Title: Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method,
Abstract: The spectral renormalization method was introduced in 2005 as an effective
way to compute ground states of nonlinear Schrödinger 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.