Coherent-State Integrals Beyond the Domain of Operator Products
Abstract
Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain eμa2eλa†2 of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.
Related
- A one-dimensional coherent domain with non-Prüfer integral closure
- Direct products of Gorenstein flat modules over coherent rings
- Generalized free wreath products and their operator algebras
- Actions of quantum groups on dual operator spaces and their crossed products
- Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products
BibTeX
@misc{cordshooli2026coherentstate,
title = {{Coherent-State Integrals Beyond the Domain of Operator Products}},
author = {Ghasem Asadi Cordshooli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19702},
eprint = {2610.19702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19702},
}
RIS
TY - UNPB
TI - Coherent-State Integrals Beyond the Domain of Operator Products
AU - Ghasem Asadi Cordshooli
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19702
UR - https://arxiv.org/abs/2610.19702
AB - Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain e^μa^2e^λa^†2 of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.
ER -