Reality of the spectrum for a spherical strip with PT-symmetric boundary conditions
For $0 < a < \pi/2$, $\alpha \in \mathbb{R}$ and $m \in \mathbb{Z}$, let $\beta = \tfrac12 \tan a$ and let $H_{a,\alpha,m}$ be the operator on $L^2(-a, a)$ given by $$H_{a,\alpha,m} u = -u'' + \frac{8m^2 - 3 - \cos(2x)}{8\cos^2 x}\, u,$$ $$D(H_{a,\alpha,m}) = \{u \in H^2(-a, a) : u'(\pm a) + (i\alpha \pm \beta)\, u(\pm a) = 0\}.$$ Krejčiřík and Siegl obtained these operators, by separation of variables and a unitary transformation, from the Laplace–Beltrami operator on the neighbourhood of width $2a$ of the equator of the unit sphere, with the boundary condition $\partial_n \psi + i\alpha \psi = 0$, where $n$ is the unit normal to the equator carried along the meridians, so it points the same way on both boundary circles. The operator $H_{a,\alpha,m}$ is not self-adjoint unless $\alpha = 0$, but it commutes with the antilinear operator $(\mathcal{PT}\psi)(x) = \overline{\psi(-x)}$, so its non-real eigenvalues come in complex-conjugate pairs. Krejčiřík and Siegl proved that the operator without the potential term has real spectrum, and that every eigenvalue of $H_{a,\alpha,m}$ with large enough real part, depending on $m$, is real and simple. Their numerical computations indicate that all eigenvalues are real.
Problem. Is $\sigma(H_{a,\alpha,m}) \subset \mathbb{R}$ for every $a \in (0, \pi/2)$, every $\alpha \in \mathbb{R}$ and every $m \in \mathbb{Z}$?
Source: Krejčiřík and Siegl (2010), Remark 4.6, who conjecture that the spectrum is real for every m; posed as an open problem by Siegl at the ESF Exploratory Workshop, Prague (2010), published in Integral Equations Operator Theory (2012).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, non-self-adjoint operators, PT-symmetric operators, Robin boundary conditions
References
@article{krejcirik2010,
author = {Krej{\v{c}}i{\v{r}}{\'\i}k, David and Siegl, Petr},
title = {{$\mathcal{PT}$}-symmetric models in curved manifolds},
journal = {J. Phys. A},
volume = {43},
number = {48},
year = {2010},
pages = {485204},
doi = {10.1088/1751-8113/43/48/485204},
eprint = {1001.2988},
archivePrefix = {arXiv},
}
@article{siegl2012,
author = {Siegl, Petr},
title = {{$\mathcal{PT}$}-symmetric {L}aplace--{B}eltrami operator in the strip on a sphere},
journal = {Integral Equations Operator Theory},
volume = {73},
number = {1},
year = {2012},
pages = {5--6},
doi = {10.1007/s00020-011-1932-x},
}
@misc{aim-spherical-pt-robin-real-spectrum,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Reality of the spectrum for a spherical strip with PT-symmetric boundary conditions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/424-spherical-pt-robin-real-spectrum.md},
note = {Entry 424 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that the separated spherical-strip operators with PT-symmetric Robin conditions have real spectrum for every half-width $0<a<\pi/2$, every real boundary parameter and every integer Fourier mode. The proof establishes strict interlacing between the associated real Robin and Dirichlet spectra at every index. A positive Laplace representation of the Fourier coefficients of $\sec^2x$ gives the comparison for nonzero integer modes, …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Real spectra for the separated spherical Robin operators v1 recorded from AIM @ 24edaad, paper/spherical_robin_real_spectrum.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00386, "Reality of the spectrum for a spherical strip with PT-symmetric boundary conditions", clopen.solutions.