  We establish necessary and sufficient conditions for complex potentials in
the Schr\"odinger equation to enable spectral singularities (SSs) and show that
such potentials have the universal form $U(x) = -w^2(x) - iw_x(x) + k_0^2$,
where $w(x)$ is a differentiable function, such that $\lim_{x\to \pm
\infty}w(x)=\mp k_0$, and $k_0$ is a nonzero real. We also find that when $k_0$
is a complex number, then the eigenvalue of the corresponding Shr\"odinger
operator has an exact solution which, depending on $k_0$, represents a coherent
perfect absorber (CPA), laser, a localized bound state, a quasi bound state in
the continuum (a quasi-BIC), or an exceptional point (the latter requiring
additional conditions). Thus, $k_0$ is a bifurcation parameter that describes
transformations among all those solutions. Additionally, in a more specific
case of a real-valued function $w(x)$ the resulting potential, although not
being $PT$ symmetric, can feature a self-dual spectral singularity associated
with the CPA-laser operation. In the space of the system parameters, the
transition through each self-dual spectral singularity corresponds to a
bifurcation of a pair of complex-conjugate propagation constants from the
continuum. The bifurcation of a first complex-conjugate pair corresponds to the
phase transition from purely real to complex spectrum.
