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Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert Schmidt with respect to a weighted $L^2$ space

Let $\langle x \rangle := (1 + |x|^2)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator \begin{equation*} T_s \varphi = \langle x \rangle^{-s} \int_{\mathbb{R}^3}...
JZS's user avatar
  • 481
5 votes
1 answer
311 views

Maximal operator estimates for the Schrödinger equation

Let $a>0$ and consider the operator $$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$ When $a=2$, the function $Tf$ solves the Cauchy problem ...
Medo's user avatar
  • 852
1 vote
0 answers
88 views

Pointwise convergence of Schrodinger's equation with potential term

A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation $$iu_t-\...
Simplyorange's user avatar
2 votes
0 answers
141 views

Mathematical study of dispersive PDEs [closed]

My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
Simplyorange's user avatar
1 vote
1 answer
136 views

Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?

We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
enihcamemit's user avatar
2 votes
0 answers
188 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
Joshua Isralowitz's user avatar
3 votes
0 answers
78 views

Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
Amir Sagiv's user avatar
  • 3,574
1 vote
0 answers
151 views

Uniform boundedness for Strichartz constants in Tao's book

This is a question from Tao's book "Nonlinear Dispersive Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm $$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
Student's user avatar
  • 333
1 vote
0 answers
68 views

When Schroedinger propagator commutes other operators?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), ...
XYZ's user avatar
  • 31
1 vote
0 answers
89 views

Eignfunctions of an elliptic operator

I am looking for a reference which makes it possible to say that one can devellop $f$ in the form of the sentence underlined by the yellow. Thank you in advance.
Fadil Kikawi's user avatar
4 votes
0 answers
433 views

Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
Felice Iandoli's user avatar