Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method

TL;DR

This paper introduces the orthogonal spectral renormalization (OSR) method for efficiently computing ground and excited states of linear and nonlinear eigenvalue problems, including complex and random potentials.

Contribution

The paper presents a novel OSR algorithm that improves convergence control, handles multiple initial guesses, and is easily implemented in higher dimensions for various eigenvalue problems.

Findings

Successfully applied to Hermitian and non-Hermitian models

Handles complex and random potentials effectively

Provides stable convergence for multiple eigenstates

Abstract

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schr\"odinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.