Integrability of the Rabi model

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.     

Integrability of the periodic KM system

We study the integrability of the periodic Kac-van Moerbeke system. We give a bi-Hamiltonian formulation and a Lax pair containing a spectral parameter. Using Griffiths aproach we linearize the system on the Jacobian of the associated spectral curve.     

On Testing Integrability

Abstract

We demonstrate, using the symbolic method together with p-adic and resultant methods, the existence of systems with exactly one or two generalized symmetries. Since the existence of one or two symmetries is often taken as a sure sign (or as the definition) of integrability, that is, the existence of symmetries on infinitely many orders, this shows that such practice is devoid of any mathematical foundation. Extensive computations show that systems with one symmetry are rather common, and with two symmetries are fairly rare, at least within the class we have been considering in this paper.     

Multi-quark hamiltonian and the nuclear force

We propose a system of simple rules which give the binding energy and approximate ground state structure of a nucleus, dependent only on the number of protons and neutrons making it up. This “nuclear code” is based on theN-quark Schrödinger equation and uses ideas familiar from atomic chemistry. We demonstrate some of the important ramifications by computing the⁴He elastic form factor. The nuclear code performs better than its atomic counterpart.     

Non-perturbative solution of the Bloch-Mathieu hamiltonian system

A non-perturbative wavefunction is obtained for the Bloch-Mathieu hamiltonian system. The explicit expression of the Floquet exponent indicates that the wavefunction depends non-analytically on E and quadratically on V₁, both demonstrating the illegitimacy of the perturbation expansion. The energy band is also given by an analytic expression instead of Kramers' indirect construction.     

On the group of the independent-particle hamiltonian operator

The detailed group structure of the independent-particle hamiltonian is derived in terms of interior and exterior products of groups describing various types of symmetry operations which commute with the hamiltonian. Modifications of this group due to external fields are examined, as are the concepts of the equivalence of particles. Simplifications due to isomorphisms and homomorphisms are outlined. Applications involving representations within this construction are examined.     

An exactly solvable anharmonic Bohr hamiltonian and its equivalent boson hamiltonian

The eigenenergies and eigenfunctions of the bound states of an anharmonic quadrupole oscillator are derived in closed form. An interacting boson model hamiltonian is found which has the same eigenspectrum.     

Many-time hamiltonian gravitation theory

It is shown that given any “good” coordinate condition in Hamiltonian general relativity one can construct an associated many-time formulation in which the constraints can be solved for some of the momenta as functionals of the remaining canonical variables. Since good coordinate conditions appear to be available for both open and closed spaces it follows that the functional wave equation for general relativity can be always put in a Tomonaga-Schwinger form. The implications of this result and some open problems are briefly discussed.     

Phipps's lagrangian and hamiltonian

In this work it is derived Phipps's Lagrangian function and Phipps's Hamiltonian function. This allows the utilization of Phipps's interaction in the Lagrangian and Hamiltonian frameworks.Dedicated to Thomas E. Phipps, Jr. on the occasion of his 70th birthday (1995).     

The molecular vibration-rotation hamiltonian

By use of commutation relations and sum rules Watson [1] has simplified the Darling and Dennison hamiltonian and obtained

The quantity H - U is thus identical to the analytical expression of the classical hamiltonian.

By reconsidering completely the question of the derivation of the quantum mechanical hamiltonian we intend to prove that the U term comes only from the particular method chosen to derive the quantum mechanical hamiltonian from its wave mechanical representations. From this we conclude that in the usual model, the quantum mechanical vibration-rotation hamiltonian has the same form as the classical one.     

On the stochastic properties of the thermodynamic hamiltonian

If the generalized momentum in the hamiltonian equations is considered as a rapidly varying stochastic function of time then a solution of the master equation is most easily obtained by applying the central limit theorem in much the same way as one does with the Langevin equation.     

Models of relativistic hamiltonian interactions

A model of a relativistically invariant Hamiltonian 2-particle interaction is given. It is classical in the sense of having 6 degrees of freedom. This model shows that the author's earlier general definition of such systems is not vacuous. In this model the forces of interaction die away as the particles are removed from each other.     

Effective hamiltonian of the nuclear moments electronic shielding

An information is given allowing to use the second quantization and the effective operator methods in the ligand field theory. The operator was constructed accounting for the interaction of the multi-shell electronic configurations through a one-particle perturbation V≪R. The expression obtained is believed to be useful in microscopic calculations and phenomenological interpretation of spectroscopic experiments. As an illustration, the effective hamiltonian of the nuclear moments electronic shielding has been deduced. It was found, in particular, that the dipolar part of the hyperfine interaction contributes to the shift of the nuclear g-factor in the systems with the electronic spin S>0.     

On hamiltonian formulation of cosmologies

Novello et al [1,2] have shown that it is possible to find a pair of canonically conjugate variables (written in terms of gauge-invariant variables) so as to obtain a Hamiltonian that describes the dynamics of a cosmological system. This opens up the way to the usual technique of quantization. Elbaz et al [4] have applied this method to the Hamiltonian formulation of FRW cosmological equations. This note presents a generalization of this approach to a variety of cosmologies. A general Schrödinger wave equation has been derived and exact solutions have been worked out for the stiff matter era for some cosmological models. It is argued that these solutions appear to hint at their possible relevance in the early phase of cosmological evolution.     

A hamiltonian approach to the roper resonance

The breathing modes along with the spin-isospin zero modes of the ω-stabilized skyrmion are quantized using Dirac's method. The result is a hamiltonian for the nucleon, the δ-isobar and the Roper with explicit meson couplings. The linear pion coupling is used to evaluate the decay width of the Roper into πN.