Moment-method quantization of a linear differential eigenvalue equation for ‖Ψ‖2

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Abstract
The one-dimensional Schrödinger equation can be reformulated in terms of a linear third-order differential eigenvalue equation for ‖Ψ(x)‖2. The bounded physical states are the unique non-negative solutions which are asymptotically zero and have finite moments. Because of this, the eigenvalue-moment methods of Handy and Bessis can be applied to ‖Ψ‖2, thereby yielding rapidly converging lower and upper bounds to the individual discrete-state eigenvalues. Two examples are considered: a one-dimensional nonlinear Schrödinger equation of the form -1/2Ψ‘+x2(1/2+a〈Ψ‖x2‖Ψ〉)Ψ=εΨ and the octic potential -1/2Ψ‘+(1/2x2+λx8)Ψ=EΨ.