PreprintCC BY 4.0arXiv
Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher
Abstract
Let B3=B(0,3)⊂ℝ3. For every sufficiently large integer k, we construct a nonzero real function uk∈ C2(B3̅) and a real potential Vk∈ L∞(B3) such that Δuk=Vkuk, uk|∂ B3=0, ord0uk=k, and |Vk|∞≤ Ck3/2. Consequently, for every sufficiently large N, there is such a Dirichlet solution with potential norm smaller than N and vanishing order at least cN2/3. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that C(1+|V|∞2/3) is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.
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BibTeX
@misc{feng2026vanishing,
title = {{Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher}},
author = {Shuowen Feng and Zaihui Gan and Renjin Jiang and Fanghua Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01615},
eprint = {2610.01615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01615},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher
AU - Shuowen Feng
AU - Zaihui Gan
AU - Renjin Jiang
AU - Fanghua Lin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01615
UR - https://arxiv.org/abs/2610.01615
AB - Let B_3=B(0,3)⊂ℝ^3. For every sufficiently large integer k, we construct a nonzero real function u_k∈ C^2(B_3̅) and a real potential V_k∈ L^∞(B_3) such that Δu_k=V_ku_k, u_k|_∂ B_3=0, ord_0u_k=k, and |V_k|_∞≤ Ck^3/2. Consequently, for every sufficiently large N, there is such a Dirichlet solution with potential norm smaller than N and vanishing order at least cN^2/3. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that C(1+|V|_∞^2/3) is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.
ER -