OP-00317 Open

Excess Floquet multiplicity for complex periodic Schrödinger equations

Note: Zhijie Chen and Chang-Shou Lin, “On algebraic multiplicity of (anti)periodic eigenvalues of Hill’s equations”, Proc. Amer. Math. Soc. 146 (2018), 3039–3047, doi:10.1090/proc/14003, give explicit Hill equations with complex potentials that appear to answer this question affirmatively. The article is not on a site the registry links papers from, so the claim is neither recorded nor checked here.

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Let $q \in L^1_{\mathrm{loc}}(\mathbb{R}; \mathbb{C})$ have period $a > 0$. For $x_0 \in [0,a]$ and $\lambda \in \mathbb{C}$ let $c(\cdot, x_0, \lambda)$ and $s(\cdot, x_0, \lambda)$ be the solutions of $-y'' + qy = \lambda y$ with $c = s' = 1$ and $c' = s = 0$ at $x_0$. Let $M(\lambda)$ be the monodromy matrix over one period and $\Delta(\lambda) = \operatorname{tr} M(\lambda) = c(x_0 + a, x_0, \lambda) + s'(x_0 + a, x_0, \lambda)$, which does not depend on $x_0$. Let $d(\lambda)$, $p(x_0, \lambda)$ and $r(x_0, \lambda)$ be the orders of $\lambda$ as a zero of the entire functions $\Delta^2 - 4$, $s(x_0 + a, x_0, \cdot)$ and $c'(x_0 + a, x_0, \cdot)$, with order $0$ where the function does not vanish, and let $p_i(\lambda) = \min_{x_0 \in [0,a]} p(x_0, \lambda)$ and $r_i(\lambda) = \min_{x_0 \in [0,a]} r(x_0, \lambda)$. Gesztesy and Weikard proved $p_i = r_i$ and $d - p_i - r_i \ge 0$ (1996). At a periodic or antiperiodic eigenvalue of geometric multiplicity one, $d > 0 = p_i + r_i$; for real $q$ this is the only way the excess $d - p_i - r_i$ can be positive.

Problem. Can $d(\lambda) > p_i(\lambda) + r_i(\lambda)$ hold at a $\lambda$ with $M(\lambda) = I$ or $M(\lambda) = -I$, that is, at a periodic or antiperiodic eigenvalue with two linearly independent periodic or antiperiodic solutions? Either an example or a proof that it cannot happen answers the question.

Source: Weikard (2015), open problem from the American Institute of Mathematics workshop on mathematical aspects of physics with non-self-adjoint operators, in the terms of Gesztesy and Weikard (1996).

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: CA, MP
Keywords: AIM Open Applied Problems, Hill operators, non-self-adjoint operators, periodic eigenvalues

References

@misc{weikard2015,
  author       = {Weikard, Rudi},
  title        = {Open problem for complex-valued periodic potentials},
  howpublished = {Open problems in non-self-adjoint spectral theory, problem 2015-27, from the American Institute of Mathematics workshop Mathematical aspects of physics with non-self-adjoint operators, San Jose, June 2015},
  year         = {2015},
  url          = {https://nsa.fjfi.cvut.cz/problems/04_AIM_2015/2015-27/2015-27_AIM_Weikard.pdf},
}
@article{gesztesy1996,
  author  = {Gesztesy, Fritz and Weikard, Rudi},
  title   = {Picard potentials and {H}ill's equation on a torus},
  journal = {Acta Math.},
  volume  = {176},
  number  = {1},
  year    = {1996},
  pages   = {73--107},
    doi     = {10.1007/BF02547336},
}
@article{chen2018,
  author  = {Chen, Zhijie and Lin, Chang-Shou},
  title   = {On algebraic multiplicity of (anti)periodic eigenvalues of {H}ill's equations},
  journal = {Proc. Amer. Math. Soc.},
  volume  = {146},
  number  = {7},
  year    = {2018},
  pages   = {3039--3047},
  doi     = {10.1090/proc/14003},
}
@misc{aim-complex-hill-excess-multiplicity,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Excess Floquet multiplicity for complex periodic Schrödinger equations}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/425-complex-hill-excess-multiplicity.md},
  note         = {Entry 425 at commit a153f46, 2026-10-13},
}

Status history

When	Change	By	Reason
2026-10-14	Needs verification → Open	—	Listed open with a note on a claim the registry cannot record.

Citing this problem

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

OP-00317, "Excess Floquet multiplicity for complex periodic Schrödinger equations", clopen.solutions.