In this talk we will see the well posedness results for the nonlinear Schr\"{o}dinger equation for the magnetic Laplacian on $\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\C^n$ with power type nonlinearity $\lambda |u|^{\alpha} u$. We establish the well posedness in certain first order Sobolev spaces associated to the twisted Laplacian. The approach is via the spectral theory of the Schr\"{o}dinger propagator for the twisted Laplacian, and local existence is proved using Strichartz estimates established for the same.Using blowup analysis and conservation laws, we conclude global well posedness in the defocussing case (\lambda>0) with $0\leq \alpha< 2/(n-1)$ and also in the focussing case (\lambda<0) with $0\leq \alpha< 2/n$.We also prove finite time blow up in the focussing case (\lambda<0) with $2/n\leq \alpha< 2/(n-1)$. In this talk we also see a Hardy-Sobolev inequality for the twisted Laplacian on $\C^n$. We also show that the inequality is optimal in the sense that weight can not be improved.

Date/Time

Venue

LH 4

Speaker

Vijay Kumar Sohani

Affiliation

IISc Bangalore

Title

Nonlinear Schrodinger equation and Hardy Sobolev inequality for the twisted Laplacian