  We report results of the analysis of the spontaneous symmetry breaking (SSB)
in the basic (actually, simplest) model which is capable to produce the SSB
phenomenology in the one-dimensional setting. It is based on the
Gross-Pitaevskii - nonlinear Schroedinger equation with the cubic
self-attractive term and a double-well-potential built as an infinitely deep
potential box split by a narrow (delta-functional) barrier. The barrier's
strength, epsilon, is the single free parameter of the scaled form of the
model. It may be implemented in atomic Bose-Einstein condensates and nonlinear
optics. The SSB bifurcation of the symmetric ground state (GS) is predicted
analytically in two limit cases, viz., for deep or weak splitting of the
potential box by the barrier. For the generic case, a variational approximation
(VA) is elaborated. The analytical findings are presented along with systematic
numerical results. Stability of stationary states is studied through the
calculation of eigenvalues for small perturbations, and by means of direct
simulations. The GS always undergoes the SSB bifurcation of the supercritical
type, as predicted by the VA at moderate values of epsilon, although the VA
fails at small epsilon, due to inapplicability of the underlying ansatz in that
case. However, the latter case is correctly treated by the approximation based
on a soliton ansatz. On top of the GS, the first and second excited states are
studied too. The antisymmetric mode (the first excited state) is destabilized
at a critical value of its norm. The second excited state undergoes the SSB
bifurcation, like the GS, but, unlike it, the bifurcation produces an unstable
asymmetric mode. All unstable modes tend to spontaneously reshape into the
asymmetric GS.
