We study the self-adjoint extensions of symmetric inverted potentials which go to $-\infty$ faster than $-|x|^{s}$ with $s>2$ as $x\to\pm\infty$. Two schemes are proposed. The first one has a strong boundary requirement that the Wronskians between any two energy eigenstate wavefunctions vanish. In the second scheme, one only imposes a weaker requirement that the Wronskian approaches to the same limit as $x\to\pm\infty$. Discrete bound state spectra with even and odd parities are obtained in both schemes. Since in the second scheme the Wronskian is not required to vanish, the energy eigenstates could be degenerate. Some explicit examples are given and analyzed.

Read more: http://arxiv.org/abs/0801.4437.