arXiv subjects

Yingdu Dong

Publications and source records attributed to Yingdu Dong.

4 recordsLinked to original sources

arXiv2026

Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential

Let \[ A_ω=-\partial_x^2+ρ\dot B_x(ω), \qquad ρ\ne0, \] be the Dirichlet Schrödinger operator on $(0,π)$ with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here $\dot B_x$ denotes the distributional derivative of Brownian motion with respect to the spatial variable $x$. For $κ\ne0$, we consider the cubic nonlinear Schrödinger equation \[ i u_t=A_ωu+κ|u|^2u. \] We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set $J$ of cardinality $b$, we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant $b$-tori. The family is parametrized by a Cantor subset of $[ν,2ν]^b$ whose relative measure tends to one as $ν\to0$.

math.PR↗

arXiv2026

High-energy asymptotics for finite-interval Schrödinger operators with Gaussian white-noise potential

We study the one-dimensional Schrödinger operator on a fixed interval with Gaussian white-noise potential, \[ H_ω=-\frac{\dd^2}{\dd x^2}+ρ\dot B_x(ω), \] under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm--Liouville operators with distributional potentials. Let $λ_n$ be the Dirichlet eigenvalues, $λ_n^+=\max\{λ_n,0\}$, and $k_n=\sqrt{λ_n^+}$. For every finite $p$, we prove the high-energy expansion \[ k_n=\frac{nπ}{L} +\fracρ{nπ}\int_0^L \sin^2\left(\frac{nπs}{L}\right)\,\dd B_s +O_{L^p(Ω)}(n^{-2}). \] Consequently, almost surely, $λ_n>0$ for all sufficiently large $n$ and, for every $\varepsilon>0$, \[ k_n=\frac{nπ}{L}+O(n^{-1+\varepsilon}). \] We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction $φ_n$, with a fixed sign convention, \[ \sup_{0\le x\le L} \left|φ_n(x)-\sqrt{\frac{2}{L}}\sin(k_n x)\right| =O(n^{-1+\varepsilon}) \] almost surely. The proofs use stochastic Prüfer coordinates, stochastic Volterra expansions, the Burkholder--Davis--Gundy inequality, and a Borel--Cantelli argument. The estimates provide a first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials.

math.SP↗

arXiv2026

Anderson localization for 1-d quasi-periodic Schrödinger operators with degenerate weights

We establish Anderson localization for 1-d discrete Schrödinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and quasi-periodic. Operators of this kind originate from distinct mathematical physics problems, which include the Frenkel-Kontorova model with impurities, the discretization of singular Sturm-Liouville operators, and the Fisher-KPP lattice equation in heterogeneous media.

math-ph↗

arXiv2022

The existence of invariant curves of a kind of almost periodic twist mappings

In this paper, we are concerned with the existence of invariant curves of the planar twist mappings where the perturbations are almost periodic. As an application, the existence of almost periodic solutions and the boundedness of all solutions for pendulum-type equations with an almost periodic external force are proved.

math.DS↗