First-principles derivation of the decay exponent β for mixed-state fraction

Derive, from first principles, the power-law exponent β governing the decay χ(e) ∝ e^β of the fraction χ(e) of mixed-type eigenstates in Bunimovich mushroom billiards, where e = (A/(4π)) k^2 is the unfolded energy (A is the billiard area and k the wavenumber), and mixed-type eigenstates are those whose Poincaré–Husimi distributions have support across both the regular and chaotic regions as identified via the M-index thresholds M_th^- ≤ M ≤ M_th^+. Establish an analytical derivation that explains the numerically observed β ≈ −1/3 across stem half-widths w ∈ [0.1, 0.9].

Background

The paper measures the fraction χ(e) of mixed-type eigenstates—states whose Poincaré–Husimi (PH) support straddles both regular and chaotic regions—using an M-index–based classification. Across stem half-widths w = 0.1–0.9, χ(e) decays as a power law in the semiclassical parameter e with a robust exponent near β ≈ −1/3.

While the Principle of Uniform Semiclassical Condensation (PUSC) implies χ(e) → 0 in the semiclassical limit, there is currently no theoretical derivation of the decay rate. The authors therefore highlight as an open problem the first-principles derivation of the observed exponent β for the Bunimovich mushroom billiard family.