We study the Cauchy problem for the nonlinear Schrödinger equation with a delta
potential, which can be written as
We show that under certain conditions, the
norm of
the solution tends to infinity in finite time. In order to prove this, we study the associated
Lagrangian and Hamiltonian, and derive an estimate of the associated variance. We also derive
several conservation laws which a classical solution of the Cauchy problem must also satisfy.
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