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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]

We study the density of states of the Pauli Hamiltonian with a Poisson random distribution of smooth finite-width vortices and we obtain classical bounds for the Lifshits tails for them. These Hamiltonians are smooth approximations to the self-adjoint extensions of the Aharonov–Bohm Hamiltonian. In this case because pairs of impurities are coupled by the magnetic field we cannot use the Laplace characteristic functional. [ABSTRACT FROM AUTHOR]

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(rij) = g(rij²) in thee spatial dimensions. In natural units &hstroke; = c = 1 the system has the semirelativistic ‘spinless-Salpeter’ Hamiltonian H = ∑I=1N √m² + pi² + ∑j>i=jNg(¦ri-rj¦²), where g is monotone increasing and has convexity gn≥0. We use ‘envelope theory’ to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N≥2. In particular, we determine the energy of the N-body oscillator g(r²) = cr² with error less than 0.15% for all m≥0, N≥2, and c>0. [ABSTRACT FROM AUTHOR]

To treat the front-form Hamiltonian approach to quantum field theory, called light cone quantum field theory, in a mathematically rigorous way, the existence of a well-defined restriction of the corresponding free fields to the hypersurface {x0+x3=0} in Minkowski space is of an essential necessity. However, even in the situation of a real scalar free field such a restriction does canonically not exist; this is called the restriction problem. Furthermore, since the beginning of light cone quantum field theory there is the problem of nonexistence of a well-defined Fock space expansion of a free quantum field in terms of light cone momenta which is called the zero-mode problem. In this paper we present solutions to these long outstanding problems where the study of the zero-mode problem (of the corresponding classical field) will lead us to a solution of the restriction problem. We introduce a new function space of “squeezed” smooth functions which can canonically be embedded into the Schwartz space S(R3). The restriction of the free field to {x0+x3=0} is canonically definable on this function space and we show that the covariant field is uniquely determined by this “tame” restriction. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study the lowest energy E of a relativistic system of N identical bosons bound by pair potentials of the form V(rij) = g(rij²) in thee spatial dimensions. In natural units &hstroke; = c = 1 the system has the semirelativistic ‘spinless-Salpeter’ Hamiltonian H = ∑I=1N √m² + pi² + ∑j>i=jNg(¦ri-rj¦²), where g is monotone increasing and has convexity gn≥0. We use ‘envelope theory’ to derive formulas for general lower energy bounds and we use a variational method to find complementary upper bounds valid for all N≥2. In particular, we determine the energy of the N-body oscillator g(r²) = cr² with error less than 0.15% for all m≥0, N≥2, and c>0. [ABSTRACT FROM AUTHOR]

We study the deformed Harry Dym and Hunter–Zheng equations with two arbitrary deformation parameters. These reduce to various other known models in appropriate limits. We show that both these systems are bi-Hamiltonian with the same Hamiltonian structures. They are integrable and belong to the same hierarchy corresponding to positive and negative flows. We present the Lax pair description for both the systems and construct the conserved charges of negative order from the Lax operator. For the deformed Harry Dym equation, we construct the nonstandard Lax representation for two special classes of values of the deformation parameters. In general, we argue that a nonstandard description will involve a pseudodifferential operator of infinite order. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we study a new hierarchy of equations derived from the system of isentropic gas dynamics equations where the pressure is a nonlocal function of the density. We show that the hierarchy of equations is integrable. We construct the two compatible Hamiltonian structures and show that the first structure has three distinct Casimirs while the second has one. The existence of Casimirs allows us to extend the flows to local ones. We construct an infinite series of commuting local Hamiltonians as well as three infinite series (related to the three Casimirs) of nonlocal charges. We discuss the zero curvature formulation of the system where we obtain a simple expression for the nonlocal conserved charges, which also clarifies the existence of the three series from a Lie algebraic point of view. We point out that the nonlocal hierarchy of Hunter–Zheng equations can be obtained from our nonlocal flows when the dynamical variables are properly constrained. © 2004 American Institute of Physics. [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]

We consider a general weak bounded finite range perturbation of a general free quantum spin Hamiltonian on a lattice. We prove that if the free Hamiltonian has a nondegenerate ground state and a spectral gap, then the perturbation also has a ground state, and estimate the localization of the spectrum in the corresponding ground state representation. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Two generalized Harry Dym equations, recently found by Brunelli, Das, and Popowicz in the bosonic limit of new supersymmetric extensions of the Harry Dym hierarchy [J. Math. Phys. 44, 4756 (2003)], are transformed into previously known integrable systems: one, into a pair of decoupled KdV equations, the other one, into a pair of coupled mKdV equations from a bi-Hamiltonian hierarchy of Kupershmidt. © 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]

We investigate the resonant states for the parabolic potential barrier known also as inverted or reversed oscillator. They correspond to the poles of meromorphic continuation of the resolvent operator to the complex energy plane. As a by-product we establish an interesting relation between parabolic cylinder functions (representing energy eigenfunctions of our system) and a class of Gel’fand distributions used in our recent paper. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We present in this article all Hamiltonian systems in E(2) that are separable in Cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg–de Vries equation and the Painlevé transcendents. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Two important notions of integrability for discrete mappings, algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities and singularity confinement is associated with the local analysis of singularities. In this article, the relationship between these two notions is explored for birational autonomous mappings. The main result of this article is that algebraically integrable mappings are shown to have the singularity confinement property. Using this result, the proof of the nonexistence of algebraic conserved quantities for a class of discrete systems is given. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The Potts model with countable set [uppercase_phi_synonym] of spin values on Z[sup d] is considered. It is proved that with respect to Poisson distribution on [uppercase_phi_synonym] the set of limiting Gibbs measures is not empty. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A class of nonautonomous coupled Korteweg–de Vries (KdV) systems in (1+1) dimensions are considered for integrability classification. Integrability of the systems is associated with the existence of a certain recursion operator. Some new integrable nonautonomous two-component KdV systems are found. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Focuses on a mathematical model of Hamiltonian system based on non-ergodicity which is induced by isotropic random perturbation. Effect of noise on ergodicity; Numerical evidences which suggests the difference in orbital separation rate of ergodicity.

We discuss the quantum Hamiltonian H[sub ...] describing a δ-sphere interaction introduced in [J. Math. Phys 30, 2275 (1989)] and formally given by HG=H[sub D] +...δ(x-R), where H[sub D] is the Dirac Hamiltonian and ... is a real 4×4 matrix defined by ... = (...). We obtain a series of new results for this model, in particular the resolvent equation, the spectral properties, the nonrelativistic limit and the various quantities related to the scattering theory. These results are generalized to the case of an asymmetric &sphere interaction and a δ-sphere plus Coulomb interaction, respectively. [ABSTRACT FROM AUTHOR]

The commutation relations of the first-order and second-order operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a "pre-Robertson condition" is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation. [ABSTRACT FROM AUTHOR]

The fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins-Miller theory, and the intrinsic characterization of the separation of the Hamilton-Jacobi equation, in a unitary and geometrical perspective. The general notion of complete integrability of first-order normal systems of PDEs leads in a natural way to completeness conditions for separated solutions of the Schrödinger equation and to the Robertson condition. Two general types of multiplicative separation for the Schrödinger equation are defined and analyzed: they are called "free" and "reduced" separation, respectively. In the free separation the coordinates are necessarily orthogonal, while the reduced separation may occur in nonorthogonal coordinates, but only in the presence of symmetries (Killing vectors). [ABSTRACT FROM AUTHOR]

We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of space-time with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space-time region. A main result is that we obtain a covariant derivation of Dirichlet, Neumann, and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface, together with the associated Hamiltonian boundary terms. As well, we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current. Our analysis uses a Noether charge method which extends and unifies several results developed in recent literature for General Relativity. As an illustration of the method, we apply it to the Maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a well-defined Hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of Minkowski space-time. [ABSTRACT FROM AUTHOR]

We recall the construction of the common eigenvectors of Gaudin Hamiltonians based on the Bethe ansatz. In the case of an arbitrary Lie algebra, this construction can be done either recursively or explicitly and we prove the equivalence of the two methods. We also prove that Bethe vectors are singular only if the Bethe equations are satisfied. In each eigenspace of the spin operator we construct additional common eigenvectors, having the same eigenvalue as the vacuum vector and which can not be obtained by the Bethe ansatz. These eigenvectors are not singular. We also recall the connection between Bethe vectors and integral solutions of the KZ equation. In an analogous way, the additional vectors lead to solutions of KZ equation which are not singular vectors and do not have an integral representation. [ABSTRACT FROM AUTHOR]

Details the separability theory of multi-Hamiltonian systems in the case when each Poisson structure has more than one Casimir variables. Construction of a hierarchy of Hamiltonian symmetries and related hierarchy of constants of motion; Integrability of Hamiltonian functions; Transformation of arbitrary canonical coordinates.

Examines the geometric quantization of first-order Hamiltonian systems using some tools of the field-antifield formalism. Coupling of massive electrodynamics with chiral fermions; Characteristic of first-order system submitted to second class constraints; Introduction of specific counterterm in the quantum field action.

Discusses the interaction of dromions for (2+1)-dimensional integrable systems. Multi-dromion solutions given for three (2+1)-dimensional integrable systems; Plots of interaction of two dromions for Hamiltonian systems; Absence of changes in the shape and velocity of the dromions after interactions; Irrotational and elastic interactions of dromions.

Presents a direct proof of the integrability of Calogero-Moser-Sutherland (CMS) model, a paradigm of solvable finite-dimensional Hamiltonian systems. Relevance of the CMS model for studying mathematical physics techniques; Methods proving the classical and quantum integrability; Definition of the CMS model.

Presents a matrix formulation based on Hamiltonian structures of constrained KP hierarchy. Derivation of the Hamiltonian structure from the matrix formulation; Generalization of structure identification to the multiconstraint case; Description of the resulting bracket as the second Gelfand-Dickey bracket.

Demonstrates that the so-called 'unavoidable quantum anomalies' can be avoided in the framework of a special nonlinear quantization scheme. Representation of the quantized Hamiltonians by nonlinear but homogeneous operators in Hilbert space; Applicability of the proposed quantization method to integrable classical dynamical systems.

Studies a second invariant for one-degree-of-freedom. Time-dependent Hamiltonians; Transformation to an autonomous Hamiltonian system.

Studies the canonical form of the Rabi Hamiltonian. Coupling of a two-level system to a single quantized boson mode; Bargmann-Fock representation; Two-dimensional eigenvalue problem.

Hamilton equations for degenerate time-dependent, and generally higher-order mechanical systems are considered and their geometric meaning is analyzed. The local dynamics of constrained systems is clarified and an algorithm for finding it is presented. © 1994 American Institute of Physics. [ABSTRACT FROM AUTHOR]

To describe constraint field models, we apply the multimomentum Hamiltonian formalism where momenta correspond to derivatives of fields with respect to all world coordinates, not only time. If a Lagrangian density is degenerate, the Euler–Lagrange equations are underdetermined and need additional gauge-type conditions which remain elusive in general. One gets these conditions automatically as a part of the Hamilton equations, but must consider a family of Hamiltonian forms associated with the same Lagrangian density in order to exhaust solutions of the Euler–Lagrange equations. The case of degenerate quadratic and affine Lagrangian densities is elaborated. As a result, we get the universal procedure of describing constraint field systems. © 1994 American Institute of Physics. [ABSTRACT FROM AUTHOR]

General adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip, and whose spectrum satisfies a gap condition are studied. An explicit rate of exponential decay is given, which is related to simple geometric quantities associated to the spectrum of the Hamiltonian, for the transition probability between the two parts of the spectrum when the evolution is taken from -∞ to +∞. [ABSTRACT FROM AUTHOR]

The properties of the canonical symmetry are investigated. The densities of the local conservation laws for the nonlinear Schrödinger equation are shown to change under the action of the canonical symmetry by total space derivatives. It is also shown how the canonical symmetry can be used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem. [ABSTRACT FROM AUTHOR]

A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case. [ABSTRACT FROM AUTHOR]

A general class of n-particle difference Calogero–Moser systems with elliptic potentials is introduced. Besides the step size and two periods, the Hamiltonian depends on nine coupling constants. We prove the quantum integrability of the model for n=2 and present partial results for n≥ 3. In degenerate cases (rational, hyperbolic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q-polynomials of Koornwinder and Macdonald. Liouville integrability of the corresponding classical systems follows as a corollary. Limit transitions lead to various well-known models such as the nonrelativistic Calogero–Moser systems associated with classical root systems and the relativistic Calogero–Moser system. [ABSTRACT FROM AUTHOR]

The procedure of geometric quantization is developed for completely integrable dynamical systems on manifolds with exact symplectic structure. These results are applied to Neumann’s nonharmonic oscillatory system on two-dimensional sphere S2. [ABSTRACT FROM AUTHOR]

The algebraic structure of the gradient-holonomic algorithm for Lax integrable dynamical systems is discussed. A generalization of the R-structure approach for the case of operator-valued affine Lie algebras is used to prove the bi-Hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions. The monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical Lie–Poisson bracket on initial affine Lie algebra with gauge central extension. As an illustration, the two-dimensional operator Benney–Kaup integrable hierarchy is considered and their bi-Hamiltonicity is proved. [ABSTRACT FROM AUTHOR]

Supergeneralization of CP(N) provided with even and odd Kählerian structures from Hamiltonian reduction are constructed. Operator Δ, which is used in Batalin–Vilkovisky quantization formalism and mechanics, which are bi-Hamiltonian under corresponding even and odd Poisson brackets, are considered. [ABSTRACT FROM AUTHOR]

In this paper, first integrals of one-dimensional time-dependent Hamiltonians, H(q,p,t), are studied. First of all, it is shown that the second first integral, J(q,H,t)=t-∫dq/Hp, valid for autonomous Hamiltonians H(q,p), can be extended to the case of nonautonomous Hamiltonians H(q,p,t), provided Hp (resp. t) is replaced with Ip [where I(q,p,t) is a first integral of H(q,p,t)] (resp. a new time T): J(q,I,T)=T-∫dq/Ip. In a second part, we derive a technique leading to a first integral, I(q,p,t), from a Lie symmetry. As an example, this technique, together with the expression of the second first integral, J(q,I,t), is used to completely integrate a special case of the linear harmonic oscillator with a time-dependent frequency. As a result, Lewis’ first integral is recovered from a Lie symmetry in a way that generalizes the recent work of Shivamoggi and Muilenberg [Phys. Lett. A 154, 1 (1991)]. [ABSTRACT FROM AUTHOR]

The bi-Hamiltonian structure of the super KP hierarchy is obtained based on the even super KP operator Λ=θ2+∑i=-2∞Uiθ-i-1, as a supersymmetric extension of the ordinary KP bi-Hamiltonian structure. It is expected to give rise to a universal super W-algebra incorporating all known extended superconformal WN algebras by reduction. Also constructed is the super BKP hierarchy by imposing a set of anti-self-dual constraints on the super KP hierarchy. [ABSTRACT FROM AUTHOR]

Path integral methods are used to derive an exact expression for the propagator for a quadratic Hamiltonian with arbitrary ordering. The van Vleck–Pauli relation is found to be a special case of our general result. [ABSTRACT FROM AUTHOR]

In this paper the structure of vortex coadjoint orbits pertaining to perfect fluids having smooth vorticities in R3, within the framework set up by J. Marsden and A. Weinstein [Physica D 7, 305–323 (1983)], in terms of an associated Hamiltonian Kähler manifold (the Clebsch manifold, described in terms of the so-called Clebsch variables) is investigated. The topological quantization of Mikhailov and Kuznetsov is related to geometric quantization. Natural candidates for the coherent states on the Clebsch manifold are also exhibited. [ABSTRACT FROM AUTHOR]

The relation between the R-matrix approach to the construction of bi-Hamiltonian structures and the reduction of Poisson manifolds is discussed. The crucial role of the reduction process is emphasized by showing, on the basis of specific examples, that the knowledge of a R matrix may not be sufficient. The reduction of the Poisson structures related with R from the entire algebra to the Lax submanifold may become an obstruction in constructing the bi-Hamiltonian structures of a specific hierarchy of integrable equations. [ABSTRACT FROM AUTHOR]

Some new results are presented on the theory of Hamiltonian systems with first-class constraints. In these systems it is possible to separate the physical part from the gauge part by transforming to canonical coordinates in which the constraints are a subset of the new momenta; this construction is accomplished by algebraic methods and the use of a set of Hamilton–Jacobi-like equations. Finally, the problem and meaning of evolution in systems with weakly vanishing Hamiltonian is commented on. [ABSTRACT FROM AUTHOR]

Two methods for the construction of alternative Lagrangians to the standard Lagrangian L=T-V where V is a central potential are presented: the direct integration of the Hessian matrix and the use of the Noether symmetries of L. The Lagrangians obtained are of the so-called nonlinear homogenous (NLH) type and the study is centered in those Lagrangians not preserving the spherical symmetry of L. Finally, the associated Hamiltonian formalism is presented. [ABSTRACT FROM AUTHOR]