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Yunfeng Shi

Publications and source records attributed to Yunfeng Shi.

At least 37 records · Page 2Linked to original sources
arXiv2024

On the spectrum of quasi-periodic Schrödinger operators on $\mathbb{Z}^d$ with $C^2$-cosine type potentials

In this paper, we establish the Anderson localization, strong dynamical localization and the $(\frac 12-)$-Hölder continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schrödinger operators with asymmetric $C^2$-cosine type potentials. We extend both the iteration scheme of \cite{CSZ23a} and the interlacing method of \cite{FV21} to handle asymmetric Rellich functions with collapsed gaps.

math-ph↗
arXiv2024

Diophantine estimates on shifts of trigonometric polynomials on $\mathbb{T}^d$

We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schrödinger and the quasi-periodic wave operators. They have applications to the nonlinear quasi-periodic Schrödinger equations (NLS) and the nonlinear quasi-periodic wave equations (NLW). One could now, for example, extend the result of Bourgain (Geom. Funct. Anal. 17(3): 682-706, 2007) to the nonlinear setting.

math-ph↗
arXiv2023

Diagonalization in a quantum kicked rotor model with non-analytic potential

In this paper we study the lattice quasi-periodic operators with power-law long-range hopping and meromorphic monotone potentials, and diagonalize the operators via a Nash-Moser iteration scheme. As applications, we obtain uniform power-law localization, uniform dynamical localization and Lipschitz continuity of the integrated density of states (IDS) for such operators. Our main motivation comes from investigating quantum suppression of chaos in a quantum kicked rotor model with non-analytical potential.

math-ph↗
arXiv2023

Localization and Regularity of the Integrated Density of States for Schrödinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential

In this paper, we study the multidimensional lattice Schrödinger operators with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative Green's function estimates, the arithmetic version of Anderson (and dynamical) localization, and the finite volume version of $(\frac 12-)$-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fröhlich-Spencer-Wittwer [\textit{Comm. Math. Phys.} 132 (1990): 5--25] to the higher lattice dimensions. We resolve the level crossing issue on eigenvalues parameterizations in the case of both higher lattice dimension and $C^2$ regular potential.

math-ph↗
arXiv2023

First-principles molten salt phase diagrams through thermodynamic integration

Precise prediction of phase diagrams in molecular dynamics (MD) simulations is challenging due to the simultaneous need for long time scales, large length scales and accurate interatomic potentials. We show that thermodynamic integration (TI) from low-cost force fields to neural network potentials (NNPs) trained using density-functional theory (DFT) enables rapid first-principles prediction of the solid-liquid phase boundary in the model salt NaCl. We use this technique to compare the accuracy of several DFT exchange-correlation functionals for predicting the NaCl phase boundary, and find that the inclusion of dispersion interactions is critical to obtain good agreement with experiment. Importantly, our approach introduces a method to predict solid-liquid phase boundaries for any material at an ab-initio level of accuracy, with the majority of the computational cost at the level of classical potentials.

cond-mat.mtrl-sci↗
arXiv2023

Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some $d$-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of \cite{Sar82, Pos83, Cra83, BLS83} are generalized to the power-law hopping case.

math-ph↗
arXiv2023

Diffusion bound for the nonlinear Anderson model

In this paper, we prove the power-law in time upper bound for the diffusion of a 1D discrete nonlinear Anderson model. We remove completely the decaying condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud. 163: 21--42, 2007.). This gives a resolution to the problem of Bourgain (Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear disordered systems. The proof uses a novel ``norm'' based on tame property of the Hamiltonian.

math.DS↗
arXiv2023

Localization for random coupled harmonic oscillators on $\mathbb{Z}^d$

In this paper we consider the localization properties of coupled harmonic oscillators in random media. Each of these oscillators is restricted to the lattice $\mathbb{Z}^d$. We show that for most states and an arbitrary choice of the random media, the long time localization for the coupled system holds in a time scale larger than the polynomial one.

math.DS↗
arXiv2022

Quantitative Green's Function Estimates for Lattice Quasi-periodic Schrödinger Operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schrödinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

math-ph↗
arXiv2021

Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

In this paper, we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of point spectrum for the $1D$ exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has the positive Lebesgue measure.

math.SP↗
arXiv2021

Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on $\mathbb{R}^d$

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schrödinger operator $H=-Δ+ λV(x),$ where $V$ is a real analytic quasi-periodic function and $λ>0$. We first show that $H$ has no eigenvalues in \textit{low energy region}. We also provide in \textit{low energy region} the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies.

math.SP↗
arXiv2020

On linear stability of KAM tori via the Craig-Wayne-Bourgain method

In this paper, we prove the Melnikov's persistency theorem by combining the traditional Kolmogorov-Arnold-Moser (KAM) technique and the Craig-Wayne-Bourgain (CWB) method. The aim of this paper is twofold. One is to establish the linear stability of the perturbed invariant tori by using the CWB method without the second Melnikov condition. The other one is to illustrate the CWB method in detail and make the CWB method more accessible.

math.DS↗
arXiv2019

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

In this paper, we study the following nonlinear Schrödinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+εf(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2π\mathbb{Z}, \end{eqnarray} where $V*$ is the Fourier multiplier defined by $\widehat{(V* u})_n=V_{n}\widehat{u}_n, V_n\in[-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\leq|ε|\ll1$, there is some $(V_n)_{n\in\mathbb{Z}}$ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \cite{BJFA2005} and Cong-Liu-Shi-Yuan \cite{CLSY} to the case that the nonlinear perturbation depends explicitly on the space variable $x$. The main difficulty here is the absence of zero momentum of the equation.

math.DS↗