Search

For the rational, elliptic, and trigonometric r-matrices, we exhibit the links between three “levels” of Poisson spaces: (a) some finite-dimensional spaces of matrix-valued meromorphic functions on the complex line, (b) spaces of spectral curves and sheaves supported on them, and (c) symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to the Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family and which are such that the Lagrangian leaves are the intersections of the symplectic leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems. [ABSTRACT FROM AUTHOR]

We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of (2n)2×(2n)2 dimensions with 2n2 free parameters (n=1,2,3,...). The simplest, 4×4 case, is treated in detail. Powerful recursion relations are constructed by explicitly giving the dependence on the spectral parameter θ of the eigenvalues of the transfer matrix at each level of coproduct sequence. A brief study of higher dimensional cases (n>=2) is presented, pointing out features of particular interest. Spin chain Hamiltonians are also briefly presented for the hierarchy. In a long final section, basic results are recapitulated with systematic analysis of their contents. Our eight vertex 4×4 case is compared to standard six vertex and eight vertex models. [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 cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys. 13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line. [ABSTRACT FROM AUTHOR]

We discuss the possibility of defining an algebraic dynamics within the settings of O*-algebras. Compared to our previous results on this subject, the main improvement here is that we are not assuming the existence of some Hamiltonian for the full physical system. We will show that, under suitable conditions, the dynamics can still be defined via some limiting procedure starting from a given regularized sequence. [ABSTRACT FROM AUTHOR]

Form-invariant Poisson brackets of hydrodynamic type with several spatial variables x1,...,xp are introduced for any commuting vector fields U1,...,Um on a manifold Mn. The incompressibility equation for certain parameters of the Poisson brackets is derived. Poisson brackets connected with an arbitrary dynamical system on Mn are constructed. A gauge transform for the Poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied. A general identity between the coefficients bkij(u) and gij(u) is found. [ABSTRACT FROM AUTHOR]

We suggest a systematic method of extension of quasiexactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As an example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasiexactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function. [ABSTRACT FROM AUTHOR]

The long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials is rigorously studied. To set the stage, the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation with time-dependent nonlinearities and potential is established. Afterward, the dynamics of NLS solitary waves in time-dependent potentials is studied. It is shown that in the space-adiabatic regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton’s equations, plus terms due to radiation damping. Finally, two physical applications are discussed: the first is adiabatic transportation of solitons and the second is the Mathieu instability of trapped solitons due to time-periodic perturbations. [ABSTRACT FROM AUTHOR]

We consider an invariant quantum Hamiltonian H=-ΔLB+V in the L2 space based on a Riemannian manifold M with a countable discrete symmetry group Γ. Typically, M is the universal covering space of a multiply connected Riemannian manifold M and Γ is the fundamental group of M. On the one hand, following the basic step of the Bloch analysis, one decomposes the L2 space over M into a direct integral of Hilbert spaces formed by equivariant functions on M. The Hamiltonian H decomposes correspondingly, with each component HΛ being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations Λ of Γ. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation Λ of Γ) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian H. We discuss these procedures in detail and show that in a sense they are mutually inverse. [ABSTRACT FROM AUTHOR]

The Faddeev–Popov gauge fixing in Yang–Mills theory is interpreted as equivariant localization. It is shown that the Faddeev–Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form, and a certain closed equivariant 4-form, which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization. [ABSTRACT FROM AUTHOR]

A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B(0,1) present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly dissipative Camassa–Holm equation. First, we try to improve other authors’ results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out. [ABSTRACT FROM AUTHOR]

This paper deals with an extension of the applications of the paper by Gelfand and Retakh [Funct. Anal. Appl. 25, 91 (1991)] on quasideterminant (QsD) algebraic method to eigenvalue problems in quantum mechanics. Using relevant identities on the free 1-mode bosonic algebra, we build characteristic QsDs associated with generalized spin-orbit Hamiltonians with a well defined representation which allows us to explicitly and straightforwardly compute analytical expressions of eigenenergies. Specific instances are provided on f-deformed generalized Jaynes–Cummings models and other Hamiltonian classes widely used in condensed matter physics. [ABSTRACT FROM AUTHOR]

We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time-of-arrival determined by Anastopoulos and Savvidou [J. Math. Phys. 47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different “attempts” of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase. [ABSTRACT FROM AUTHOR]

We discuss dimensional reduction for Hamiltonian systems which possess nonconstant Poisson brackets between pairs of coordinates and between pairs of momenta. The associated Jacobi identities imply that the dimensionally reduced brackets are always constant. Some examples are given alongside the general theory. [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]

In this paper we explore general conditions which guarantee that the geodesic flow on a two-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a two-dimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a secind rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leading-order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for (1+1)-dimensional systems, there are three possible types of conformal Killing tensors and, therefore, three distinct separability structures in contrast to the single standard Hamilton-Jacobi-type separation in the positive definite case. One of the new separability structures is the complex/harmonic type which is characterized by complex separation variables. The other new type is the linear/null separation which occurs when the conformal Killing tensor has a null eigenvector. [ABSTRACT FROM AUTHOR]

A self-contained approach to studying the unitary evolution of coupled qubits is introduced, capable of addressing a variety of physical systems described by exchange Hamiltonians containing Rabi terms. The method automatically determines both the Weyl chamber steering trajectory and the accompanying local rotations. Particular attention is paid to the case of anisotropic exchange with tracking controls, which is solved analytically. It is shown that, if computational subspace is well isolated, any exchange interaction can always generate high fidelity, single-step controlled-NOT (CNOT) logic, provided that both qubits can be individually manipulated. The results are then applied to superconducting qubit architectures, for which several CNOT gate implementations are identified. The paper concludes with consideration of two CNOT gate designs having high efficiency and operating with no significant leakage to higher-lying noncomputational states. [ABSTRACT FROM AUTHOR]

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. [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]

It is possible to tell if two or more eigenvalues of a matrix are equal without calculating the eigenvalues. We use this property to detect (avoided) crossings in the spectra of Hamiltonians representable by matrices. This approach provides a pedagogical introduction to (avoided) crossings, is capable of handling realistic Hamiltonians analytically, and offers a way to visualize crossings that is sometimes superior to that provided by the spectrum directly. We illustrate the method using the Breit-Rabi Hamiltonian to describe the hyperfine-Zeeman structure of the ground-state hydrogen atom in a uniform magnetic field. [ABSTRACT FROM AUTHOR]

A big-isotropic structure E is an isotropic subbundle of TM+T*M, endowed with the metric defined by pairing. The structure E is said to be integrable if the Courant bracket [X,Y]∈ΓE, ∀X,Y∈ΓE. Then, necessarily, one also has [X,Z]∈ΓE⊥, ∀Z∈ΓE⊥ [Vaisman, I., “Isotropic subbundles of TM+T*M,” Int. J. Geom. Methods Mod. Phys. 4, 487–516 (2007)]. A weak-Hamiltonian dynamical system is a vector field XH such that (XH,dH)∈ΓE⊥(H∈C∞(M)). We obtain the explicit expression of XH and of the integrability conditions of E under the regularity condition dim(prT*ME)=const. We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) [Dalsmo, M. and van der Schaft, A. J., “On representations and integrability of mathematical structures in energy conserving physical systems,” SIAM J. Control Optim. 37, 54–91 (1998)] may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems. [ABSTRACT FROM AUTHOR]

The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstraß ζ function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community. [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]

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain. [ABSTRACT FROM AUTHOR]

The reduced Hamiltonian system on T*(SU(3)/SU(2)) is derived from a Riemannian geodesic motion on the SU(3) group manifold parametrized by the generalized Euler angles and endowed with a bi-invariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space SU(3)/SU(2)≃S5 is not isometric or even geodesically equivalent to the standard Riemannian metric on the five-sphere S5 embedded into R6. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new completely integrable wave equation: mt+mx(u2-ux2)+2m2ux=0, m=u-uxx. The equation is derived from the two dimensional Euler equation and is proven to have Lax pair and bi-Hamiltonian structures. This equation possesses new cusp solitons—cuspons, instead of regular peakons ce-|x-ct| with speed c. Through investigating the equation, we develop a new kind of soliton solutions—“W/M”-shape-peaks solitons. There exist no smooth solitons for this integrable water wave equation. [ABSTRACT FROM AUTHOR]

Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach applied to Poisson algebras. A proof that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal is given. All results are illustrated by several examples. [ABSTRACT FROM AUTHOR]

We show that the three-body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one variable. [ABSTRACT FROM AUTHOR]

The necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block-diagonal form with k mutually interacting blocks. Applications to the perturbations of the Benney system and to the Hamiltonian systems of partial differential equations are presented. The algebraic identities connecting the Nijenhuis tensors NB(A)(u,v) and NA(u,v) and Haantjes tensors HB(A)(u,v) and HA(u,v) are discovered. [ABSTRACT FROM AUTHOR]

We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨χ,u⟩χ such that for β0 Hβ0≔M+β0P has a simple eigenvalue E0 embedded in its absolutely continuous spectrum, with corresponding eigenvector ψ. Let Hω be a rank one perturbation of the operator Hβ0, namely, Hω =M+(β0+ω)P. Under suitable conditions, the operator Hω has no point spectrum in a neighborhood of Eω, for ω small. Here, we study the evolution of the state χ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)= ʃ∞-∞ ∣ ⟨ψ,e-iHωtψ⟩∣² dt. By perturbation theory, we prove that τω(ψ) is is finite for ω≠0, and that for ω small it is of order ω-2. Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in ℝn. [ABSTRACT FROM AUTHOR]

In this paper we perform an exact study of “quantum fidelity” (also called Loschmidt echo) for the time-periodic quantum harmonic oscillator of the following Hamiltonian: H⁁g(t):(P2/2)+f(t)(Q2/2)+(g2/Q2), when compared with the quantum evolution induced by H⁁0(t) (g=0), in the case where f is a T-periodic function and g a real constant. The reference (initial) state is taken to be an arbitrary “generalized coherent state” in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of t=0, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times tk. We discuss the result when the classical motion induced by Hamiltonian H⁁0(t) is assumed to be stable versus unstable. [ABSTRACT FROM AUTHOR]

It is well known that nonholonomic systems obeying D’Alembert’s principle are described on the Hamiltonian side, after using the Legendre transformation, by the so-called almost-Poisson brackets. In this paper we define the Lagrangian and Hamiltonian sides of a class of generalized nonholonomic systems (GNHS), obeying a generalized version of D’Alembert’s principle, such as rubber wheels (like some simplified models of pneumatic tires) and certain servomechanisms (like the controlled inverted pendulum), and show that corresponding equations of motion can also be described in terms of a bracket. We present essentially all possible brackets in terms of which the mentioned equations can be written down, which include the brackets that appear in the literature, and point out those (if any) that are naturally related to each system. In particular, we show there always exists a Leibniz bracket related to a GNHS, and conversely, that every Leibniz system is a GNHS. The control of the inverted pendulum on a cart is studied as an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper we provide a complete description of the first integrals of the classical Bianchi IX system that can be described by a general class of formal power series. As a corollary we also obtain a complete description of some of its analytic first integrals in a neighborhood of the origin. In particular, we prove that the system is not completely integrable by analytic first integrals. [ABSTRACT FROM AUTHOR]

A four-dimensional family of skew-symmetric solutions of the Jacobi equations for Poisson structures is characterized. As a consequence, previously known types of Poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions. Additionally, it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions, such as their Casimir invariants, symplectic structure and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a limited sample of known finite-dimensional Poisson structures. Moreover, the results developed are valid globally in phase space, thus ameliorating the usual scope of Darboux theorem which is of local nature. [ABSTRACT FROM AUTHOR]

We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1/λ. We also give alternative, elementary proofs of these identities by direct computations. [ABSTRACT FROM AUTHOR]

The necessary conditions are derived for existence of a bi-Hamiltonian structure for a given hydrodynamic type system. One of the conditions is the vanishing of its Haantjes tensor. A theorem is proved on the canonical forms of the generic bi-Hamiltonian systems. The Schouten (2,1)-tensor Sijk is connected with any Hamiltonian system of hydrodynamic type. The complete symmetry of the (3,0)-tensor Sijk is demonstrated. The necessary conditions for existence of a single nondegenerate Hamiltonian structure are obtained in terms of the special differential k-forms Ωp1...pk(u1,...,uk). [ABSTRACT FROM AUTHOR]

For applications of group theory in quantum mechanics, one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some Hamiltonian of interest. Here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite-dimensional irreps of the SO(5) Lie algebra in an SO(3) basis. The SO(3) subgroup of SO(5) is defined by regarding SO(5) as linear transformations of the five-dimensional space of an SO(3) irrep of angular momentum two. A need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations. The algorithm has been implemented in MAPLE, and some tables of results are presented. [ABSTRACT FROM AUTHOR]

When an eigenvector of a semibounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-type inequalities and can be applied to non-necessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrödinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the ground-state energy of a system made of an arbitrary number of attractive Coulombian particles. [ABSTRACT FROM AUTHOR]

We study a new hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the elastic beam equations we also obtain a nonstandard Lax representation. [ABSTRACT FROM AUTHOR]

In the jet-bundle description of time-dependent mechanics there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior choice of a connection. This situation is analyzed by Echeverría-Enríquez et al. [J. Phys. A 28, 5553–5567 (1995)]. The aim of this paper is to extend the results in that paper to first order field theory, using the k-cosymplectic formalism described by de León and co-workers [J. Math. Phys. 39, 876–893 (1998); 42, 2092–2104 (2001)]. If the trivial configuration bundle of a Lagrangian system is endowed with one connection, different from the trivial one given by the product structure, we study the consequences on the geometric elements of the theory, the dynamical equations and the variational principles. [ABSTRACT FROM AUTHOR]

We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing explicit expressions for the Poisson bracket between two Poisson forms. [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 consider the non-Hermitian eigenvalue problems -u″(z)±(iz)mu(z)=λu(z), m>=3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane. We prove that the eigenvalues λ are all real and positive. [ABSTRACT FROM AUTHOR]