We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schr\"odinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness range, set $\sigma=\frac12-\sqrt{\frac14+a}$. We obtain the free strong decay rate for $2<p<3/\sigma$ and the limiting Lorentz estimate from $L^{(3/\sigma)',1}$ to $L^{3/\sigma,\infty}$. The restriction $p<3/\sigma$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev--Lorentz estimates adapted to $\mathcal L_a$.
While unconditional uniqueness for the cubic nonlinear Schr\"odinger equation (NLS) on $\Bbb R^3$ is known in the scaling-subcritical Sobolev spaces $\dot H^s(\Bbb R^3)$ for $1/2<s<1$, the critical case $s=1/2$ has remained open. We resolve this problem by adapting Kato's bootstrap argument to the critical space $\dot...
We establish global well-posedness and sharp higher-order decay estimates for the fractional compressible Navier--Stokes--Poisson system in $\R^{3}$ with fractional Lam\'e--Navier dissipation $\Lame^\alpha u$ with $\alpha\in[1/2,1]$, and Riesz coupling $\Lambda^{2\beta}\phi=\rho-\bar{\rho}$ with $\beta\in[0,1]$, where...
We study positive solutions of the partially confined stationary nonlinear Schr\"odinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for po...
Zhijie Chen, Youjun Wang, Jian-Jun Zhang et al.· 0 citations
Let $\Omega\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-\Delta+V$ uniquely determines every complex-valued potential $V\in L^p(\Omega)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. The pro...
We study critical Strichartz estimates for the resonant hyperbolic Schr\"odinger flow on the waveguide $\mathbb{R} \times \mathbb{T}^2$. For initial data localized to frequencies at most $N$, we prove that the unit-time $L^2\to L^{10/3}$ operator norm has sharp order $N^{1/5}$. We further track its dependence on the ob...
We study weighted and Sobolev estimates for two fractional Schr\"odinger operators on $\mathbb{R}^n$, where $n\geq3$ and $0<s<1$. For the additive operator $L=(-\Delta)^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse H\"older class, $RH_q$, we establish a critical-radius form inequali...
Aparajita Dasgupta, Drishti Garg· 0 citations
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