Simon’s multidimensional weighted square-integrability conjecture
For $d \ge 2$ let $V : \mathbb{R}^d \to \mathbb{R}$ be bounded and measurable with $$\int_{\mathbb{R}^d} \frac{|V(x)|^2}{(1+|x|)^{d-1}}\,dx < \infty,$$ and let $H = -\Delta + V$ on $H^2(\mathbb{R}^d) \subset L^2(\mathbb{R}^d)$. In dimension one, Deift and Killip proved that for every $V \in L^2(\mathbb{R})$ the absolutely continuous spectrum of $-d^2/dx^2 + V$ is $[0,\infty)$, with multiplicity two; as Simon notes, this gives the statement below for spherically symmetric $V$. Denisov proved the analogue for the massless Dirac operator on $\mathbb{R}^3$: for bounded $V$ satisfying the condition above, its absolutely continuous spectrum is all of $\mathbb{R}$.
Problem. Is it true that for every $d \ge 2$ and every such $V$, the absolutely continuous part of $H$ has infinite multiplicity at Lebesgue-almost every energy $E > 0$? Additional singular spectrum is not excluded.
Source: Simon (2000), Problem 8; see Simon (2019), Conjecture 20.2, and Denisov and Kiselev (2007), (5.1). Stated for bounded potentials by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- absolutely continuous spectrum, AIM Open Applied Problems, decaying potentials, Schrödinger operators
References
@incollection{simon2000,
author = {Simon, Barry},
title = {Schr{\"o}dinger operators in the twenty-first century},
booktitle = {Mathematical Physics 2000},
editor = {Fokas, Athanassios and Grigoryan, Alexander and Kibble, Tom and Zegarlinski, Boguslaw},
publisher = {Imperial College Press},
address = {London},
year = {2000},
pages = {283--288},
doi = {10.1142/9781848160224_0014},
}
@article{simon2019,
author = {Simon, Barry},
title = {Tosio {K}ato's work on non-relativistic quantum mechanics, {P}art 2},
journal = {Bull. Math. Sci.},
volume = {9},
number = {1},
year = {2019},
pages = {1950005},
doi = {10.1142/S166436071950005X},
eprint = {1711.00528},
archivePrefix = {arXiv},
}
@incollection{denisov2007,
author = {Denisov, Sergey A. and Kiselev, Alexander},
title = {Spectral properties of {S}chr{\"o}dinger operators with decaying potentials},
booktitle = {Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday},
series = {Proc. Sympos. Pure Math.},
volume = {76, Part 2},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2007},
pages = {565--589},
doi = {10.1090/pspum/076.2/2307748},
}
@article{deift1999,
author = {Deift, Percy and Killip, Rowan},
title = {On the absolutely continuous spectrum of one-dimensional {S}chr{\"o}dinger operators with square summable potentials},
journal = {Comm. Math. Phys.},
volume = {203},
number = {2},
year = {1999},
pages = {341--347},
doi = {10.1007/s002200050615},
}
@article{denisov2004,
author = {Denisov, Sergey A.},
title = {On the absolutely continuous spectrum of {D}irac operator},
journal = {Comm. Partial Differential Equations},
volume = {29},
number = {9-10},
year = {2004},
pages = {1403--1428},
doi = {10.1081/PDE-200037758},
eprint = {math-ph/0309033},
archivePrefix = {arXiv},
}
@misc{aim-simon-weighted-l2,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Simon’s multidimensional weighted square-integrability conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/097-simon-weighted-l2.md},
note = {Entry 097 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-00381, "Simon’s multidimensional weighted square-integrability conjecture", clopen.solutions.