Irreducibility of complex Fermi varieties
Let $d \ge 2$, let $V \in C^\infty(\mathbb{R}^d; \mathbb{R})$ be $\mathbb{Z}^d$-periodic, and for $k \in \mathbb{C}^d$ let $H(k) = (-i\nabla + k)^2 + V$ on the torus $\mathbb{R}^d/\mathbb{Z}^d$, with domain $H^2$. For real $E$ the complex Fermi variety $$F_E = \{ k \in \mathbb{C}^d : E \text{ is an eigenvalue of } H(k) \}$$ is invariant under translation by $2\pi\mathbb{Z}^d$, and $F_E / 2\pi\mathbb{Z}^d$ is a complex analytic set. Bättig, Knörrer and Trubowitz proved it irreducible for separable potentials and, for $d = 3$, for potentials of the form $V_1(x_1) + V_2(x_2, x_3)$ (1991). Kuchment and Vainberg showed for $d = 2, 3$ that irreducibility at every energy inside the spectral bands would exclude eigenvalues embedded in the bands of $-\Delta + V + W$ for every sufficiently fast decaying perturbation $W$ (2000). For the discrete periodic Schrödinger operator on $\mathbb{Z}^d$, Liu proved irreducibility at every energy when $d \ge 3$, and when $d = 2$ at every energy except the mean of the potential (2022).
Problem. For every such $V$, is there a discrete set $S_V \subset \mathbb{R}$ such that $F_E / 2\pi\mathbb{Z}^d$ is irreducible, that is, not the union of two proper closed analytic subsets, for every real $E \notin S_V$?
Source: Kuchment (2016), Conjecture 5.35, allowing a discrete set of exceptional energies; for every energy, Kuchment and Vainberg (2000), Conjecture 12, who attribute it to Bättig, Knörrer and Trubowitz (1991).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, embedded eigenvalues, Fermi varieties, periodic Schrödinger operators
References
@article{kuchment2016,
author = {Kuchment, Peter},
title = {An overview of periodic elliptic operators},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {53},
number = {3},
year = {2016},
pages = {343--414},
doi = {10.1090/bull/1528},
eprint = {1510.00971},
archivePrefix = {arXiv},
}
@article{kuchment2000,
author = {Kuchment, Peter and Vainberg, Boris},
title = {On absence of embedded eigenvalues for {S}chr{\"o}dinger operators with perturbed periodic potentials},
journal = {Comm. Partial Differential Equations},
volume = {25},
number = {9-10},
year = {2000},
pages = {1809--1826},
doi = {10.1080/03605300008821568},
eprint = {math-ph/9904016},
archivePrefix = {arXiv},
}
@article{battig1991,
author = {B{\"a}ttig, Daniel and Kn{\"o}rrer, Horst and Trubowitz, Eugene},
title = {A directional compactification of the complex {F}ermi surface},
journal = {Compos. Math.},
volume = {79},
number = {2},
year = {1991},
pages = {205--229},
}
@article{liu2022,
author = {Liu, Wencai},
title = {Irreducibility of the {F}ermi variety for discrete periodic {S}chr{\"o}dinger operators and embedded eigenvalues},
journal = {Geom. Funct. Anal.},
volume = {32},
number = {1},
year = {2022},
pages = {1--30},
doi = {10.1007/s00039-021-00587-z},
eprint = {2006.04733},
archivePrefix = {arXiv},
}
@misc{aim-complex-fermi-irreducibility,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Irreducibility of complex Fermi varieties}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/095-complex-fermi-irreducibility.md},
note = {Entry 095 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00335, "Irreducibility of complex Fermi varieties", clopen.solutions.