Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.08437

An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian

Fabio Bagarello·Emanuele Balistreri·Sergey Kuzhel
View original source ↗Read source document ↗Export metadata ↓

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 $\mathfrak{su}(1,1)$, and of a generalized version of it relevant in presence of non Hermitian generators.

Explore related subjects

math-phmath.MPquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fabio Bagarello, Emanuele Balistreri, Sergey Kuzhel. 2026-10-06. An unified approach for the (inverted) quantum harmonic oscillator and the Berry-Keating hamiltonian. https://arxiv.org/abs/2610.08437

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$

In this paper, we construct the Hamiltonian systems attached to generic $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connections with an arbitrary number of unramified poles of arbitrary orders, under the assumption that every leading polar coefficient is regular semisimple. In particular, we propose the Lax pairs and Hamiltonian evolutions expressed in terms of irregular times and monodromies associated to the poles as well as $g$ pairs of Darboux coordinates defined as the apparent singularities arising in the oper gauge. Moreover, we also provide a reduction of the isomonodromic deformations to a subset of $g$ non-trivial isomonodromic deformations. This reduction is equivalent to a map reducing the set of irregular times to only $g$ non-trivial isomonodromic times. We apply our construction to all genus-one cases covered by these hypotheses and recover the standard Painlevé equations $2$--$6$. We finally make the connection with the topological recursion and the quantization of classical spectral curves from this perspective.

math-ph↗
arXiv2026

Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlevé $1$ case $(g=1)$ and the next two elements of the Painlevé $1$ hierarchy.

math-ph↗
arXiv2026

The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

Two approaches to the Painlevé I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is established, identifying these setups explicitly. In particular, this yields new expressions for the Lax matrices and Hamiltonians.

math-ph↗