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Yilin Song

Publications and source records attributed to Yilin Song.

At least 19 recordsLinked to original sources
arXiv2026

Scattering Theory For 3D Cubic Damped Magnetic Schr\"odinger Equation

We consider the three-dimensional defocusing cubic nonlinear Schr\"odinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schr\"odinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

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arXiv2026

Generic ill-posedness for Schr\"odinger equation with power-type nonlinearity on $\mathbb{S}^2$

In this article, we investigate the local well-posedness of the nonlinear Schr\"odinger equation on the two-dimensional sphere $\mathbb{S}^2$: \begin{align*} i\partial_tu+\Delta_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to be gauge-invariant. More presicely, there exists a function $V\in C^\infty(\mathbb{C},\mathbb{R})$ such that $F=\frac{\partial V}{\partial \bar{z}}$. Moreover, $V(z)$ obeys \begin{equation}\label{H-11} V(e^{i\theta}z)=V(z),\,\,\theta\in\Bbb R,\,\,z\in\Bbb C, |\partial_z^{k_1}\partial_{\bar{z}}^{k_2}V(z)|\leq C_{k_1,k_2}(1+|z|)^{1+\alpha-k_1-k_2},\tag{H-1} \end{equation} for some $\alpha\geq3.$ The main contribution of this paper is the new lower bound of threshold of local well-posedness $s_c(\mathbb{S}^2,\alpha)$. Specifically, under assumption \eqref{H-11}, we prove that for $\alpha \geq 3$, the equation is ill-posed in $H^s(\mathbb{S}^2)$ with $s < 1 - \frac{2}{\alpha-1}$ in the sense that the norm inflation occurs. Combined with the well-posedness in Yang [Sci. China Math. 58 (2015), 1023-1046], the exact threshold $s_c(\mathbb{S}^2,\alpha)$ for $\alpha\geq5$ is $1-\frac{2}{\alpha-1}$, which matches the scaling-critical regularity as the Euclidean setting. Moreover, for $\alpha \in [3, \frac{11}{3})$, we show that the solution map is not uniformly continuous in the range $0 < s < \frac14$ for the power-type nonlinearity $F(u)=|u|^{\alpha-1}u$, which lies strictly above the scaling-invariant threshold. This provides a new characterization of the ill-posedness regime for all $\alpha \geq 3$, extending an earlier result of Burq-G\'erard-Tzvetkov [Math. Res. Lett. 9 (2002), 323-335]. Our result can also be regarded as a Schr\"odinger counterpart of Xia [Int. Math. Res. Not. (2021), 15533-15554].

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arXiv2026

Geometric uncertainty principles for Schr\"odinger evolutions on negatively curved manifolds

In this paper, we study the uncertainty principle for Schr\"odinger equations with a bounded time-independent potentials on certain Cartan-Hadamard manifolds endowed with an asymptotic hyperbolic metric in dimensions $n\geq2$. The classical Hardy uncertainty principle in Euclidean space, as developed in the works of Escauriaza-Kenig-Ponce-Vega (JEMS, 2008; Duke Math. J., 2010), reveals a rigidity phenomenon for solution $u$ to Schr\"odinger equations: sufficiently strong Gaussian decay at two distinct times yields $u\equiv0$. In this work, we show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation. Our approach follows a general strategy of Escauriaza-Kenig-Ponce-Vega, where the underlying geometry brings an essential change. This enables us to establish new Carleman estimates and logarithmic convexity. Unlike the Euclidean setting, the hyperbolic geometry exhibits exponential volume growth and nontrivial geodesic escape at infinity, which fundamentally alters the propagation mechanism of Schr\"odinger evolutions. Based on the newly-built virial identities and an approximation argument, we derive the logarithmic convexity. The main difficulty in proving the logarithmic convexity is the lack of convolution structure on general manifolds. By making use of the exponential map and Jacobi field, we define a new mollifier on curved geometry. Meanwhile, to establish the Carleman estimate adapted to hyperbolic space, we introduce a new weight function adapted to the curved manifold. Our results highlight the role of curvature in shaping quantitative uniqueness properties for dispersive equations.

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arXiv2026

Magnetic uncertainty in variable geometry

In this paper, we study Hardy-type uncertainty principles and unique continuation properties for linear covariant Schrodinger equations with variable coefficients in the presence of bounded electric and magnetic potentials. Under suitable smallness assumptions on the leading coefficients, we prove that any solution exhibiting super-quadratic exponential decay at two distinct times must vanish identically. Under an additional structural assumption on the coefficient matrix $G$, we further establish a Hardy-type result at the quadratic exponential scale. We also obtain an analogous uniqueness result for the heat equation with variable-coefficient magnetic perturbations. Our results unify and extend previous works in two directions: they recover the constant-coefficient covariant case treated by Barcelo-Fanelli-Gutierrez-Ruiz-Vilela when $G=I$, and the variable-coefficient non-magnetic case considered by Federico-Li-Yu when $A=0$. The proofs combine logarithmic convexity arguments with Carleman estimates adapted to variable-coefficient covariant Schr\"odinger and parabolic flows. Although our approach follows the general strategy introduced by Escauriaza-Kenig-Ponce-Vega, substantial new difficulties arise from the interaction between the variable metric and the magnetic structure, which requires new weight functions and refined commutator estimates.

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arXiv2026

Threshold Scattering for the Energy-Critical NLS with a Repulsive Inverse Square Potential

We study the threshold scattering problem for the energy-critical nonlinear Schr\"odinger equation with a repulsive inverse-square potential $\frac{a}{|x|^2} > 0$ in dimensions $d= 4, 5, 6$. On the energy level surface determined by the ground state of the energy-critical NLS without potential, we show that, despite the absence of a ground state in this setting, a strong form of rigidity persists below the kinetic threshold. Specifically, we prove that any solution on this energy surface with kinetic energy strictly below that of the ground state is global and scatters to zero. Our approach combines refined modulation analysis, a center-translated global Virial estimate, and a bootstrap argument to control the modulation parameters.

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arXiv2026

On the pointwise convergence of NLS flow on $ \S^2 $

In this paper, we study the almost everywhere convergence of the cubic nonlinear Schr\"odinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^2. \end{equation*} Inspired by the randomization method and the ansatz introduced by Burq, Camps, Sun, and Tzvetkov [Preprint, arXiv:2404.18229], we prove almost sure pointwise convergence almost everywhere for the nonlinear solution at very low regularity. This extends Compaan-Luc\`a-Staffilani [Int. Math. Res. Not. IMRN, (1) (2021), 596--647] to the spherical setting. We also provide a new necessary condition for the associated $L^p$ maximal estimate for the linear Schr\"odinger equation on $\S^2$. More precisely, we show that the $L^p$ maximal estimate fails for $s<\frac{1}{2}-\frac{1}{2p}$ with $p\ge 2$. In the special case $p=3$, our result matches the corresponding range in the $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449].

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arXiv2026

A Paley-Wiener type uniqueness result for the electromagnetic Schr\"odinger equation

In this paper, we establish a Paley-Wiener type uncertainty principle for Schr\"odinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+\Delta_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where $\Delta_A=(\nabla-iA)^2$ denotes the magnetic Schr\"odinger operator. Specifically, under suitable assumptions on $A$ and $V$, we show that if a solution $u$ exhibits linear exponential decay and support property in one spatial direction at times $t=0$ and $t=1$ respectively, then $u$ must vanish identically. This result extends the theorem of Kenig-Ponce-Vega [Ann. Sci. \'Ec. Norm. Sup\'er. (4) 47 (2014), 539-557] to the case $A\neq0$. We overcome the difficulty brought by the magnetic potential which breaks the translation invariance in the leading term of Hamiltonian $H=\Delta_A+V$. As a direct consequence, we also obtain a uniqueness result for a class of semi-linear Schr\"odinger equation with electromagnetic potentials.

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arXiv2026

Dynamics of focusing nonlinear Schr\"odinger equation with partial harmonic confinement in higher dimensions

We study the following focusing intercritical nonlinear Schr\"odinger equation with partial harmonic confinement: \begin{equation*} \begin{cases} i\partial_t u+\Delta_{z}u-y^2 u =- |u|^{\alpha}u,\quad t\in \mathbb{R},\newline u(0,z)= u_0(z), \ z=(x,y)\in \mathbb{R}^d\times \mathbb{R}, \end{cases} \end{equation*} where $d \geq 1 $ is an integer and the exponent $\alpha$ satisfies \begin{equation}\label{assumption} \frac{4}{d}< \alpha<\begin{cases} \frac{4}{d-1}, \,\,\, \text{if} ~~ d\geq 2; \newline + \infty,\,\,\, \text{if} ~~ d=1. \end{cases} \end{equation} For this model, A. Ardia and R. Carles [Comm. Math. Sci. 19 (2021), 993-1032] established a sharp scattering result below the ground state threshold in dimensions $d \leq 4$ via the concentration-compactness and rigidity argument. However, their approach breaks down in higher dimensions due to the lack of smoothness in the nonlinearity. In this paper, we introduce a new strategy that removes this dimensional restriction and extend their results to higher dimensions by circumventing the concentration-compactness principle. The main ingredients of our work are the interaction Morawetz-Dodson-Murphy estimates and an alternative variational characterization of the ground state threshold.

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arXiv2026

Proximal Learning for Trials With External Controls: A Case Study in HIV Prevention

With the advent of effective pre-exposure prophylaxis agents, active-controlled HIV prevention trials have become a common study design. Nevertheless, estimating absolute efficacy relative to a placebo remains important. In this paper, we introduce a novel application of proximal causal inference methods to estimate the counterfactual cumulative HIV incidence under placebo for participants in an active-controlled trial of cabotegravir, using external control data from a placebo-controlled trial with similar eligibility criteria. We leverage baseline sexually transmitted infection status and geographic region as negative control outcome and exposure variables, respectively. We address two key challenges: unmeasured differences in HIV risk between trials and statistical difficulties arising from low HIV incidence rates in both studies. To overcome these challenges, we develop two proximal inference approaches: (1) a semiparametric inverse probability of censoring weighting estimator, and (2) a two-stage regression-based strategy tailored to low-event-rate settings. Our theoretical and numerical investigations demonstrate these methods yield reliable estimates of the counterfactual one-year cumulative HIV incidence under placebo, and provide robust evidence of the superior efficacy of cabotegravir compared with placebo. These findings highlight the potential of proximal inference methods to estimate placebo-controlled effects in both single-arm and active-controlled trials by leveraging external controls.

stat.ME↗
arXiv2026

Demystifying Prediction Powered Inference

Machine learning predictions are increasingly used to supplement incomplete or costly-to-measure outcomes in fields such as biomedical research, environmental science, and social science. However, treating predictions as ground truth introduces bias while ignoring them wastes valuable information. Prediction-Powered Inference (PPI) offers a principled framework that leverages predictions from large unlabeled datasets to improve statistical efficiency while maintaining valid inference through explicit bias correction using a smaller labeled subset. Despite its potential, the growing PPI variants and the subtle distinctions between them have made it challenging for practitioners to determine when and how to apply these methods responsibly. This paper demystifies PPI by synthesizing its theoretical foundations, methodological extensions, connections to existing statistics literature, and diagnostic tools into a unified practical workflow. Using the MOSAIKS housing price data, we show that PPI variants produce tighter confidence intervals than complete-case analysis, but that double-dipping, i.e. reusing training data for inference, leads to anti-conservative confidence intervals and below-nominal coverage. Under missing-not-at-random mechanisms, all methods, including classical inference using only labeled data, yield biased estimates. We provide a decision flowchart linking assumption violations to appropriate PPI variants, a summary table of representative methods, and practical diagnostic strategies for evaluating core assumptions. By framing PPI as a general recipe rather than a single estimator, this work bridges methodological innovation and applied practice, helping researchers responsibly integrate predictions into valid inference.

stat.ML↗
arXiv2025

Control and stabilization problem for a class of fourth-order nonlinear Schr\"odinger equation on boundaryless compact manifold

In this paper, we study the stabilization property and large time controllability for a class of fourth-order Schr\"odinger equations on a compact manifold without boundary in dimensions $1\leq d\leq5$: \begin{align} i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=-|u|^{2k}u, \,\,x\in M\tag{4NLS}\label{4NLS1} \end{align} where $k\in\mathbb{N}$ and $\beta\in\mathbb{R}_{+}$ when $1\leq d\leq 4$ but $\beta\in\mathbb{Q}_{+}$ for $d=5$. We adapt the strategy in Macia [Vietnam J. Math. (2021)] to establish observability and the propagation of singularities. Moreover, we use these propagation estimates to deduce the unique continuation property for $\eqref{4NLS1}$. By the classical Hilbert Uniqueness Method (HUM) and the Picard iteration, the stabilization and large time controllability hold under the Geometric Control Condition (GCC) and the Unique Continuation Property (UCP) for the linearized equation. To obtain the controllability and stabilization at the $H^2$ level with $d=5$, we will focus on $M=\Bbb S^5$ with $k=1$. Our results extend those of Laurent [SIAM J. Math. Anal. (2009)] and Capistrano Filho-Pampu [Math. Z., 2022].

math.AP↗
arXiv2025

Asymptotic behavior of mass-critical Schrödinger equation in $ \mathbb{R}$

In this paper, we study the long-time behavior for the mass-critical nonlinear Schrödinger equation on the line \[ i\partial_t u + \partial_x^2 u = |u|^4 u, u(0, x) = u_0 \in L_x^2(\Bbb R). \] The global well-posedness and scattering for this equation was solved in Dodson [Amer. J. Math. (2016)]. Inspired by the pioneering work of Killip-Visan-Zhang [Amer. J. Math. (2021)], we show that solution can be approximated by a finite-dimensional Hamiltonian system. This system is the nonlinear Schrödinger equation on the rescaled torus $\Bbb R/(L_n\Bbb Z)$ with Fourier truncated nonlinear term. To prove this, we introduce the Fourier truncated mass-critical NLS on $\mathbb{R}$. First, we establish the uniformly global space-time bound for this truncated model on $\mathbb{R}$. Second, we show that the truncated NLS on rescaled torus can be approximated by the truncated equation on $\Bbb R$. Then, using the Gromov theorem, we can show the non-squeezing property for the truncated NLS on torus. The last step to show the non-squeezing property for original NLS is to connect the solution with truncated nonlinearity and a single equation in $\mathbb{R}$, which can be done by performing the nonlinear profile decomposition. Our second result is to study the homogenization of the mass-critical inhomogeneous NLS, where we add a $L^\infty$ function $h(nx)$ in front of the nonlinear term. Based on the method of Ntekoume [Comm. PDE, (2020)], we give the sufficient condition on $h$ such that the scattering holds for this inhomogeneous model and show that the solution to inhomogeneous converges to the homogeneous model when $n\to\infty$. As a corollary, we can transfer the non-squeezing property from homogeneous model to inhomogeneous.

math.AP↗
arXiv2025

On Growth of Sobolev norms for cubic Schr\"odinger equation with harmonic potential in dimensions $d=2,3$

In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schr\"odinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-\Delta+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three.

math.AP↗
arXiv2025

Scattering theory for the defocusing 3d NLS in the exterior of a strictly convex obstacle

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation $iu_t + Δ_Ωu = |u|^αu$ in the exterior domain $Ω$ of a smooth, compact and strictly convex obstacle in $\mathbb{R}^3$. It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^3))$ with $s_c := \frac{3}{2} - \frac{2}α \in (0, \frac{3}{2})$, then $u$ is global and scatters. In this paper, assuming that this conjecture holds, we prove that if $u$ is a solution to the nonlinear Schrödinger equation in exterior domain $Ω$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(Ω))$ with $s_c \in \left[\frac{1}{2}, \frac{3}{2}\right)$, then $u$ is global and scatters. The proof of the main results relies on the concentration-compactness/rigidity argument of Kenig and Merle [Invent. Math. {\bf 166} (2006)]. The main difficulty is to construct minimal counterexamples when the scaling and translation invariance breakdown on $Ω$. To achieve this, two key ingredients are required. First, we adopt the approach of Killip, Visan, and Zhang [Amer. J. Math. {\bf 138} (2016)] to derive the linear profile decomposition for the linear propagator $e^{itΔ_Ω}$ in $\dot{H}^{s_c}(Ω)$. The second ingredient is the embedding of the nonlinear profiles. More precisely, we need to demonstrate that nonlinear solutions in the limiting geometries, which exhibit global spacetime bounds, can be embedded back into $Ω$. Finally, to rule out the minimal counterexamples, we will establish long-time Strichartz estimates for the exterior domain NLS, along with spatially localized and frequency-localized Morawetz estimates.

math.AP↗
arXiv2024

Improved global well-posedness for the cubic NLS on two-dimensional waveguide $\R\times\T$

In this article, we show that the solution to defocusing cubic nonlinear Schr\"odinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+\Delta_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in $H^s(\R\times\T)$ with $s>\frac{1}{2}$. The proof is based on the $I$-method. Inspired by Colliander-Keel-Staffilani-Takaoka-Tao [Discrete Contin. Dyn. Syst. 21 (2008), 665-686], we construct the modified energy to improve the energy increment. The main difficulty lies in controlling the resonant interactions caused by the modified energy. To this end, we establish refined bilinear Strichartz estimates with angular truncation on the rescaled waveguide, thereby generalizing results previously obtained by Takaoka [J. Differ. Equa. 394 (2024), 296-319]. Furthermore, we demonstrate polynomial growth of $H^s$ with $\frac{1}{2} < s < 1$. Our result extends the recent work of Deng-Fan-Yang-Zhao-Zheng [J. Func. Anal. 287 (2024), 110595].

math.AP↗
arXiv2024

Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on $\Bbb T^3$

In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global.

math.AP↗
arXiv2024

On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement

In this article, we study the scattering theory for the two dimensional defocusing quintic nonlinear Schrödinger equation(NLS) with partial harmonic oscillator which is given by \begin{align}\label{NLS-abstract} \begin{cases}\tag{PHNLS} i\partial_tu+(\partial_{x_1}^2+\partial_{x_2}^2)u-x_2^2u=|u|^4u,&(t,x_1,x_2)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R},\\ u(0,x_1,x_2)=u_0(x_1,x_2). \end{cases} \end{align} First, we establish the linear profile decomposition for the Schrödinger operator $e^{it(\partial_{x_1}^2+\partial_{x_2}^2-x_2^2)}$ by utilizing the classical linear profile decomposition associated with the Schrödinger equation in $L^2(\mathbb{R})$. Then, applying the normal form technique, we approximate the nonlinear profiles using solutions of the new-type quintic dispersive continuous resonant (DCR) system. This allows us to employ the concentration-compactness/rigidity argument introduced by Kenig and Merle in our setting and prove scattering for equation (PHNLS) in the weighted Sobolev space. The second part of this paper is dedicated to proving the scattering theory for this mass-critical (DCR) system. Inspired by Dodson's seminal work [B. Dodson, Amer. J. Math. 138 (2016), 531-569], we develop long-time Strichartz estimates associated with the spectral projection operator $Π_n$, along with low-frequency localized Morawetz estimates, to address the challenges posed by the Galilean transformation and spatial translation.

math.AP↗
arXiv2024

Adjusting for Incomplete Baseline Covariates in Randomized Controlled Trials: A Cross-World Imputation Framework

In randomized controlled trials, adjusting for baseline covariates is often applied to improve the precision of treatment effect estimation. However, missingness in covariates is common. Recently, Zhao & Ding (2022) studied two simple strategies, the single imputation method and missingness indicator method (MIM), to deal with missing covariates, and showed that both methods can provide efficiency gain. To better understand and compare these two strategies, we propose and investigate a novel imputation framework termed cross-world imputation (CWI), which includes single imputation and MIM as special cases. Through the lens of CWI, we show that MIM implicitly searches for the optimal CWI values and thus achieves optimal efficiency. We also derive conditions under which the single imputation method, by searching for the optimal single imputation values, can achieve the same efficiency as the MIM.

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