Spectral Theory of an Exactly Solvable Quantum-Optical Oscillator
Abstract
We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line (−∞,Ω], where Ω is the atomic resonance frequency, while the spectrum above Ω consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy ωn = ((3πg n)/4)1/2 + Ω/4 + O(n−1/2). The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above Ω and an inverted oscillator below Ω.
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BibTeX
@misc{johar2026spectral,
title = {{Spectral Theory of an Exactly Solvable Quantum-Optical Oscillator}},
author = {Sam Johar and Joseph Kraisler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18421},
eprint = {2610.18421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18421},
}
RIS
TY - UNPB
TI - Spectral Theory of an Exactly Solvable Quantum-Optical Oscillator
AU - Sam Johar
AU - Joseph Kraisler
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.18421
UR - https://arxiv.org/abs/2610.18421
AB - We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line (−∞,Ω], where Ω is the atomic resonance frequency, while the spectrum above Ω consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy ω_n = ((3πg n)/4)^1/2 + Ω/4 + O(n^−1/2). The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above Ω and an inverted oscillator below Ω.
ER -