Pleijel's theorem for Schr\"odinger operators with radial potentials

TL;DR

This paper extends Pleijel's theorem to Schrödinger operators with radial potentials, analyzing nodal domain counts for eigenfunctions in different potential regimes, including unbounded and negative potentials.

Contribution

It generalizes Pleijel's theorem to a broader class of Schrödinger operators with radial potentials, beyond the Laplacian and harmonic oscillator cases.

Findings

Established asymptotic bounds for nodal domains with radial potentials.

Analyzed eigenfunctions for potentials tending to infinity and zero.

Extended Pleijel's theorem to new classes of Schrödinger operators.

Abstract

In 1956 $\AA$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs naturally for Schr"\odinger operators. The first significant result has been obtained recently by the first author for the harmonic oscillator. The purpose of this paper is to consider more general potentials which are radial. We will analyze the case when the potential tends to $+\infty$ and the case when the potential is negative and tends to zero, where the considered eigenfucntion are associated to the eigenvalues below the essential spectrum.