  We study the self-adjoint extensions of the Hamiltonian operator with
symmetric potentials which go to $-\infty$ faster than $-|x|^{2p}$ with $p>1$
as $x\to\pm\infty$. In this extension procedure, one requires the Wronskian
between any states in the spectrum to approach to the same limit as
$x\to\pm\infty$. Then the boundary terms cancel and the Hamiltonian operator
can be shown to be hermitian. Discrete bound states with even and odd parities
are obtained. Since the Wronskian is not required to vanish asymptotically, the
energy eigenstates could be degenerate. Some explicit examples are given and
analyzed.
