nonlinear Schr\"odinger equation with a delta potential
global well-posedness of the Cauchy problem
orbital stability and instability of the ground state
Hokkaido University Preprint Series in Mathematics
We study nonlinear Schr\"odinger equation with a delta-function impurity in one space dimension. Global well-posedness is proved for the Cauchy problem in $L^2(\R)$ under subcritical nonlinearity, as well as under critical nonlinearity with smallness assumption on the data. In the attractive case, orbital stability and instability of the ground state is proved in $H^1(\R)$.