Completeness of modes for slowly growing imaginary Schrödinger potentials
For $0 < \alpha \le 1/2$ let $A_\alpha$ be the operator $A_\alpha u = -u'' + i x^\alpha u$ on $L^2(0,\infty)$ with domain $$D(A_\alpha) = \{u \in L^2(0,\infty) : u \in H^2_{\mathrm{loc}}([0,\infty)),\ -u'' + i x^\alpha u \in L^2(0,\infty),\ u(0) = 0\}.$$ More generally, for $\alpha > 0$ and $c \in \mathbb{C}$ with $|\arg c| < \pi$, the operator $L_{c,\alpha} = -d^2/dx^2 + c x^\alpha$ with the same domain and Dirichlet condition has compact inverse and simple eigenvalues; for $0 < |\arg c| < \pi$ its eigenfunctions do not form a Riesz basis of $L^2(0,\infty)$. Their completeness in $L^2(0,\infty)$ was known for $|\arg c| < 2\pi\alpha/(\alpha + 2)$ (Almog and Helffer; Savchuk and Shkalikov), which for $c = i$ means $\alpha > 2/3$. Tumanov proved completeness for $|\arg c| < \theta_0(\alpha)$, where $\theta_0$ is continuous on $(0,2)$ with $2\pi\alpha/(\alpha + 2) < \theta_0(\alpha) < \pi\alpha$; in particular, for $c = i$ it holds for every $\alpha > \alpha_0$, for some $\alpha_0 < 2/3$.
Problem. Is the closed linear span of the eigenfunctions of $A_\alpha$ equal to $L^2(0,\infty)$ for every $0 < \alpha \le 1/2$? Only completeness is asked, not a Riesz basis or any other stable expansion.
Source: Almog (2015), open problem 1.1 of the American Institute of Mathematics workshop on mathematical aspects of physics with non-self-adjoint operators, for 0 < α ≤ 2/3, on the line or on the half-line with a Dirichlet condition; stated on the half-line for 0 < α ≤ 1/2 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- CA, FA
- Keywords
- AIM Open Applied Problems, completeness of eigenfunctions, non-self-adjoint operators, Schrödinger operators
References
@misc{almog2015nsa,
author = {Almog, Yaniv},
title = {Completeness of eigenfunctions for {S}chr{\"o}dinger operators with complex potentials},
howpublished = {List of open problems, AIM workshop ``Mathematical aspects of physics with non-self-adjoint operators'', American Institute of Mathematics, San Jose, California, 8--12 June 2015},
year = {2015},
url = {https://aimath.org/pastworkshops/nonselfadjointproblems.pdf},
note = {Problem 1.1},
}
@article{tumanov2022,
author = {Tumanov, Sergey},
title = {Completeness theorem for the system of eigenfunctions of the complex {S}chr{\"o}dinger operator {$\mathscr{L}_{c,\alpha} = -d^2/dx^2 + cx^\alpha$}},
journal = {J. Differential Equations},
volume = {319},
year = {2022},
pages = {80--99},
doi = {10.1016/j.jde.2022.02.010},
eprint = {2101.01680},
archivePrefix = {arXiv},
}
@article{savchuk2017,
author = {Savchuk, A. M. and Shkalikov, A. A.},
title = {Spectral properties of the complex {A}iry operator on the half-line},
journal = {Funct. Anal. Appl.},
volume = {51},
number = {1},
year = {2017},
pages = {66--79},
doi = {10.1007/s10688-017-0168-1},
eprint = {1702.00627},
archivePrefix = {arXiv},
}
@article{almog2015cpde,
author = {Almog, Yaniv and Helffer, Bernard},
title = {On the spectrum of non-selfadjoint {S}chr{\"o}dinger operators with compact resolvent},
journal = {Comm. Partial Differential Equations},
volume = {40},
number = {8},
year = {2015},
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doi = {10.1080/03605302.2015.1025978},
eprint = {1410.5122},
archivePrefix = {arXiv},
}
@article{tumanov2021,
author = {Tumanov, Sergey},
title = {Completeness theorem for the system of eigenfunctions of the complex {S}chr{\"o}dinger operator {$\mathscr{L}_c = -d^2/dx^2 + cx^{2/3}$}},
journal = {J. Funct. Anal.},
volume = {280},
number = {7},
year = {2021},
pages = {108820},
doi = {10.1016/j.jfa.2020.108820},
eprint = {1903.10594},
archivePrefix = {arXiv},
}
@misc{aim-imaginary-sublinear-potential-completeness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Completeness of modes for slowly growing imaginary Schrödinger potentials}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/428-imaginary-sublinear-potential-completeness.md},
note = {Entry 428 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-00355, "Completeness of modes for slowly growing imaginary Schrödinger potentials", clopen.solutions.