# Eigenfunction Behavior and Adaptive Finite Element Approximations of Nonlinear Eigenvalue Problems in Quantum Physics

## TL;DR

This paper studies nonlinear eigenvalue problems from quantum physics, proving eigenfunctions are non-polynomial and demonstrating that adaptive finite element methods converge even with coarse initial meshes.

## Contribution

It refines the unique continuation property and shows convergence of adaptive finite element approximations for nonlinear eigenvalue problems.

## Key findings

- Eigenfunctions cannot be polynomial on any open set.
- Adaptive finite element methods converge without fine initial meshes.
- Applicable to certain linear eigenvalue problems as well.

## Abstract

In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that the eigenfunction cannot be a polynomial on any open set, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing result.