  Historically, most of the quantum mechanical results have originated in one
dimensional model potentials. However, Von-Neumann's Bound states in the
Continuum (BIC) originated in specially constructed, three dimensional,
oscillatory, central potentials. One dimensional version of BIC has long been
attempted, where only quasi-exactly-solvable models have succeeded but not
without instigating degeneracy in one dimension. Here, we present an exactly
solvable bottomless exponential potential barrier $V(x)=-V_0[\exp(2|x|/a)-1]$
which for $E<V_0$ has a continuum of non-square-integrable, definite-parity,
degenerate states. In this continuum, we show a surprising presence of discrete
energy, square-integrable, definite-parity, non-degenerate states. For $E>V_0$,
there is again a continuum of complex scattering solutions $\psi(x)$ whose real
and imaginary parts though solutions of Schr{\"o}dinger equation yet their
parities cannot be ascertained as $C\psi(x)$ is also a solution where $C$ is an
arbitrary complex non-real number.
