A universal form of complex potentials with spectral singularities

TL;DR

This paper derives a universal form for complex potentials in the Schrödinger equation that exhibit spectral singularities, revealing how these potentials can describe phenomena like CPA-lasers, bound states, and phase transitions.

Contribution

It establishes necessary and sufficient conditions for spectral singularities and introduces a universal potential form with a bifurcation parameter $k_0$ that governs various spectral phenomena.

Findings

Universal potential form for spectral singularities.

Exact solutions representing CPA, laser, and bound states.

Bifurcation parameter $k_0$ controls spectral transitions.

Abstract

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.