An optimal Weyl remainder for interior transmission eigenvalues
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected domain with $C^\infty$ boundary, and fix a constant $n > 1$. A number $z \ne 0$ is an interior transmission eigenvalue if there is a nonzero pair $(u, v)$ with $$\Delta u + n z u = 0, \quad \Delta v + z v = 0 \quad \text{in } \Omega, \qquad u = v, \quad \partial_\nu u = \partial_\nu v \quad \text{on } \partial\Omega.$$ These are the nonzero eigenvalues of the closed operator $A(u, v) = (-n^{-1} \Delta u, -\Delta v)$ on $L^2(\Omega) \oplus L^2(\Omega)$, whose domain consists of the pairs with distributional Laplacians in $L^2(\Omega)$ and with equal Dirichlet traces and equal normal traces on $\partial\Omega$. Count each with its algebraic multiplicity, the rank of the Riesz projection of $A$ at it, and let $N(r)$ be the number of transmission eigenvalues with $|z| \le r^2$. Let $\omega_d$ be the volume of the unit ball in $\mathbb{R}^d$. Petkov and Vodev proved Weyl asymptotics for $N(r)$ with a remainder governed by eigenvalue-free regions, and Vodev's eigenvalue-free regions give $$N(r) = \frac{\omega_d}{(2\pi)^d} |\Omega| \bigl(1 + n^{d/2}\bigr) r^d + O_\varepsilon\bigl(r^{d-1+\varepsilon}\bigr) \qquad \text{for every } \varepsilon > 0.$$
Problem. Is $$N(r) = \frac{\omega_d}{(2\pi)^d} |\Omega| \bigl(1 + n^{d/2}\bigr) r^d + O_{\Omega, n}\bigl(r^{d-1}\bigr) \qquad (r \to \infty)$$ for every such domain $\Omega$ and every constant $n > 1$? No convexity of $\Omega$ is assumed.
Source: Vodev (2017), Conjecture 1, for general coefficients; see also Vodev (2018), after Corollary 2.3. Stated by AIM — Open Applied Problems in the isotropic case with constant index of refraction.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, interior transmission eigenvalues, non-self-adjoint operators, Weyl's law
References
@article{vodev2017rims,
author = {Vodev, Georgi},
title = {Asymptotic behavior of the transmission eigenvalues},
journal = {RIMS K{\^o}ky{\^u}roku},
volume = {2045},
year = {2017},
pages = {61--66},
url = {https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2045-04.pdf},
}
@article{vodev2018,
author = {Vodev, Georgi},
title = {High-frequency approximation of the interior {D}irichlet-to-{N}eumann map and applications to the transmission eigenvalues},
journal = {Anal. PDE},
volume = {11},
number = {1},
year = {2018},
pages = {213--236},
doi = {10.2140/apde.2018.11.213},
eprint = {1701.04668},
archivePrefix = {arXiv},
}
@article{petkov2017,
author = {Petkov, Vesselin and Vodev, Georgi},
title = {Asymptotics of the number of the interior transmission eigenvalues},
journal = {J. Spectr. Theory},
volume = {7},
number = {1},
year = {2017},
pages = {1--31},
doi = {10.4171/JST/154},
eprint = {1403.3949},
archivePrefix = {arXiv},
}
@misc{aim-transmission-eigenvalue-sharp-weyl-remainder,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An optimal Weyl remainder for interior transmission eigenvalues}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/429-transmission-eigenvalue-sharp-weyl-remainder.md},
note = {Entry 429 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove a Weyl formula with remainder $O(r^{d-1})$ for the interior transmission problem with a constant refractive index $n>1$ in an arbitrary bounded smooth domain in $\mathbb R^d$, $d\ge2$. The count includes every nonzero complex energy of modulus at most $r^2$, with its algebraic multiplicity for the original maximal-domain transmission operator. The leading coefficient is $\omega_d(2\pi)^{-d}|\Omega|(1+n^{d/2})$. The argument combines the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | A sharp Weyl remainder for constant-index transmission eigenvalues v1 recorded from AIM @ 24edaad, 429-transmission-eigenvalue-sharp-weyl-remainder/weyl_remainder.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00395, "An optimal Weyl remainder for interior transmission eigenvalues", clopen.solutions.