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Pleijel's theorem for Schrödinger operators

Philippe Charron, Corentin Léna

TL;DR

This work extends Pleijel's asymptotic bound on nodal domains to a broad class of Schrödinger operators $H_V=-\Delta+V$ in $\mathbb{R}^d$, including non-radial potentials. By combining an IMS localization-based partition with refined Faber–Krahn inequalities and Weyl-type counting via Dirichlet bracketing, the authors derive a Pleijel-type bound $\limsup_{n\to\infty} \frac{\mu(f_n)}{n} \le \gamma_d$ for real eigenfunctions under explicit growth/singularity conditions on $V$ (Cases A and B). The results cover positive-coefficient combinations of potentials with polynomial growth and potentials with Coulomb-type singularities, including both increasing-at-infinity and vanishing-at-infinity regimes, and they recover known nonradial generalizations in special cases. The approach unifies and extends previous radial-only methods, providing a framework that could adapt to manifolds and broader Weyl-law contexts, thereby deepening our understanding of nodal-domain distribution in quantum-mechanical operators.

Abstract

We are concerned in this paper with the real eigenfunctions of Schrödinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant's theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel's Theorem on the Dirichlet Laplacian and were obtained for some classes of Schrödinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.

Pleijel's theorem for Schrödinger operators

TL;DR

This work extends Pleijel's asymptotic bound on nodal domains to a broad class of Schrödinger operators in , including non-radial potentials. By combining an IMS localization-based partition with refined Faber–Krahn inequalities and Weyl-type counting via Dirichlet bracketing, the authors derive a Pleijel-type bound for real eigenfunctions under explicit growth/singularity conditions on (Cases A and B). The results cover positive-coefficient combinations of potentials with polynomial growth and potentials with Coulomb-type singularities, including both increasing-at-infinity and vanishing-at-infinity regimes, and they recover known nonradial generalizations in special cases. The approach unifies and extends previous radial-only methods, providing a framework that could adapt to manifolds and broader Weyl-law contexts, thereby deepening our understanding of nodal-domain distribution in quantum-mechanical operators.

Abstract

We are concerned in this paper with the real eigenfunctions of Schrödinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant's theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel's Theorem on the Dirichlet Laplacian and were obtained for some classes of Schrödinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.

Paper Structure

This paper contains 30 sections, 20 theorems, 127 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Past results
  4. Aim of this paper
  5. Main results
  6. Constraints and conditions
  7. Outline of the proof
  8. Acknowledgments
  9. Preliminary results
  10. Lower semiboundedness and the Friedrichs extension
  11. Partition and Rayleigh quotient estimates
  12. Potentials growing at infinity
  13. Construction of the partition
  14. Nodal count estimates
  15. Weyl's law for positive exponents
  16. ...and 15 more sections

Key Result

Proposition 1.6

In case A, $\sigma(H_V)=\{\lambda_n\}_{n\ge1}$, with each $\lambda_n$ an eigenvalue of finite multiplicity and $\lambda_n\to+\infty$. In case B, and with each $\lambda_n$ an eigenvalue of finite multiplicity and $\lambda_n\to 0$. In both cases, the eigenvalues are numbered in increasing order with account of multiplicity.

Figures (3)

  • Figure 1: Localization of a nodal domain.
  • Figure 2: A covering hypercube.
  • Figure 3: Partition for case B.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7: Case A
  • Corollary 1.8
  • Theorem 1.9: Case B
  • Corollary 1.10
  • ...and 27 more