$, $ε\in \R$ and $\varpi\in\R^d$ satisfying a suitable Diophantine condition, we establish polynomial long-time stability of polynomially weighted $\ell^2$-norm $$\|q(t)\|_s:=\left(\sum_{n\in{\Z^d}}|q_n|^2 (1+|n|^2)^{s}\right)^{\frac{1}{2}},\quad s>0. $$ More precisely, given any $M_*\in\N^*$, for phase parameters $x$ belonging to an almost full-measure subset of $\R/\Z$, if $|ε|$ is sufficiently small, then solutions $q(t)$ of Eq. (\ref{eq-abs}) with high-order weighted $\ell^2$-norm $\|q(0)\|_s$ of sufficiently small size $\varepsilon$ satisfy $$\|q(t)\|_s=\CO(\varepsilon),\quad \forall \ |t|\leq ε^{-1}\varepsilon^{-M_*}. $$ The proof relies on a Birkhoff normal form procedure.

math-ph↗

arXiv2026

Generalized Reducibility and Growth of Sobolev Norms

We introduce the concept of {\it generalized reducibility}, which provides a flexible framework for analyzing the long-time behavior of solutions to quadratic quantum Hamiltonians. As an application of this notion, for many prescribed sub-exponential growth rates $f(t)$, either monotone or oscillatory, we explicitly construct time-decaying perturbations of the one-dimensional quantum harmonic oscillator such that the Sobolev norms of solutions grow at the rate $f(t)$.

math.AP↗

arXiv2025

Local rigidity of group actions of isometries on compact Riemannian manifolds

In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$. We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition. Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis's theorem on group actions satisfying Kazhdan's property (T).

math.DS↗

arXiv2025

Symplectic Normal Form and Growth of Sobolev Norm

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schrödinger representations. Each pattern of Sobolev norm behavior corresponds to a specific $n-$dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2. When applied to periodically or quasi-periodically forced $n-$dimensional quantum harmonic oscillators, we identify novel growth rates for the $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{λst}$ (with $λ>0$) and $t^{(2n-1)s}+ ιt^{2ns}$ (with $ι\geq 0$). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3. As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the ``fastest" growth, as articulated in Theorem 1.4.

math.AP↗

arXiv2024

Local rigidity of actions of isometries on compact real analytic Riemannian manifolds

In this article, we consider analytic perturbations of isometries of an analytic Riemannian manifold M. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on M to the same group of isometry it is a perturbation of. Our result relies on a "Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffemorphisms of the circle that are small perturbations of rotations.

math.DS↗

arXiv2024

Almost reducibility and oscillatory growth of Sobolev norms

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-{\rm i}\partial_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the $\CH^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal H}^s-$norm of the solution is $o(t^s)$ as $t\to\infty$ but arbitrarily ``close" to $t^s$ in an oscillatory way.

math.AP↗

arXiv2022

FocalClick: Towards Practical Interactive Image Segmentation

Interactive segmentation allows users to extract target masks by making positive/negative clicks. Although explored by many previous works, there is still a gap between academic approaches and industrial needs: first, existing models are not efficient enough to work on low power devices; second, they perform poorly when used to refine preexisting masks as they could not avoid destroying the correct part. FocalClick solves both issues at once by predicting and updating the mask in localized areas. For higher efficiency, we decompose the slow prediction on the entire image into two fast inferences on small crops: a coarse segmentation on the Target Crop, and a local refinement on the Focus Crop. To make the model work with preexisting masks, we formulate a sub-task termed Interactive Mask Correction, and propose Progressive Merge as the solution. Progressive Merge exploits morphological information to decide where to preserve and where to update, enabling users to refine any preexisting mask effectively. FocalClick achieves competitive results against SOTA methods with significantly smaller FLOPs. It also shows significant superiority when making corrections on preexisting masks. Code and data will be released at github.com/XavierCHEN34/ClickSEG

cs.CV↗

arXiv2022

Geometry of hyperbolic Cauchy-Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of CR singularities stable under perturbation~: {\it elliptic} and {\it hyperbolic}. Elliptic case was studied by Moser-Webster \cite{moser-webster} who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to {\it normal form} from which lots of geometric features can be read off. In this article we focus on perturbations of {\it hyperbolic} quadrics. As was shown by Moser-Webster \cite{moser-webster}, such a surface can be transformed to a formal {\it normal form} by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a {\it non-degenerate} real analytic surface $M$ in $(\mathbb{C}^2,0)$ having a {\it hyperbolic} CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting $M$ along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $\{τ_1,τ_2\}$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $\{z_1z_2=const\}$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

math.CV↗

arXiv2022

Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree $2$ in $(x,-{\rm i}\partial_x)$, with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the $t^{2s}-$polynomial growth of ${\mathcal H}^s-$norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.

math.AP↗

arXiv2021

1-d Quantum Harmonic Oscillator with Time Quasi-periodic Quadratic Perturbation: Reducibility and Growth of Sobolev Norms

For a family of 1-d quantum harmonic oscillator with a perturbation which is $C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$ and $-{\rm i}\partial_x$ with coefficients quasi-periodically depending on time $t$, we show the reducibility (i.e., conjugation to time-independent) for a.e. $E$. As an application of reducibility, we describe the behaviors of solution in Sobolev space: -- Boundedness w.r.t. $t$ is always true for "most" $E\in{\mathcal I}$. -- For "generic" time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. $t$ occur for $E$ in a "small" part of ${\mathcal I}$. Concrete examples are given for which the growths of Sobolev norm do occur.

math.AP↗

arXiv2019

Dispersive estimate for quasi-periodic Schrödinger operators on 1-$d$ lattices

Consider the one-dimensional discrete Schrödinger operator $H_θ$: $$(H_θ q)_n=-(q_{n+1}+q_{n-1})+ V(θ+nω) q_n \ , \quad n\in Z \ ,$$ with $ω\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=( R/2πZ)^d$. For $V$ sufficiently small, we prove the dispersive estimate: {for every $ϕ\in\ell^1( Z)$,} $$ \| e^{-{\rm i}tH_θ}ϕ\|_{\ell^\infty} \leq K_0 \frac{ |\ln\varepsilon_0|^{a(\ln\ln(2+\langle t\rangle))^2 d}} {\langle t\rangle^{\frac13}} \|ϕ\|_{\ell^1} \ , \quad \langle t \rangle:=\sqrt{1+t^2} \ ,$$ {with $a$ and $K_0$ two absolute constants} and $\varepsilon_0$ an analytic norm of $V$. The estimate holds for every $θ\in T^d$.

math-ph↗

arXiv2017

Asymptotics of spectral gaps of quasi-periodic Schrödinger operators

For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schödinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's conjecture \cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al \cite{BDGL, DGL1, dgsv, Go} and Kotani \cite{Kot97}.

math.DS↗

arXiv2016

Ballistic Transport and Absolute Continuity of One-Frequency Schrödinger Operators

For the solution $u(t)$ to the discrete Schrödinger equation $${\rm i}\frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(θ+ nα)u_n(t), \quad n\in\Z,$$ with $α\in\R\setminus\Q$ and $V\in C^ω(\T,\R)$, we consider the growth rate with $t$ of its diffusion norm $\langle u(t)\rangle_{p}:=\left(\sum_{n\in\Z}(n^{p}+1) |u_n(t)|^2\right)^\frac12$, and the (non-averaged) transport exponents $$β_u^{+}(p) := \limsup_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}, \quad β_u^{-}(p):= \liminf_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}.$$ We will show that, if the corresponding Schrödinger operator has purely absolutely continuous spectrum, then $β_{u}^{\pm}(p)=1$, provided that $u(0)$ is well localized.

math.DS↗

arXiv2016

Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

For the solution $q(t)=(q_n(t))_{n\in\mathbb Z}$ to one-dimensional discrete Schrödinger equation $${\rm i}\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(θ+nω) q_n, \quad n\in\mathbb Z,$$ with $ω\in\mathbb R^d$ Diophantine, and $V$ a small real-analytic function on $\mathbb T^d$, we consider the growth rate of the diffusion norm $\|q(t)\|_{D}:=\left(\sum_{n}n^2|q_n(t)|^2\right)^{\frac12}$ for any non-zero $q(0)$ with $\|q(0)\|_{D}<\infty$. We prove that $\|q(t)\|_{D}$ grows {\it linearly} with the time $t$ for any $θ\in\mathbb T^d$ if $V$ is sufficiently small.

math-ph↗