Weakly bound states in heterogeneous waveguides: a calculation to fourth order

TL;DR

This paper extends the calculation of weakly bound states in heterogeneous waveguides to fourth order, revealing conditions under which localized states form below the continuum threshold.

Contribution

It provides a general expression for the fourth order energy correction and applies it to a solvable model, confirming the formation of localized states.

Findings

Fourth order correction can lower energy below the continuum threshold.

Localized states can form when second and third order corrections vanish.

The derived formula matches exact results in a solvable model.

Abstract

We have extended a previous calculation of the energy of a weakly heterogeneous waveguide to fourth order in the density perturbation, deriving its general expression. For particular configurations where the second and third orders both vanish, we discover that the fourth order contribution lowers in general the energy of the state, below the threshold of the continuum. In these cases the waveguide possesses a localized state. We have applied our general formula to a solvable model with vanishing second and third orders reproducing the exact expression for the fourth order.