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We establish necessary and sufficient conditions for complex potentials inthe Schr\"odinger equation to enable spectral singularities (SSs) and show thatsuch 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\"odingeroperator has an exact solution which, depending on $k_0$, represents a coherentperfect absorber (CPA), laser, a localized bound state, a quasi bound state inthe continuum (a quasi-BIC), or an exceptional point (the latter requiringadditional conditions). Thus, $k_0$ is a bifurcation parameter that describestransformations among all those solutions. Additionally, in a more specificcase of a real-valued function $w(x)$ the resulting potential, although notbeing $PT$ symmetric, can feature a self-dual spectral singularity associatedwith the CPA-laser operation. In the space of the system parameters, thetransition through each self-dual spectral singularity corresponds to abifurcation of a pair of complex-conjugate propagation constants from thecontinuum. The bifurcation of a first complex-conjugate pair corresponds to thephase transition from purely real to complex spectrum.

A universal form of complex potentials with spectral singularities