Spectral algorithms for multiple scale localized eigenfunctions in infinitely long, slightly bent quantum waveguides

Abstract

A “bent waveguide” in the sense used here is a small perturbation of a two-dimensional rectangular strip which is infinitely long in the down-channel direction and has a finite, constant width in the cross-channel coordinate. The goal is to calculate the smallest (“ground state”) eigenvalue of the stationary Schrödinger equation which here is a two-dimensional Helmholtz equation, ψxx+ψyy+Eψ=0 where E is the eigenvalue and homogeneous Dirichlet boundary conditions are imposed on the walls of the waveguide. Perturbation theory gives a good description when the “bending strength” parameter ϵ is small as described in our previous article (Amore et al., 2017) and other works cited therein. However, such series are asymptotic, and it is often impractical to calculate more than a handful of terms. It is therefore useful to develop numerical methods for the perturbed strip to cover intermediate ϵ where the perturbation series may be inaccurate and also to check the pertubation expansion when ϵ is small. The perturbation-induced change-in-eigenvalue, δ≡E(ϵ)−E(0), is O(ϵ2). We show that the computation becomes very challenging as ϵ→0 because (i) the ground state eigenfunction varies on both O(1) and O(1∕ϵ) length scales and (ii) high accuracy is needed to compute several correct digits in δ, which is itself small compared to the eigenvalue E. The multiple length scales are not geographically separate, but rather are inextricably commingled in the neighborhood of the boundary deformation. We show that coordinate mapping and immersed boundary strategies both reduce the computational domain to the uniform strip, allowing application of pseudospectral methods on tensor product grids with tensor product basis functions. We compared different basis sets; Chebyshev polynomials are best in the cross-channel direction. However, sine functions generate rather accurate analytical approximations with just a single basis function. In the down-channel coordinate, X∈[−∞,∞], Fourier domain truncation using the change of coordinate X=sinh(Lt) is considerably more efficient than rational Chebyshev functions TBn(X;L). All the spectral methods, however, yielded the required accuracy on a desktop computer.