Eigenstates and instabilities of chains with embedded defects

TL;DR

This paper develops a general method for analyzing eigenstates and stability of one-dimensional Schrödinger lattices with embedded defects, applicable to linear, nonlinear, and non-Hermitian cases, revealing oscillatory instabilities.

Contribution

It introduces a matching solution approach for eigenvalue problems in lattices with embedded defects, applicable to both linear and nonlinear, Hamiltonian and non-Hermitian systems.

Findings

Semi-analytical spectrum computation for linear defects.

Method for linearization spectrum around nonlinear stationary states.

Identification of oscillatory instabilities in nonlinear states.

Abstract

We consider the eigenvalue problem for one-dimensional linear Schr\"odinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples we discuss both linear and nonlinear, Hamiltonian and $\cP \cT$-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.