The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

The Gross–Pitaevskii equation is solved by analytic methods for an external double-well potential, that is, an infinite square well plus a \(\delta\)-function central barrier. We find solutions that have the symmetry of the non-interacting Hamiltonian as well as asymmetric solutions that bifurcate from the symmetric solutions for attractive interactions and from the antisymmetric solutions for repulsive interactions. We present a variational approximation to the asymmetric state as well as an approximate numerical approach. We compare with other approximate methods. Stability of the states is considered.