A universal form of localized complex potentials with spectral singularities

Abstract

Abstract We establish necessary and sufficient conditions for localized complex potentials in the Schrödinger equation to enable spectral singularities (SSs) and show that such potentials have the universal form U ( x ) = − w 2 ( x ) − i w x ( x ) + k 0 2 , where w(x) is a differentiable function, such that lim x → ± ∞ w ( x ) = ∓ k 0 , and k 0 is a non-zero real. We also find that when k 0 is a complex number, then the eigenvalue of the corresponding Shrödinger 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   symmetric, can feature a self-dual SS associated with the CPA-laser operation. At this moment, the complex potential has exactly two coexisting SSs at different points of the continuous spectrum. In the space of the system parameters, the transition through each self-dual SS 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.