Sharp asymptotic structure of the equilibrium layer and gap

Determine whether the constrained equilibrium measure \(\lambda_\tau\) has, at the start of the tail, an extension gap of width \(g\tau\) and a physical-interval saturated layer of width \(l\tau\) at each endpoint, and establish whether the \(O(\tau^{3/2})\) remainder in the initial-slope expansion is sharp.

Background

The constrained equilibrium measure is proved to possess a nontrivial layer at the ends of the physical interval and a gap in the extension. The proof of the initial-slope theorem constructs comparison profiles with widths proportional to τ\tau, with explicit constants gg and ll, and obtains an error of order τ3/2\tau^{3/2}.

Numerical equilibrium measures exhibit the predicted widths, but the paper does not establish that the actual minimizer has those precise first-order widths or that the stated remainder order cannot be improved.

References

The second question is the structure of the equilibrium measure at the start of the tail: the proof of Theorem~\ref{thm:initial_slope} uses a gap of width $g\tau$ and a layer of width $l\tau$ at each end of the physical interval, and the computed measures $\lambda_\tau$ have these widths, but that $\lambda_\tau$ has them, beyond the existence of a layer and a gap (Corollary~\ref{cor:layer}), is not proved, and the order $\tau{3/2}$ of the remainder in eq:initial_slope is not known to be sharp.

eq:initial_slope:

∣rω(τ)+s(ω) τ∣≤Cτ3/2(0<τ≤τ0).\bigl|r_\omega(\tau) + s(\omega)\,\tau\bigr| \le C\tau^{3/2} \qquad (0 < \tau \le \tau_0).

— Numerical Stability of Extended-Domain Spectral Collocation: Floating-Point Regularization and an Equilibrium Law for the Frame Constants  (2610.13222 - Wu, 9 Oct 2026) in Conclusion, Section 6; discussion following Theorem 4.13