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We give a brief overview of a simple and unified way, called the prepotential approach, to treat both exact and quasi-exact solvabilities of the one-dimensional Schr\"odinger equation. It is based on the prepotential together with Bethe…

Quantum Physics · Physics 2024-04-29 Choon-Lin Ho

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear…

Analysis of PDEs · Mathematics 2019-09-11 N. S. Papageorgiou , V. D. Rădulescu , D. D. Repovš

A two-interacting-qubit quantum Rabi-like model with vanishing transverse fields on the qubit-pair is studied. Independently of the coupling regime, this model can be exactly and unitarily reduced to two independent single-spin quantum Rabi…

An appropriate rational approximation to the eigenfunction of the Schr\"{o}dinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of Hankel determinants. The…

Mathematical Physics · Physics 2009-11-13 P. Amore , F. M. Fernandez

The symmetric quantum Rabi model (QRM) is integrable due to a discrete $\mathbb{Z}_2$-symmetry of the Hamiltonian. This symmetry is generated by a known involution operator, measuring the parity of the eigenfunctions. An experimentally…

Mathematical Physics · Physics 2023-02-21 Cid Reyes-Bustos , Daniel Braak , Masato Wakayama

Multi-qubit quantum Rabi model, which is a fundamental model describing light-matter interaction, plays an important role in various physical systems. In this paper, we propose a theoretical method to simulate multi-qubit quantum Rabi model…

Quantum Physics · Physics 2019-02-27 Jialun Li , Gangcheng Wang , Ruoqi Xiao , Chunfang Sun , Chunfeng Wu , Kang Xue

We apply a mixed quantum-classical (MQC) approach to the quantum Rabi model, involving a classical optical field coupled self-consistently to a quantum two-level system. Under the rotating wave approximation, we analytically show this…

Quantum Physics · Physics 2025-06-13 Ming-Hsiu Hsieh , Roel Tempelaar

Rabi oscillations typify the inherent nonlinearity of optical excitations in quantum dots. Using an integral kernel formulation to solve the 3D Maxwell-Bloch equations in ensembles of up to $10^4$ quantum dots, we observe features in Rabi…

Atomic and Molecular Clusters · Physics 2017-09-13 Connor Glosser , B. Shanker , Carlo Piermarocchi

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…

Number Theory · Mathematics 2013-02-19 Bryce Kerr

In the setting of exponential investors and uncertainty governed by Brownian motions we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential-quadratic…

Portfolio Management · Quantitative Finance 2014-09-30 Jin Hyuk Choi , Kasper Larsen

We discuss a universal algebraic approach to quasi-exactly solvable models which allows us to interpret them as constrained Hamiltonian systems with a finite number of physical states. Using this approach we reproduce well-known…

Mathematical Physics · Physics 2009-12-18 Sergey Klishevich

The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators…

High Energy Physics - Theory · Physics 2013-04-08 J. Harnad , P. Winternitz

New integrable multi-atom matter-radiation models with and without rotating wave approximation (RWA) are constructed and exactly solved through algebraic Bethe ansatz. The models with RWA are generated through ancestor model approach in an…

Exactly Solvable and Integrable Systems · Physics 2009-11-10 Anjan Kundu

We employ a squeeze operator transformation approach to solve the anisotropic quantum Rabi model that includes a diamagnetic term. By carefully adjusting the amplitude of the diamagnetic term, we demonstrate that the anisotropic Rabi model…

We investigate the quantum Jaynes-Cummings model - a particular case of the Gaudin model with one of the spins being infinite. Starting from the Bethe equations we derive Baxter's equation and from it a closed set of equations for the…

High Energy Physics - Theory · Physics 2011-02-16 Olivier Babelon , Dmitri Talalaev

The Rabi Hamiltonian, describing the interaction between a two-level atomic system and a single cavity mode of the electromagnetic field, is one of the fundamental models in quantum optics. The model becomes exactly solvable by considering…

Quantum Physics · Physics 2021-08-04 Giovanni Scala , Karolina Słowik , Paolo Facchi , Saverio Pascazio , Francesco Pepe

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…

Analysis of PDEs · Mathematics 2025-12-23 Philip Korman

We investigate first- and second-order quantum phase transitions of the anisotropic quantum Rabi model, in which the rotating- and counter-rotating terms are allowed to have different coupling strength. The model interpolates between two…

Quantum Physics · Physics 2017-12-06 Maoxin Liu , Stefano Chesi , Zu-Jian Ying , Xiaosong Chen , Hong-Gang Luo , Hai-Qing Lin

The occurrence of a second-order superradiant quantum phase transition is brought to light in a quantum system consisting of two interacting qubits coupled to the same quantized field mode. We introduce an appropriate thermodynamic-like…

Quantum Physics · Physics 2025-02-06 R. Grimaudo , G. Falci , A. Messina , E. Paladino , A. Sergi , E. Solano , D. Valenti

In this paper we show that several authors have derived wrong physical conclusions from a gross misunderstanding of the exact eigenvalues and eigenfunctions of a conditionally-solvable quantum-mechanical model. It consists of an eigenvalue…

Quantum Physics · Physics 2024-12-17 Francisco M. Fernández