Purely magnetic tunneling conjecture for elliptical domains

Let $\Omega$ be an ellipse with boundary $\Gamma$ of length $2L$, and let $s$ denote the curvilinear coordinate. Let $\lambda_1(h)$ and $\lambda_2(h)$ be the first two eigenvalues, let $\kappa(s)$ be the boundary curvature, and write $\kappa_{\max}$ and $\kappa_{\min}$ for its maximum and minimum. Let $C_1$, $\mu_1”(\xi_0)$, and $\xi_0$ be the constants from the magnetic boundary spectral reduction, and set

$$k_2=-\kappa”\left(\frac L2\right),\qquad \gamma_0=\frac{|\Omega|}{|\Gamma|}=\frac{|\Omega|}{2L}.$$

Define

$$\mathsf{S}=\sqrt{\frac{2C_1}{\mu_1”(\xi_0)}}\int_{-\frac L2}^{\frac L2}\sqrt{\kappa_{\max}-\kappa(s)}\,\mathrm{d}s,$$

and

$$\mathsf{A}=\exp\left(-\int_{[\frac L2,L]}\frac{\partial_s\sqrt{\kappa_{\max}-\kappa(s)}-\sqrt{\frac{k_2}{2}}}{\sqrt{\kappa_{\max}-\kappa(s)}}\,\mathrm{d}s\right).$$

The points $s=-\frac L2$ and $s=\frac L2$ correspond to the right and left points of maximal curvature, respectively. Purely magnetic tunneling conjecture. There exists $\alpha_0\in\mathbb{R}$ such that

$$\lambda_2(h)-\lambda_1(h)=h^{\frac{13}{8}}\mathsf{A}\frac{2^{\frac52}C_1^{\frac34}}{\sqrt{\pi}}\left(k_2\mu_1”(\xi_0)\right)^{\frac14}\left(\kappa_{\max}-\kappa_{\min}\right)^{\frac12}\left|\cos\left(L\left(\frac{\gamma_0}{h}-\frac{\xi_0}{h^{\frac12}}-\alpha_0\right)\right)\right|\mathrm{e}^{-\mathsf{S}/h^{\frac14}}+o\left(h^{\frac{13}{8}}\right)\mathrm{e}^{-\mathsf{S}/h^{\frac14}}\quad\text{as }h\to0.$$

This gives an explicit asymptotic formula for the exponentially small splitting between the first two eigenvalues in a purely magnetic tunneling problem for an ellipse. The formula was numerically checked in the paper and is recalled from an earlier conjecture; the theoretical formula is presented as an open question rather than a proved result.