Explain the low-energy disagreement in semiclassical scaling

Determine the origin of the poorer agreement between the predicted semiclassical scaling and the numerical results at smaller scaled energy separations, particularly for u=1 in the generalized quarter-Sinai billiard.

Background

The paper tests the predicted energy scaling of off-diagonal matrix-element variances in a generalized quarter-Sinai billiard by examining traces at fixed scaled separation u=|ω|/√Ē. For u=2, 3, and 4, the numerical data agree well with the predicted Ē{-1/2} scaling. For u=1, however, the scaling is absent at lower energies and emerges only above approximately Ē=3×104. The authors explicitly state that the origin of this poorer agreement is unresolved, leaving open the explanation of the finite-energy behavior at small scaled separation.

References

For $u=1$, the semiclassical scaling emerges for $\bar E>3\times 104$ but is absent at lower energies due to finite-energy effects. The origin of the poorer agreement at smaller $u$ remains unclear.

— Semiclassical scaling of eigenstate thermalization in single-particle chaotic systems  (2609.20052 - Ye et al., 17 Sep 2026) in Section "Numerical results in a generalized quarter-Sinai billiard"