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Award ID contains: 1763074

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  1. ; ; (, Annals of PDE)
    Full Text Available 
  2. ; ; (, Inventiones mathematicae)
    null (Ed.)
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  3. (, International Mathematics Research Notices)
    null (Ed.)
    Abstract We study the scattering problem for the nonlinear Schrödinger equation $$i\partial _t u + \Delta u = |u|^p u$$ on $$\mathbb{R}^d$$, $$d\geq 1$$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $$\Sigma$$ holds and the wave operator is well defined on $$\Sigma$$. We show that there exists $$0<\beta <p$$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $$L^2\to L^2$$ of class $$C^{1+\beta }$$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology. 
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  4. ; ; (, SIAM Journal on Mathematical Analysis)
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  5. ; (, RIMS kokyuroku bessatsu)
    We present two adaptations of an argument of Sonin, which is known to be a powerful tool for obtaining both qualitative and quantitative information about special functions; see [12]. Our particular applications are as follows: (i) We give a rigorous formulation and proof of the following assertion about focusing NLS in any dimension: The spatial envelope of aspherically symmetric soliton in arepulsive potential is a non‐increasing function of the radius. (ii) Driven by the question of determining the most stably singular matrix, we determine the location of the maximal eigenvalue density of an ncross n GUE matrix. Strikingly, in even dimensions, this maximum is not at zero. 
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