Subunit-moment eigenvalue bounds for complex Schrödinger potentials
Let $d \ge 1$, $0 < \gamma < 1$, with $\gamma \ge 1/2$ when $d = 1$, and $\sigma > d/2$. For complex-valued $V \in L^{\gamma + d/2}(\mathbb{R}^d)$ let $H = -\Delta + V$ be the m-sectorial operator on $L^2(\mathbb{R}^d)$ defined by its quadratic form. Its spectrum in $\mathbb{C} \setminus [0, \infty)$ consists of isolated eigenvalues $E_j$ of finite algebraic multiplicity, listed with algebraic multiplicity, and $\delta(z) = \operatorname{dist}(z, [0, \infty))$. For real $V$ the inequality below is the Lieb–Thirring inequality, since then $\delta(E_j) = |E_j|$. For $\gamma \ge 1$ and every $\sigma > d/2$ it was proved by Demuth, Hansmann and Katriel, who obtained it from the bound of Frank, Laptev, Lieb and Seiringer for eigenvalues outside a sector around $[0, \infty)$. At $\sigma = d/2$ it fails: Bögli and Štampach for $d = 1$ and $\gamma \ge 1/2$, and Bögli, Petpradittha and Štampach for $d \ge 2$, gave counterexamples to the question of Demuth, Hansmann and Katriel. Bögli and Petpradittha extended the bound of Demuth, Hansmann and Katriel, for every $\sigma > d/2$, to $\gamma > 1/2$ when $d = 1$, $\gamma > 0$ when $d = 2$ and $\gamma \ge 0$ when $d \ge 3$.
Problem. Is there a constant $C_{d,\gamma,\sigma}$, independent of $V$, such that $$\sum_j |E_j|^{-\sigma} \delta(E_j)^{\gamma + \sigma} \le C_{d,\gamma,\sigma} \int_{\mathbb{R}^d} |V(x)|^{\gamma + d/2} \, dx$$ for every complex-valued $V \in L^{\gamma + d/2}(\mathbb{R}^d)$, the left side being read as a series of nonnegative terms when there are infinitely many eigenvalues?
Source: Cuenin and Frank (2024), Question 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, complex potentials, Lieb–Thirring inequalities, non-self-adjoint operators
References
@misc{cuenin2024,
author = {Cuenin, Jean-Claude and Frank, Rupert L.},
title = {Open problem: violation of locality for {S}chr{\"o}dinger operators with complex potentials},
year = {2024},
eprint = {2409.11285},
archivePrefix = {arXiv},
}
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eprint = {0908.2188},
archivePrefix = {arXiv},
}
@article{frank2006,
author = {Frank, Rupert L. and Laptev, Ari and Lieb, Elliott H. and Seiringer, Robert},
title = {{L}ieb--{T}hirring inequalities for {S}chr{\"o}dinger operators with complex-valued potentials},
journal = {Lett. Math. Phys.},
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eprint = {math-ph/0605017},
archivePrefix = {arXiv},
}
@article{bogli2021,
author = {B{\"o}gli, Sabine and {\v{S}}tampach, Franti{\v{s}}ek},
title = {On {L}ieb--{T}hirring inequalities for one-dimensional non-self-adjoint {J}acobi and {S}chr{\"o}dinger operators},
journal = {J. Spectr. Theory},
volume = {11},
number = {3},
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eprint = {2004.09794},
archivePrefix = {arXiv},
}
@article{bogli2025,
author = {B{\"o}gli, Sabine and Petpradittha, Sukrid and {\v{S}}tampach, Franti{\v{s}}ek},
title = {On {L}ieb--{T}hirring inequalities for multidimensional {S}chr{\"o}dinger operators with complex potentials},
journal = {Integral Equations Operator Theory},
volume = {97},
number = {3},
year = {2025},
pages = {25},
doi = {10.1007/s00020-025-02810-0},
eprint = {2510.02192},
archivePrefix = {arXiv},
}
@misc{bogli2026,
author = {B{\"o}gli, Sabine and Petpradittha, Sukrid},
title = {On the discrete spectrum of non-selfadjoint operators with applications to {S}chr{\"o}dinger operators with complex potentials},
year = {2026},
eprint = {2602.02359},
archivePrefix = {arXiv},
}
@misc{aim-complex-schrodinger-subunit-moments,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Subunit-moment eigenvalue bounds for complex Schrödinger potentials}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/422-complex-schrodinger-subunit-moments.md},
note = {Entry 422 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-00417, "Subunit-moment eigenvalue bounds for complex Schrödinger potentials", clopen.solutions.