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The energy-critical NLS with inverse-square potential

Autor: Killip, R., Miao, C., Visan, M., Zhang, J., Zheng, J.

We consider the defocusing energy-critical nonlinear Schr\"odinger equation with inverse-square potential $iu_t = -\Delta u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14 +\frac1{25}$.

Externí odkaz: http://arxiv.org/abs/1509.05822

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Sobolev spaces adapted to the Schr\'odinger operator with inverse-square potential

Autor: Killip, R., Miao, C., Visan, M., Zhang, J., Zheng, J.

We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with those defi

Externí odkaz: http://arxiv.org/abs/1503.02716

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The focusing energy-critical nonlinear Schr\'odinger equation in dimensions five and higher

Autor: Killip, R., Visan, M.

We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabl

Externí odkaz: http://arxiv.org/abs/0804.1018

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Global existence and scattering for rough solutions to generalized nonlinear Schr\'odinger equations on $\R$

Autor: Colliander, J., Holmer, J., Visan, M., Zhang, X.

We consider the Cauchy problem for a family of semilinear defocusing Schr\"odinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction

Externí odkaz: http://arxiv.org/abs/math/0612452

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A counterexample to dispersive estimates for Schr\'odinger operators in higher dimensions

Autor: Goldberg, M., Visan, M.

Publikováno v: Comm. Math. Phys. 266 no. 1 (2006), 211-238.

In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^\alpha$, $\alpha < \frac{n-3}2$, for which the solutions to the linear Schr\"odinger equation in $\R^n$, $$ -i\partial_t u = - \Delta u + Vu

Externí odkaz: http://arxiv.org/abs/math/0508206

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Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\'odinger equation in $\R^{1+4}$

Autor: Ryckman, E., Visan, M.

We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow tho

Externí odkaz: http://arxiv.org/abs/math/0501462

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