All Questions
Tagged with schrodinger-operators harmonic-analysis
11 questions
1
vote
2
answers
225
views
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}...
- 6,035
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 ...
- 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
$$iu_t-\...
- 265
2
votes
0
answers
141
views
Mathematical study of dispersive PDEs [closed]
My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
- 265
1
vote
1
answer
136
views
Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?
We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
- 39k
2
votes
0
answers
188
views
Schrodinger operator with matrix potential
This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
3
votes
0
answers
78
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Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions
Consider the following Schrödinger equation
$$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$
where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
- 3,574
1
vote
0
answers
151
views
Uniform boundedness for Strichartz constants in Tao's book
This is a question from Tao's book "Nonlinear Dispersive
Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm
$$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
- 333
1
vote
0
answers
68
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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), ...
- 31
1
vote
0
answers
89
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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.
- 135
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}
...
- 3,574