Zeros of real valued Eigenfunctions

TL;DR

This paper establishes a uniqueness result for real-valued eigenfunctions of symmetric operators on manifolds, showing that at most one eigenspace contains such an eigenfunction with a zero set of dimension less than n-1.

Contribution

It proves a new uniqueness theorem for real-valued eigenfunctions with small zero sets on manifolds, extending understanding of eigenfunction zero structures.

Findings

At most one eigenspace contains a real eigenfunction with zero set dimension less than n-1.

Provides conditions under which eigenfunctions are uniquely characterized by their zero sets.

Extends classical eigenfunction zero set theory to more general symmetric operators on manifolds.

Abstract

Suppose we are given a symmetric operator T acting on a subspace of L2{M,m} where M is a connected manifold and m is a measure positive on open sets. Then there is at most one eigenspace that contains a real valued eigenfunction whose set of zeros has dimension less than n minus one, where n is the dimension of M.