Classical and quantum billiard inside the square with gravitational field

Daniel Jaud

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper derives the Airy-function determinant condition at y=0,L, applies the large negative-argument asymptotics of Ai and Bi to reduce the determinant to a sine of a phase difference, obtains the Bohr–Sommerfeld–type condition ε^{3/2} − (ε − L/R)^{3/2} = (3π/2) r, and then performs a second-order expansion in w = L/(R ε) to produce a closed-form approximation for εy,r and Ey,r. The candidate solution follows the same steps and arrives at the identical formulas and asymptotics. Minor additions (e.g., monotonicity/uniqueness and the O(r−2) constant term mgL/2 in the large-r expansion) are consistent refinements.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

Both the paper and the candidate solution correctly apply Airy-function methods to quantify the spectrum in a gravitationally tilted infinite square well. The arguments are standard but well executed. Minor revisions would clarify the exact asymptotic regime required for using the oscillatory forms of Ai/Bi, give a brief derivation of the determinant-to-sine reduction, and comment on the accuracy introduced by truncating the small-w expansion.