Eigenfunctions on a Stadium Associated with Avoided Crossings of Energy Levels

TL;DR

This work studies the spectral behavior of a quantum stadium billiard, focusing on avoided crossings of energy levels associated with the Dirichlet Laplacian eigenproblem $-$\nabla^2 u=\lambda u$ inside the stadium with $u|_{\partial}=0$. Approach: an updated numerical pipeline (building on FORTRAN code) computes selected eigenpairs for each stadium configuration determined by $a$ (with $r=1$) and uses Mathematica to render surface and contour plots of even-even eigenfunctions to analyze how eigenfunctions evolve as $a$ varies. Findings: the curves $f_3$, $f_4$, and $f_5$ show avoided crossings near $a\approx0.785$ and $a\approx1.51$, and the corresponding contour plots reveal a swap of characteristic features across the crossings. Significance: provides visual, qualitative insight into eigenfunction restructuring near spectral avoided crossings in chaotic billiard-like domains and complements traditional non-crossing arguments such as the Wigner–von Neumann principle.

Abstract

The authors examine graphical properties of eigenfunctions with stadium boundaries associated with avoided crossings of energy levels.