Solvable model of bound states in the continuum (BIC) in one dimension

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.