Ground states on a fractured strip and one dimensional reduction

We consider the nonlinear Schr''odinger equation on a strip with Neumann boundary conditions and a delta condition on the $x$-axis. First, we show the existence of ground states as minimizers of the action or of the energy under suitable constraints. Second, we prove that the energy minimizers converge to the ground state on the line with a delta condition as the amplitude of the strip shrinks to zero.