Full-order resonance growth for every nonzero real compact potential
Let $d \ge 3$ be odd and let $V \in L^\infty(\mathbb{R}^d)$ be real-valued, compactly supported and not zero almost everywhere. The resolvent $$R_V(z) = (-\Delta + V - z^2)^{-1} : L^2_{\mathrm{comp}}(\mathbb{R}^d) \to L^2_{\mathrm{loc}}(\mathbb{R}^d)$$ continues meromorphically from $\operatorname{Im} z > 0$ to all of $\mathbb{C}$; its poles are the resonances of $V$. Let $N_V(r)$ be the number of resonances $z$ with $|z| \le r$, counted with algebraic multiplicity. Zworski proved that $N_V(r) \le C r^d$ for every bounded compactly supported potential, and Christiansen and Hislop proved that $\limsup_{r \to \infty} \log N_V(r) / \log r = d$ for generic real-valued, and for generic complex-valued, bounded compactly supported potentials. Christiansen and Sá Barreto proved that $\limsup_{r \to \infty} N_V(r)/r > 0$ for every nonzero real-valued $V \in C_c^\infty(\mathbb{R}^d)$. Smith and Zworski proved that every $V$ as above has at least one resonance, and infinitely many if in addition $V \in H^{(d-3)/2}(\mathbb{R}^d)$. Christiansen found complex-valued compactly supported potentials with no resonances at all, in odd dimensions $d \ge 3$.
Problem. Are there, for every such $d$ and $V$, constants $c_V > 0$ and $r_V < \infty$ such that $N_V(r) \ge c_V r^d$ for every $r \ge r_V$?
Source: Zworski (2017), Section 2.7, Conjecture 1, for every odd dimension; stated for odd d ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, scattering resonances, Schrödinger operators
References
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author = {Zworski, Maciej},
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doi = {10.1007/s13373-017-0099-4},
eprint = {1609.03550},
archivePrefix = {arXiv},
}
@article{zworski1989duke,
author = {Zworski, Maciej},
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}
@article{christiansen2005,
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archivePrefix = {arXiv},
}
@article{christiansen1999,
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@article{sabarreto2001,
author = {S{\'a} Barreto, Ant{\^o}nio},
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doi = {10.3233/ASY-2001-456},
}
@article{smith2016,
author = {Smith, Hart F. and Zworski, Maciej},
title = {Heat traces and existence of scattering resonances for bounded potentials},
journal = {Ann. Inst. Fourier (Grenoble)},
volume = {66},
number = {2},
year = {2016},
pages = {455--475},
doi = {10.5802/aif.3016},
eprint = {1411.5420},
archivePrefix = {arXiv},
}
@article{christiansen2006,
author = {Christiansen, T. J.},
title = {Schr{\"o}dinger operators with complex-valued potentials and no resonances},
journal = {Duke Math. J.},
volume = {133},
number = {2},
year = {2006},
pages = {313--323},
doi = {10.1215/S0012-7094-06-13324-0},
}
@misc{aim-full-order-potential-resonances,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Full-order resonance growth for every nonzero real compact potential}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/116-full-order-potential-resonances.md},
note = {Entry 116 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-00347, "Full-order resonance growth for every nonzero real compact potential", clopen.solutions.