Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space
Abstract
We study the stationary spectrum of a coherently driven single-mode Kerr cavity in Bargmann--Fock space. For positive Kerr coupling the Hamiltonian is self-adjoint, bounded below, and has compact resolvent. For nonzero drive, a minimal-solution continued fraction yields an exact scalar spectral condition, with all eigenvalues simple. A Liouville transformation identifies the eigenvalue equation with a degenerate double-confluent Heun problem and classifies its singularities. An Olver-type Volterra construction gives normalized sectorial solutions at infinity with explicit error bounds. In the strong-drive regime, the leading turning-point energy coefficient is proved, whereas the next coefficient remains conditional on a stated global-connection conjecture; independent Bogoliubov and complex-WKB calculations provide consistency checks. A formal weak-drive expansion is verified from the Bargmann recurrence, and its upper-edge tridiagonal series is summed in closed form. Exploratory WKB iteration portraits are included only as visual aids and do not establish the conjectural global connection.
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BibTeX
@misc{janowicz2026operatortheoretic,
title = {{Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space}},
author = {Maciej Janowicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16969},
eprint = {2609.16969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16969},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space
AU - Maciej Janowicz
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16969
UR - https://arxiv.org/abs/2609.16969
AB - We study the stationary spectrum of a coherently driven single-mode Kerr cavity in Bargmann--Fock space. For positive Kerr coupling the Hamiltonian is self-adjoint, bounded below, and has compact resolvent. For nonzero drive, a minimal-solution continued fraction yields an exact scalar spectral condition, with all eigenvalues simple. A Liouville transformation identifies the eigenvalue equation with a degenerate double-confluent Heun problem and classifies its singularities. An Olver-type Volterra construction gives normalized sectorial solutions at infinity with explicit error bounds. In the strong-drive regime, the leading turning-point energy coefficient is proved, whereas the next coefficient remains conditional on a stated global-connection conjecture; independent Bogoliubov and complex-WKB calculations provide consistency checks. A formal weak-drive expansion is verified from the Bargmann recurrence, and its upper-edge tridiagonal series is summed in closed form. Exploratory WKB iteration portraits are included only as visual aids and do not establish the conjectural global connection.
ER -