  We study an anti-symmetric (square) well and barrier potential of
depth/height $(V_0)$ placed between two rigid walls. Unlike the usual
double-well, here the closely lying sub-barrier doublets need not be the lowest
ones in the spectrum. When $V_0$ admits certain calculable values, $E=0$ or
$E=V_0$ or both could become energy eigenvalues of the special eigenstate which
emerge only if one seeks a linear solution of Schr{\"o}dinger equation in the
appropriate regions.
