Quantum defect analysis of the eigenvalue spectrum of the Newton-Schrödinger equation

We point out that quantum defect theory is the appropriate framework to explain and understand the behavior of the solutions of the Newton-Schrödinger equation. We find that, beyond ordinary quantum defect theory, the nonlinearity of the equation induces novel features, in particular a strong state dependence of the quantum defects. We show how this can be compensated by a rescaling of the energy unit.

Figure 1

(Color) Self-trapping potentials (top) generated by the eigenfunctions of the radial Newton-Schrödinger equation, and renormalized radial eigenfunctions (bottom), for $n=5–15$.

Figure 2

(Color) Integrand $p$ (radial momentum) of the classical action integral as a function of $r$ for the eigenstates with $n=5–15$ of the radial Newton-Schrödinger equation (top). For comparison, $p\left(r\right)$ for motion at the same energies in a pure Coulomb potential is also shown (bottom). The state dependence of $p$ at short ranges in the Newton-Schrödinger case, as compared to the Coulomb case, can be clearly seen. The state dependence gives rise to a linear increase of the short-range contributions to the action integral, and thus the quantum defects, with $n$.