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Transition probabilities for a class of two level systems described by explicitly time dependent Hamiltonians are considered. Provided only that the approach to the infinite time limit is non-trivial falling at least as fast as 1/t for large t, the transition probability takes a particularly simple form depending only on the value of Hamiltonian parameters in this limit [ABSTRACT FROM AUTHOR]

The averaging theory of first order is applied to study a generalized Yang-Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang-Mills Hamiltonian systems, in the sense of Liouville-Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class C1. This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion. [ABSTRACT FROM AUTHOR]

We show how to apply to Hamiltonian differential systems recent results for studying the periodic orbits of a differential system using the averaging theory. We have chosen two classical integrable Hamiltonian systems, one with the Hooke potential and the other with the Kepler potential, and we study the periodic orbits which bifurcate from the periodic orbits of these integrable systems, first perturbing the Hooke Hamiltonian with a nonautonomous potential, and second perturbing the Kepler problem with an autonomous potential. [ABSTRACT FROM AUTHOR]

The dynamics of N point vortices in a fluid is described by the Helmholtz–Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for N=2 or 3 but chaotic dynamics for N>3. Here we consider a generalization of the HK equations to describe the dynamics of magnetic vortices within a collective-coordinate approximation. In particular, we analyze in detail the dynamics of a system of three magnetic vortices by a suitable generalization of the solution for three point vortices in an ordinary fluid obtained by Gröbli more than a century ago. The significance of our results for the dynamics of ferromagnetic elements is briefly discussed. [ABSTRACT FROM AUTHOR]

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of δ-pseudodifferential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors are given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by Δ-differential counterparts of Ablowitz–Kaup–Newell–Segur and Kaup–Broer hierarchies. [ABSTRACT FROM AUTHOR]

Hamiltonian n-dimensional Lotka–Volterra systems are introduced that have n-1 conserved quantities. The explicit integrability in quadratures is demonstrated. [ABSTRACT FROM AUTHOR]

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of nonautonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories. [ABSTRACT FROM AUTHOR]

The usual canonical Hamiltonian or Lagrangian formalism of classical mechanics applied to macroscopic systems describes energy conserving adiabatic motion. If irreversible diabatic processes are to be included, the law of increasing entropy must also be considered. The notion of entropy then enters into the classical mechanical formalism. The resulting general formulation and its physical consequences are explored. [ABSTRACT FROM AUTHOR]

In the context of the rigorous analysis of Bose-Einstein condensation, recent achievements have been obtained in the form of asymptotic results when some appropriate scaling is performed in the Hamiltonian, and the limit of infinite number of particles is taken. In particular, two modified thermodynamic limits of infinite dilution turned out to provide an insight in this analysis, the so-called Gross-Pitaevskii limit and the related Thomas-Fermi limit. Here such scalings are discussed with respect to their physical and mathematical motivations and to the currently known results obtained within this framework. [ABSTRACT FROM AUTHOR]

We present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural Hamiltonian systems defined in a flat pseudo-Riemannian space of arbitrary dimension and signature. We then apply our algorithm to explore the integrability of two physical systems. First, we study the Holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta. Second, we analyze a class of cylindrically symmetric potentials in three-dimensional Euclidean space and recover known families of second-order maximally superintegrable systems. [ABSTRACT FROM AUTHOR]

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing. [ABSTRACT FROM AUTHOR]

We prove the integrability of the general quantum Hamiltonian systems governed by an arbitrary non-skew-symmetric, so(3)-valued, nondynamical classical r-matrix with spectral parameters. We consider the most interesting example of these quantum integrable systems, namely, the so(3) “generalized Gaudin systems” in detail. In the case of an arbitrary r-matrix which is “diagonal” in the sl(2) basis we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique. [ABSTRACT FROM AUTHOR]

We consider the Potts model with competing two-step interactions and spin values 1, 2, 3 on a semi-infinite Cayley tree of order 2. We describe ground states and verify the Peierls condition for the model. Using a contour argument we show the existence of three different Gibbs measures associated with translation invariant ground states. [ABSTRACT FROM AUTHOR]

In this paper we study systematically the question of supersymmetrization of the nonlocal gas equation. We obtain both the N=1 and the N=2 supersymmetric generalizations of the system which are integrable. We show that both the systems are bi-Hamiltonian. While the N=1 supersymmetrization allows the hierarchy of equations to be extended to negative orders (local equations), we argue that this is not the case for the N=2 supersymmetrization. In the bosonic limit, however, the N=2 system of equations lead to a new coupled integrable system of equations. [ABSTRACT FROM AUTHOR]

Consistent tensor products on auxiliary spaces, hereafter denoted “fusion procedures,” and commuting transfer matrices are defined for general quadratic algebras, nondynamical and dynamical, inspired by results on reflection algebras. Applications of these procedures then yield integer-indexed families of commuting Hamiltonians. [ABSTRACT FROM AUTHOR]

We give a definition and discuss main properties of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems. The definition relies on a technical construction which is a definition of a relative Hopf invariant for divergence-free vector fields in multiconnected domains of the following type: (flat domain in R2)×(circle). We prove correctness of the definitions, and discuss ergodic interpretation of the Hamiltonian Hopf invariant, together with its relation with the Calabi invariant. [ABSTRACT FROM AUTHOR]

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops Γ of a fixed length L, formally given by -Δ-αδ(x-Γ) with α>0. It is shown that the ground state of this operator is locally maximized by a circular Γ. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of Γ. [ABSTRACT FROM AUTHOR]

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable. [ABSTRACT FROM AUTHOR]

We study the spectrum in such a PT-symmetric square well (of a diameter L≼∞) where the “strength of the non-Hermiticity” is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance ℓ

We investigate integrable two-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly integrable systems. In the case of a quadratic second invariant, we recover the classical strongly integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates. [ABSTRACT FROM AUTHOR]

A bi-Hamiltonian system is a system of differential equations which can be written in Hamiltonian form in two distinct ways. The applications of Gelfand–Zakharevich bi-Hamiltonian structure, which is an extension of a Poisson–Nijenhuis structure on phase space, has been extensively explored by Falqui, Magri, and Pedroni in the context of separation of variables. It is well known that the integrable Hamiltonian systems defined by the Adler–Kostant–Symes (AKS) scheme contains bi-Hamiltonian structure. In this paper we unveil the connection between Adler–Kostant–Symes formalism applied to loop algebra and the Gelfand–Zakharevich bi-Hamiltonian structure by superposition the results of Fordy and Kulish in the AKS scheme. We also study the commuting flows of the AKS hierachy and its connection to the Zakharov–Shabat hierarchy. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

General Natanzon potentials related to the SO(2,2) group are studied. The S-matrices for systems under consideration are related to intertwining operators between Weyl equivalent most degenerate principal series representations of SO(2,2).© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The problem of the evolution of SU(2) and SU (1,1) states is analyzed from a unified point of view. A generalized Rabi matrix for a time-dependent SU (2)-SU (1,1) coherence preserving Hamiltonian is also derived. [ABSTRACT FROM AUTHOR]