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arXiv · 2609.16969

Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space

Maciej Janowicz
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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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Maciej Janowicz. 2026-09-15. Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space. https://arxiv.org/abs/2609.16969

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