A renormalized Lee Model: resolvent and spectral analysis
Abstract
We construct and analyze a renormalized three-dimensional massive Lee model; its renormalized Hamiltonian is defined intrinsically through an abstract construction which explicitly provides the operator domain, action and a Kre\uın-like resolvent formula. Conservation of the total excitation number reduces the spectral analysis of the full Hamiltonian to a sector-wise study of the spectral profile of an explicit operator pencil, playing the role of an operator-valued Weyl function. We construct essential spectrum components in a sector generated by lower excitation sectors and obtain the inclusion in the expected sector-wise HVZ type formula; the full formula is proved in the first two sectors and, under explicit conditions, in every sector. Below the free or one-excitation threshold, the point spectrum in every higher sector is described by a bounded self-adjoint nonnegative Birman--Schwinger operator. In every sector with at least two excitations we prove absence of sub-threshold spectrum for arbitrary coupling when the renormalized gap is not smaller than the boson mass, and under an explicit weak-coupling condition when it is smaller. In the latter case we also derive a sufficient condition for the existence of a simple bound state in the two-excitation sector.
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BibTeX
@misc{cacciapuoti2026renormalized,
title = {{A renormalized Lee Model: resolvent and spectral analysis}},
author = {Claudio Cacciapuoti and Federica Muscolino and Diego Noja and Andrea Posilicano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34980},
eprint = {2609.34980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34980},
}
RIS
TY - UNPB
TI - A renormalized Lee Model: resolvent and spectral analysis
AU - Claudio Cacciapuoti
AU - Federica Muscolino
AU - Diego Noja
AU - Andrea Posilicano
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34980
UR - https://arxiv.org/abs/2609.34980
AB - We construct and analyze a renormalized three-dimensional massive Lee model; its renormalized Hamiltonian is defined intrinsically through an abstract construction which explicitly provides the operator domain, action and a Kre\uın-like resolvent formula. Conservation of the total excitation number reduces the spectral analysis of the full Hamiltonian to a sector-wise study of the spectral profile of an explicit operator pencil, playing the role of an operator-valued Weyl function. We construct essential spectrum components in a sector generated by lower excitation sectors and obtain the inclusion in the expected sector-wise HVZ type formula; the full formula is proved in the first two sectors and, under explicit conditions, in every sector. Below the free or one-excitation threshold, the point spectrum in every higher sector is described by a bounded self-adjoint nonnegative Birman--Schwinger operator. In every sector with at least two excitations we prove absence of sub-threshold spectrum for arbitrary coupling when the renormalized gap is not smaller than the boson mass, and under an explicit weak-coupling condition when it is smaller. In the latter case we also derive a sufficient condition for the existence of a simple bound state in the two-excitation sector.
ER -