On Delusive Nodes of Free Oscillations

TL;DR

This paper reviews several theorems and a conjecture related to nodal sets of Laplacian eigenfunctions, revealing inaccuracies and partial correctness, while highlighting Arnold's contributions to understanding spectral problems in mathematical physics.

Contribution

It critically examines existing theorems and conjectures on nodal sets, clarifying their inaccuracies and emphasizing Arnold's influential role in the field.

Findings

All reviewed assertions are incorrect or only partly correct.

Analysis improved understanding of spectral problems for the Laplacian.

Highlights Arnold's significant contributions to the area.

Abstract

Two theorems and one conjecture about nodal sets of eigenfunctions arising in various spectral problems for the Laplacian are reviewed. It occurred that all these assertions are incorrect or only partly correct, but their analysis has brought better understanding of the corresponding area of mathematical physics. The contribution made by V. I. Arnold is emphasized.