Finite time blowup for the nonlinear Schrödinger equation with a delta potential

We study the Cauchy problem for the nonlinear Schrodinger equation with a delta potential, which can be written as iut + Au + (|u|2 sigma + c8)u = 0. We show that under certain conditions, the L infinity norm of the solution tends to infinity in finite time. In order to prove this, we study the associated Lagrangian and Hamil-tonian, and derive an estimate of the associated variance. We also derive several con-servation laws which a classical solution of the Cauchy problem must also satisfy.