Absolute continuity for smooth periodic scalar elliptic operators
Let $d \ge 3$, let $A = (a_{ij})$ be a real symmetric $C^\infty$, $\mathbb{Z}^d$-periodic matrix field with $cI \le A(x) \le CI$ for some $0 < c \le C < \infty$, and let $V$ be real, smooth and periodic. Let $H$ be the self-adjoint operator on $L^2(\mathbb{R}^d)$ associated with the quadratic form $$q[u] = \int_{\mathbb{R}^d} \bigl( \nabla\bar u \cdot A \nabla u + V |u|^2 \bigr) \, dx, \qquad u \in H^1(\mathbb{R}^d).$$ For every periodic elliptic operator the singular continuous spectrum is empty, and an eigenvalue occurs only if a band function is constant on an open set. For $A = I$, Thomas proved that the spectrum is absolutely continuous (in dimension three; the proof extends to all dimensions); Birman and Suslina extended this to $A = \omega(x) A_0$ with $\omega$ a positive periodic function and $A_0$ a constant matrix, and for $d = 2$ it holds for every smooth $A$ (Morame). For the Laplace–Beltrami operator $-\Delta_g + V$ of a smooth periodic Riemannian metric $g$ in dimension $d \ge 3$, absolute continuity is known when the operator is invariant under the reflection $x_1 \mapsto -x_1$ (Friedlander), and when a conformal multiple of $g$ is a product of one Euclidean direction and a metric in the remaining variables (Krupchyk and Uhlmann). Filonov constructed a second-order periodic elliptic operator with non-empty point spectrum whose leading coefficients are just below the regularity needed for unique continuation.
Problem. For every $d \ge 3$ and all such $A$ and $V$, is every spectral measure of $H$ absolutely continuous with respect to Lebesgue measure? No reflection symmetry or product structure is assumed.
Source: Kuchment (2016), Conjecture 6.13, for self-adjoint second-order periodic elliptic scalar operators with smooth coefficients, after Birman and Suslina (1999), who conjectured absolute continuity for the same quadratic form with a merely bounded measurable metric. Stated by AIM — Open Applied Problems for divergence-form operators with a potential in dimension d ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, periodic elliptic operators, spectral measures
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},
}
@inproceedings{suslina2000,
author = {Suslina, Tatiana},
title = {Absolute continuity of the spectrum of periodic operators of mathematical physics},
booktitle = {Journ{\'e}es {\'E}quations aux d{\'e}riv{\'e}es partielles},
year = {2000},
pages = {Exp. No. XVIII, 1--13},
doi = {10.5802/jedp.582},
}
@article{thomas1973,
author = {Thomas, Lawrence E.},
title = {Time dependent approach to scattering from impurities in a crystal},
journal = {Comm. Math. Phys.},
volume = {33},
number = {4},
year = {1973},
pages = {335--343},
doi = {10.1007/BF01646745},
}
@article{birman1999,
author = {Birman, M. Sh. and Suslina, T. A.},
title = {A periodic magnetic {H}amiltonian with a variable metric. {T}he problem of absolute continuity},
journal = {Algebra i Analiz},
volume = {11},
number = {2},
year = {1999},
pages = {1--40},
note = {In Russian; English translation in St. Petersburg Math. J. 11 (2000), no. 2, 203--232},
}
@article{morame1998,
author = {Morame, Abderemane},
title = {Absence of singular spectrum for a perturbation of a two-dimensional {L}aplace--{B}eltrami operator with periodic electromagnetic potential},
journal = {J. Phys. A},
volume = {31},
number = {37},
year = {1998},
pages = {7593--7601},
doi = {10.1088/0305-4470/31/37/017},
}
@article{friedlander2002,
author = {Friedlander, L.},
title = {On the spectrum of a class of second order periodic elliptic differential operators},
journal = {Comm. Math. Phys.},
volume = {229},
number = {1},
year = {2002},
pages = {49--55},
doi = {10.1007/s00220-002-0675-6},
}
@article{krupchyk2017,
author = {Krupchyk, Katsiaryna and Uhlmann, Gunther},
title = {Absolute continuity of the periodic {S}chr{\"o}dinger operator in transversal geometry},
journal = {J. Eur. Math. Soc.},
volume = {19},
number = {2},
year = {2017},
pages = {531--550},
doi = {10.4171/JEMS/673},
eprint = {1312.2989},
archivePrefix = {arXiv},
}
@article{filonov2001,
author = {Filonov, N.},
title = {Second-order elliptic equation of divergence form having a compactly supported solution},
journal = {J. Math. Sci. (N.Y.)},
volume = {106},
number = {3},
year = {2001},
pages = {3078--3086},
doi = {10.1023/A:1011379807662},
}
@misc{aim-periodic-elliptic-absolute-continuity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Absolute continuity for smooth periodic scalar elliptic operators}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/094-periodic-elliptic-absolute-continuity.md},
note = {Entry 094 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-00368, "Absolute continuity for smooth periodic scalar elliptic operators", clopen.solutions.