Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation

TL;DR

This paper investigates the emergence of bound states below the continuum in a solvable PT-symmetric discrete Schrödinger model, revealing exact solutions and explicit formulas for bound states with multiple couplings.

Contribution

It introduces a solvable PT-symmetric discrete Schrödinger model with multiple couplings, providing explicit bound-state solutions and analyzing their emergence from the continuum edge.

Findings

Exact bound-state formulas for up to J=7 couplings.

Bound states can emerge from the continuum with infinitesimal interaction changes.

The model is exactly solvable for all finite coupling counts J.

Abstract

At the lower edge of the energy continuum the birth of an isolated quantum bound state is studied as caused by an infinitesimally small change of the interaction. In our model a single, asymptotically free massive quantum particle is assumed moving along a discretized real line of coordinates, $x \in \mathbb{Z}$. The motion is assumed controlled by a weakly nonlocal 2J-parametric external potential which is non-Hermitian but PT-symmetric. Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the bound-state energy is treated as a variable real parameter while the value of one of the couplings (responsible for the existence of the bound state) is determined via the standard secular equation. It is found that in such an arrangement the model is exactly solvable at all of the finite counts J of the couplings. For illustration, the explicit closed bound-state formulae are presented up to J=7.