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arXiv · 2609.37704

On the zero sets of the Bargmann Fock space

Shengzhao Hou·Pan Ma
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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.

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Shengzhao Hou, Pan Ma. 2026-09-29. On the zero sets of the Bargmann Fock space. https://arxiv.org/abs/2609.37704

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