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1. CMB Online first
Almost Sure Global Well-posedness for Fractional Cubic SchrÃ¶dinger Equation on Torus In a previous paper, we proved that $1$-d periodic fractional
SchrÃ¶dinger equation with cubic nonlinearity is locally well-posed
in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for
$s\gt \frac{10\alpha-1}{12}$. In this paper we define an invariant
probability measure $\mu$ on $H^s$ for $s\lt \alpha-\frac{1}{2}$,
so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$
such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally
well-posed for initial data in $\Omega$. We see that this fills
the gap between the local well-posedness and the global well-posedness
range in almost sure sense for $\frac{1-\alpha}{2}\lt \alpha-\frac{1}{2}$,
i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.
Keywords:NLS, fractional Schrodinger equation, almost sure global wellposedness Category:35Q55 |
2. CMB 2011 (vol 56 pp. 378)
Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases In this paper, we consider the Gross-Pitaevskii equation for the
trapped dipolar quantum gases. We obtain the sharp criterion for the
global existence and finite time blow up in the unstable regime by
constructing a variational problem and the so-called invariant
manifold of the evolution flow.
Keywords:Gross-Pitaevskii equation, sharp threshold, global existence, blow up Categories:35Q55, 35A05, 81Q99 |
3. CMB 2010 (vol 53 pp. 737)
On the Negative Index Theorem for the Linearized Non-Linear SchrÃ¶dinger Problem
A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky,
and Vougalter based on the variational principle for the
quadratic form of a self-adjoint operator.
It is the negative index theorem for a linearized NLS operator in
three dimensions.
Categories:35Q55, 81Q10 |
4. CMB 2007 (vol 50 pp. 35)
A Singular Critical Potential for the SchrÃ¶dinger Operator Consider a real potential $V$ on
$\RR^d$, $d\geq 2$, and the Schr\"odinger equation:
\begin{equation}
\tag{LS} \label{LS1} i\partial_t u +\Delta u -Vu=0,\quad
u_{\restriction t=0}=u_0\in L^2.
\end{equation}
In this paper, we investigate the minimal local regularity of $V$
needed to get local in time dispersive estimates (such as local in
time Strichartz estimates or local smoothing effect with gain of
$1/2$ derivative) on solutions of \eqref{LS1}. Prior works
show some dispersive properties when $V$ (small at infinity) is in
$L^{d/2}$ or in spaces just a little larger but with a smallness
condition on $V$ (or at least on its negative part).
In this work, we prove the critical character of these results by
constructing a positive potential $V$ which has compact support,
bounded outside $0$ and of the order $(\log|x|)^2/|x|^2$ near $0$.
The lack of dispersiveness comes from the existence of a sequence
of quasimodes for the operator $P:=-\Delta+V$.
The elementary construction of $V$ consists in sticking together
concentrated, truncated potential wells near $0$. This yields a
potential oscillating with infinite speed and amplitude at $0$,
such that the operator $P$ admits a sequence of quasi-modes of
polynomial order whose support concentrates on the pole.
Categories:35B65, 35L05, 35Q40, 35Q55 |
5. CMB 2005 (vol 48 pp. 3)
Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a
Riemanian manifold with piecewise smooth boundary and suppose that the
billiard associated to the geodesic flow reflecting on the boundary
according to the laws of geometric optics is ergodic.
We prove that the boundary value of the eigenfunctions of the Laplace
operator with reasonable boundary conditions are asymptotically
equidistributed in the boundary, extending previous results by
G\'erard and Leichtnam as well as Hassell and Zelditch,
obtained under the additional assumption of the convexity of~$M$.
Categories:35Q55, 35BXX, 37K05, 37L50, 81Q20 |