Emergence and localization of exceptional points in an exactly solvable toy model
Miloslav Znojil
TL;DR
This paper studies the emergence and localization of exceptional points (EPs) in an exactly solvable, discrete, PT-symmetric quantum square-well model with complex Robin boundaries. It demonstrates exact solvability via a Chebyshev-polynomial Ansatz for eigenvectors and constructs a Dieudonné-compatible metric $\Theta$ to preserve unitarity under a non-Hermitian Hamiltonian $H^{(N)}(z)$ for $z\in\mathcal{D}$. For even $N$, EPs can be localized analytically through a boundary-controlled parametrization $z(t)=i\sqrt{1-r^2(t)}$, with non-Hermiticity active for $|r(t)|\le 1$ while the spectrum remains real for all $r(t)\in\mathbb{R}$. The work also clarifies the absence of EP degeneracies in some odd-$N$ one-parameter reductions and uncovers a multi-band spectral structure in another one-parameter family, highlighting rich parameter-controlled non-Hermitian dynamics.
Abstract
The most elementary non-Hermitian quantum square-well problem with real spectrum is considered. The Schroedinger equation is required discrete and endowed with PT-symmetric Robin (i.e., two-parametric) boundary conditions. Some of the rather enigmatic aspects of impact of the variability of the parameters on the emergence of the Kato's exceptional-point (EP) singularities is clarified. In particular, the current puzzle of the apparent absence of the EP degeneracies at the odd-matrix dimensions in certain simplified one-parametric cases is explained. A not quite expected existence of a multi-band spectral structure in another simplified one-parametric family of models is also revealed.