OP-00359 Open

Nonempty spectrum for a magnetic complex Airy operator at every field strength

Note: Kaiwen Jin, “Nonclosedness of an ordinary Sobolev restriction of a magnetic Airy operator” (AIM, October 2026, https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/research/solutions/423-ordinary-sobolev-magnetic-airy-domain-clarification/paper/domain-correction.pdf), shows that the ordinary Sobolev domain printed in Almog’s 2015 AIM problem sheet gives a nonclosed operator. The statement here uses the closed Dirichlet realization, so the note clarifies the source and does not bear on the question.

Submit a result

Let $\Omega = \{(x,y) \in \mathbb{R}^2 : y > 0\}$ and, for real $c \ne 0$, let $A_c$ be the Dirichlet realization of $$-\partial_x^2 - \left(\partial_y - \frac{i x^2}{2}\right)^2 + i c y$$ in $L^2(\Omega)$, with domain $D(A_c) = H^1_0(\Omega) \cap \{u : A_c u \in L^2(\Omega)\}$, where $A_c u$ is taken in the sense of distributions. It arises in the linearization of the time-dependent Ginzburg–Landau equations near the normal state, for a superconductor filling a half-plane under an electric current perpendicular to its boundary and the magnetic field this current induces. The same operator on all of $\mathbb{R}^2$ has compact resolvent and empty spectrum for every $c \ne 0$ (Almog, Helffer and Pan). On the half-plane $A_c$ has compact resolvent, and Almog, Helffer and Pan showed that when $\sigma(A_c) \ne \varnothing$ the semigroup $e^{-tA_c}$ decays exponentially at the rate $\inf_{z \in \sigma(A_c)} \operatorname{Re} z$, and that $\sigma(A_c) \ne \varnothing$ for $|c|$ large enough. In a sequel they showed that $\sigma(A_c) \ne \varnothing$ also for $|c|$ small enough.

Problem. Is $\sigma(A_c) \ne \varnothing$ for every real $c \ne 0$?

Source: Almog, Helffer and Pan (2013), who conjecture that the spectrum is nonempty (as reported by Helffer, 2011); posed by Almog (2015) as open problem 1.2 of the American Institute of Mathematics workshop on mathematical aspects of physics with non-self-adjoint operators.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: AP, MP
Keywords: AIM Open Applied Problems, magnetic Schrödinger operators, non-self-adjoint operators, superconductivity

References

@article{almog2013,
  author  = {Almog, Yaniv and Helffer, Bernard and Pan, Xing-Bin},
  title   = {Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field},
  journal = {Trans. Amer. Math. Soc.},
  volume  = {365},
  number  = {3},
  year    = {2013},
  pages   = {1183--1217},
  doi     = {10.1090/S0002-9947-2012-05572-3},
}
@article{helffer2011,
  author  = {Helffer, Bernard},
  title   = {On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current},
  journal = {Confluentes Math.},
  volume  = {3},
  number  = {2},
  year    = {2011},
  pages   = {237--251},
  doi     = {10.1142/S1793744211000308},
}
@misc{almog2015nsa,
  author       = {Almog, Yaniv},
  title        = {Magnetic {S}chr{\"o}dinger operator},
  howpublished = {List of open problems, AIM workshop ``Mathematical aspects of physics with non-self-adjoint operators'', American Institute of Mathematics, San Jose, California, 8--12 June 2015},
  year         = {2015},
  url          = {https://aimath.org/pastworkshops/nonselfadjointproblems.pdf},
  note         = {Problem 1.2},
}
@article{almog2010,
  author  = {Almog, Yaniv and Helffer, Bernard and Pan, Xing-Bin},
  title   = {Superconductivity near the normal state under the action of electric currents and induced magnetic fields in {$\mathbb{R}^2$}},
  journal = {Comm. Math. Phys.},
  volume  = {300},
  number  = {1},
  year    = {2010},
  pages   = {147--184},
  doi     = {10.1007/s00220-010-1111-y},
}
@article{almog2012,
  author  = {Almog, Yaniv and Helffer, Bernard and Pan, Xing-Bin},
  title   = {Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field, {P}art {II}: {T}he large conductivity limit},
  journal = {SIAM J. Math. Anal.},
  volume  = {44},
  number  = {6},
  year    = {2012},
  pages   = {3671--3733},
    doi     = {10.1137/110860598},
}
@article{almog2015cpde,
  author        = {Almog, Yaniv and Helffer, Bernard},
  title         = {On the spectrum of non-selfadjoint {S}chr{\"o}dinger operators with compact resolvent},
  journal       = {Comm. Partial Differential Equations},
  volume        = {40},
  number        = {8},
  year          = {2015},
  pages         = {1441--1466},
  doi           = {10.1080/03605302.2015.1025978},
  eprint        = {1410.5122},
  archivePrefix = {arXiv},
}
@misc{aim-magnetic-airy-half-plane-spectrum,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Nonempty spectrum for a magnetic complex Airy operator at every field strength}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/423-magnetic-airy-half-plane-spectrum.md},
  note         = {Entry 423 at commit a153f46, 2026-10-13},
}

Status history

When	Change	By	Reason
2026-10-18	Partially resolved → Open	Kaiwen Jin	"Nonclosedness of an ordinary Sobolev restriction of a magnetic Airy operator" withdrawn by its author.
2026-10-18	Open → Partially resolved	Kaiwen Jin	Nonclosedness of an ordinary Sobolev restriction of a magnetic Airy operator v1 announced.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00359, "Nonempty spectrum for a magnetic complex Airy operator at every field strength", clopen.solutions.