Hasson, R. and Richards, D.
A scaling law for the energy levels of a nonlinear Schrödinger equation.
Journal of Physics B: Atomic, Molecular and Optical Physics, 34(9) pp. 1805–1813.
It is shown that the energy levels of the one-dimensional nonlinear Schrödinger, or Gross-Pitaevskii, equation with the homogeneous trap potential x2p, p≥1, obey an approximate scaling law and as a consequence the energy increases approximately linearly with the quantum number. Moreover, for a quadratic trap, p = 1, the rate of increase of energy with the quantum number is independent of the nonlinearity: this prediction is confirmed with numerical calculations. It is also shown that the energy levels computed using a variational approximation do not satisfy this scaling law.
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