PreprintCC BY 4.0arXiv
On the zero sets of the Bargmann Fock space
Abstract
At the critical density, the classical density theory of the Bargmann--Fock space no longer determines exact zero sets. We show that this failure is substantial: the complete radial counting function together with any prescribed finite collection of angular moments does not determine exact realizability, and the finite-dimensional relation between the dimension of the vanishing space and the zero-insertion capacity breaks down in infinite dimension. For arbitrary locally finite multisets, we establish a variational characterization of exact zero sets in terms of a renormalized logarithmic potential and relative entropy. This characterization yields sharp criteria for several classes of critical configurations.
Related
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BibTeX
@misc{hou2026zero,
title = {{On the zero sets of the Bargmann Fock space}},
author = {Shengzhao Hou and Pan Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37704},
eprint = {2609.37704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the zero sets of the Bargmann Fock space
AU - Shengzhao Hou
AU - Pan Ma
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37704
UR - https://arxiv.org/abs/2609.37704
AB - At the critical density, the classical density theory of the Bargmann--Fock space no longer determines exact zero sets. We show that this failure is substantial: the complete radial counting function together with any prescribed finite collection of angular moments does not determine exact realizability, and the finite-dimensional relation between the dimension of the vanishing space and the zero-insertion capacity breaks down in infinite dimension. For arbitrary locally finite multisets, we establish a variational characterization of exact zero sets in terms of a renormalized logarithmic potential and relative entropy. This characterization yields sharp criteria for several classes of critical configurations.
ER -