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

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

2000-12-14

arxiv.org/abs/quant-ph/0012068v1

Physics

Quantum Physics

10 pages, 4 figures. Submitted to J. Phys. B

Scientific paper

10.1088/0953-4075/34/9/315

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.

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