Linear spectrum problem

For Eq. (1) in the absence of nonlinear term (g = 0), we assume that ψ(x, z) = Φ(x)e^−iλz, then we have Eq. (3), which with the -symmetric potential (2), as |x| → ∞, reduces to , whose characteristic equation is , whose roots, in general, are complex numbers and complicated. For example, if λ = 1/(3β²) and β > 0 (without loss of generality), we have its three roots Λ₁ = i/β, , which probably lead to the result that the corresponding eigenfunctions should satisfy periodic boundary conditions. Additionally, if β depends on the space x, e.g., β(x) = β₀ exp(−x²), and V(x), W(x) are given by Eq. (9), then we have β(x), W(x) → 0 and V(x) → x²/2 as |x| → ∞. Thus for this case Eq. (3) reduces to as |x| → ∞, where the condition is used, and we have the asymptotic solutions . Based on the standard conditions of wave function, we only take , which generally corresponds to zero boundary conditions and discrete spectra. Therefore, in order to verify these results, we use the Fourier collocation method^59,60,61 to numerically study the above-mentioned linear spectrum problems and obtain the agreeable conclusions as ones by the theoretical analysis.