A Scaling Law for the Energy Levels of a Nonlinear Schrodinger Equation

It is shown that the energy levels of the one-dimensional nonlinear Schrodinger, or Gross-Pitaevskii, equation with the homogeneous trap potential $x^{2p}$, $p\geq 1$, 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.