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

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) it is 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.

Figure 1

Eigenfunctions ${\psi }_0$ and ${\psi }_{10}$ of the one-dimensional harmonic oscillator (46) calculated with the orthogonal spectral renormalization method in comparison with the exact solution.

Figure 2

Spectrum of the first eight eigenstates of the anharmonic oscillator (49) calculated with the orthogonal spectral renormalization in comparison with a numerically accurate calculation from [49].

Figure 3

Eigenfunctions of the particle in the box problem (50) with $a=1$ and $V_0=20$. The OSR solution is compared with the analytical solution.

Figure 4

Spectrum of the first nine $\mathcal{PT}$-symmetric states of the complex extended harmonic oscillator model (55). The OSR solution is compared with a numerically accurate solution. The algorithm fails to converge for the ground state for small $\epsilon$ close to 1 and for higher excited states for larger $\epsilon$.

Figure 5

Spectrum of a Bose-Einstein condensate in a harmonic trap in dependence of the nonlinearity parameter $g$. The chemical potentials $E$ are calculated with the OSR method and compared with a numerically exact calculation in an oscillator basis.

Figure 6

Wave functions of the Gross-Pitaevskii equation (56) for the ground state and the first two excited states calculated with the OSR and in an harmonic oscillator basis (exact).

Figure 7

Spectrum of the double well [Eq. (58)] calculated with the OSR method in comparison with a solution from [22].

Figure 8

Spectrum of the lowest states of the two dimensional BEC in a harmonic trap with gain and loss [Eq. (61)]. The OSR solution is compared with a calculation in a harmonic oscillator basis [30].