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An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian
Abstract
We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of su(1,1), and of a generalized version of it relevant in presence of non Hermitian generators.
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BibTeX
@misc{bagarello2026unified,
title = {{An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian}},
author = {Fabio Bagarello and Emanuele Balistreri and Sergey Kuzhel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08437},
eprint = {2610.08437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08437},
}
RIS
TY - UNPB
TI - An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian
AU - Fabio Bagarello
AU - Emanuele Balistreri
AU - Sergey Kuzhel
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08437
UR - https://arxiv.org/abs/2610.08437
AB - We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of su(1,1), and of a generalized version of it relevant in presence of non Hermitian generators.
ER -