Periodic orbit theory in fractal drum

TL;DR

This paper investigates the spectral properties of pseudointegrable fractal drums using periodic orbit theory, revealing behavior akin to chaotic systems and confirming results with eigenvalue calculations.

Contribution

It applies periodic orbit theory to fractal drums, demonstrating spectral rigidity behavior similar to chaotic systems and validating findings with eigenvalue data.

Findings

Spectral rigidity $ ext{Δ}_3(L)$ decreases to small values

Results align well with direct eigenvalue calculations

Behavior approaches that of chaotic systems

Abstract

The level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory. We find that the spectral rigidity $\Delta_3(L)$, which is a measure for the correlations between the eigenvalues, decreases to quite small values (as compared to systems with only small boundary roughness), thereby approaching the behavior of chaotic systems. The periodic orbit results are in good agreement with direct calculations of $\Delta_3(L)$ from the eigenvalues.