# P11148: High-Angular-Momentum Doublet Splitting on a Nearly Circular Elliptic Cone

- ID: `P11148`
- Reference: `high-angular-momentum-doublet-splitting-elliptic-cone`
- Page: https://theoremdb.org/statements/P11148
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- Record maturity: Reviewed problem

## The problem

Fix \(a>0\) and \(h>0\). For real \(\varepsilon\) near \(0\), let \(\Sigma_\varepsilon\) be the lateral surface in \(\mathbb R^3\) obtained by joining the apex \(A=(0,0,h)\) to the base ellipse
\[
\Gamma_\varepsilon=\{(ae^\varepsilon\cos\theta,ae^{-\varepsilon}\sin\theta,0):\theta\in\mathbb R/2\pi\mathbb Z\}.
\]
Equip \(\Sigma_\varepsilon\) with its induced metric, impose Dirichlet boundary condition on \(\Gamma_\varepsilon\), and take the Friedrichs realization of the Laplace-Beltrami operator at the conic tip. Pull the family to \((0,1)\times S^1\) using
\[
X_\varepsilon(r,\theta)=(rae^\varepsilon\cos\theta,rae^{-\varepsilon}\sin\theta,h(1-r)).
\]
At \(\varepsilon=0\), put
\[
L=\sqrt{a^2+h^2},\qquad \kappa=\frac aL\in(0,1).
\]
The circular cone has double eigenvalues
\[
\lambda_{m,n}(0)=\frac{j_{m/\kappa,n}^2}{L^2},\qquad m,n\ge1,
\]
carried by the angular factors \(\cos(m\theta)\) and \(\sin(m\theta)\), where \(j_{\nu,n}\) is the \(n\)-th positive zero of \(J_\nu\). Fix \(m,n\) and exclude values of \(\kappa\) for which \(\lambda_{m,n}(0)\) coincides with any circular-cone eigenvalue outside this two-dimensional angular doublet. Let \(\lambda^c_{m,n}(\varepsilon)\) and \(\lambda^s_{m,n}(\varepsilon)\) denote the two analytic eigenvalue branches selected by the reflection parities continuing \(\cos(m\theta)\) and \(\sin(m\theta)\).

Prove that, outside a discrete exceptional set of \(\kappa\) depending on \(m,n\), all Taylor coefficients of
\[
\lambda^c_{m,n}(\varepsilon)-\lambda^s_{m,n}(\varepsilon)
\]
through order \(m-1\) vanish and the order-\(m\) coefficient is nonzero; equivalently,
\[
\lambda^c_{m,n}(\varepsilon)-\lambda^s_{m,n}(\varepsilon)
=c_{m,n}(\kappa)\varepsilon^m+O(\varepsilon^{m+1}),
\qquad c_{m,n}(\kappa)\neq0
\]
generically. Then determine the sharp high-angular-momentum asymptotics of \(c_{m,n}(\kappa)\) for each fixed radial index \(n\) as \(m\to\infty\). In particular, determine whether there is an explicit \(\Gamma_n(\kappa)\) (or \(\Gamma(\kappa)\) independent of fixed \(n\)) such that
\[
\log|c_{m,n}(\kappa)|=m\Gamma_n(\kappa)+o(m),
\]
and if not, determine the correct leading asymptotic scale. Finally, compare the resulting splitting with the neighboring circular-cone spectral gap and characterize the asymptotic regimes of sequences \(\varepsilon_m\to0\) for which the elliptic doublet splitting is spectrally negligible or spectrally resolved.

## Status

The reviewed record remains open.

## Research packet

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## References

No external mathematical reference has been recorded for this problem.