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Search: MSC category 35Q55 ( NLS-like equations (nonlinear Schrodinger) [See also 37K10] )

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1. CMB 2011 (vol 56 pp. 378)

Ma, Li; Wang, Jing
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

2. CMB 2010 (vol 53 pp. 737)

Vougalter, Vitali
On the Negative Index Theorem for the Linearized Non-Linear Schrödinger 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

3. CMB 2007 (vol 50 pp. 35)

Duyckaerts, Thomas
A Singular Critical Potential for the Schrödinger 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

4. CMB 2005 (vol 48 pp. 3)

Burq, N.
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

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