Fast Drift and Diffusion in a Class of Isochronous Systems with the Windows Method

The aim of the paper is to deal with some peculiar difficulties arising from the use of the geometrical tool known as windows method in the context of the well known problem of Arnold’s diffusion for isochronous nearly-integrable Hamiltonian systems. Despite the simple features of the class of systems at hand, it is possible to show how the absence of an anisochrony term leads to several substantial differences in the application of the method, requiring some additional devices, such as non-equally spaced transition chains and variable windows. As a consequence, we show the existence of a set of unstable orbits, whose drifting time matches, up to a constant, the one obtained via variational methods.     

On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.     

Effective Hamiltonians for holes in antiferromagnets: a new approach to implement forbidden double occupancy

A coherent state representation for the electrons of ordered antiferromagnets is used to derive effective Hamiltonians for the dynamics of holes in such systems. By an appropriate choice of these states, the constraint of forbidden double occupancy can be implemented rigorously. Using these coherent states, one arrives at a path integral representation of the partition function of the systems, from which the effective Hamiltonians can be read off. We apply this method to the t-J model on the square lattice and on the triangular lattice. In the former case, we reproduce the well-known fermion-boson Hamiltonian for a hole in a collinear antiferromagnet. We demonstrate that our method also works for non-collinear antiferromagnets by calculating the spectrum of a hole in the triangular antiferromagnet in the self-consistent Born approximation and by comparing it with numerically exact results. Received: 23 December 1997 / Accepted: 17 March 1998     

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystem of E ⁽¹⁾ ₈ inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by B?cklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure. Received: 18 September 1999 / Accepted: 29 January 2001     

Fock-tani representation for the quantum theory of reactive collisions

A previously-developed second-quantization representation for nonrelativistic systems of composite bound states and their constituents is extended to any number of composite species. The result is an explicit representation for the kinematics and dynamics of reactive scattering processes (rearrangement collisions). A single Hamiltonian in the new representation simultaneously exhibits the various possible scattering and reaction channels, thus circumventing problems associated with “different Hamiltonians for different channels”. The “unperturbed” Hamiltonian describes the free propagation of all possible bound composites as well as their unbound constituents, while the interaction Hamiltonian describes only true scattering and reaction processes, each term corresponding to a collision process with specified initial and final bound composites and/or unbound constituents. The key to the generalization is a suitable ordering of the composite species and a corresponding ordered-product form of the unitary transformation to the new representation.     

A General Variational Algorithm to Calculate Vibrational Energy Levels of Tetraatomic Molecules

A general variational method for calculating vibrational energy levels of tetraatomic molecules is presented. The quantum mechanical Hamiltonian of the system is expressed in a set of coordinates defined by three orthogonalized vectors in the body-fixed frame without any dynamical approximation. The eigenvalue problem is solved by a Lanczos iterative diagonalization algorithm, which requires the evaluation of the action of the Hamiltonian operator on a vector. The Lanczos recursion is carried out in a mixed grid/basis set, i.e., a direct product discrete variable representation (DVR) for the radial coordinates and a nondirect product finite basis representation (FBR) for the angular coordinates. The action of the potential energy operator on a vector is accomplished via a pseudo-spectral transform method. Six types of orthogonal coordinates are implemented in this algorithm, which is capable of describing most four-atom systems with small and/or large amplitude vibrational motions. Its application to the molecules H₂CO, NH₃, and HOOH and the van der Waals cluster He₂Cl₂ is discussed.     

A Feynman-Kac formula for the quantum Heisenberg ferromagnet. I

The Hamiltonian of the (anisotropic) quantum Heisenberg (anti-) ferromagnet on an arbitrary finite lattice is lifted to a Hamiltonian acting on sections of the bundle obtained by twisting a certain line bundle over the classical spin configuration space (which is a Kähler manifold) with the Dolbeault complex. This procedure is extended fromSU(2) to arbitrary compact semi-simple Lie groups and arbitrary irreducible representations. The Bott-Borel-Weil theorem gives a heat kernel representation for the original partition function in an external magnetic field. TheU(1)-gauged local Hamiltonian is the sum of the free, supersymmetric, twisted Dolbeault Laplace operator (multiplied by the inverse of an arbitrary small mass parameter) plus the lifted Hamiltonian.The resulting (Euclidean) Lagrangian is nonlocal and describes bosons which do and fermions which do not propagate through the lattice. All fields couple to the external magnetic field. The Lagrangian contains Yukawa and Luttinger type interactions.     

关于Birkhoff逆问题中Santilli方法的研究

研究构造Birkhoff动力学函数的Santilli方法.首先,基于Cauchy-Kovalevskaya型方程解的存在性定理,采用反证法证明自治系统总有自治Birkhoff表示;其次,给出更简洁的方法证明Santilli第二方法可以被简化;找到Santilli第三方法中所隐含的一种等量关系,提出改进的Santilli第三方法,并研究该方法的MATLAB程序化计算;最后,总结全文并对结果进行讨论.     

A generalized internal axis method for high barrier tunneling problems,as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of $\text{n}{}^2\text{2}$ coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

Reduction of the semisimple 1:1 resonance

The method of “averaging” is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated “reduced” Hamiltonian on a “reduced space”. This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case. The reduced space will be a 2-sphere in R³, and the reduced differential equations will be Euler's equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on “degeneracies” due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.     

An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability

A systematic method for deducing the Liouville integrability of a finite-dimensional Hamiltonian system reduced from an infinite-dimensional Hamiltonian system is proposed within the framework of the zero-curvature representation theory. Also a systematic way to treat the higher-order constraints and to obtain the associated infinitely many hierarchies of finite-dimensional integrable Hamiltonian systems is presented.     

Calculation of an atom–surface scattering model using the S-matrix KVP

For the first time, we solve the model elastic atom–surface scattering problem with the S-matrix Kohn variational principle (KVP). The KVP consists of Hamiltonian matrix equations over a basis that includes both scattering and L² functions. For ease of evaluation, we choose the L² basis to be a pointwise representation (e.g. a discrete variable representation). Also for efficient solution, we use the reduced dimensional eigenbasis found by diagonalizing the Hamiltonian in the pointwise representation using the procedure of successive diagonalization and truncation. It is found that even upon further optimization, the KVP method appears to be too slow for solving large-scale single energy problems.The method is demonstrated to be efficient however, if the S-matrix elements for many different energies is desired.     

Subdynamics and nonintegrable systems

Work on the application of Poincaré's theorem to large classical or quantum systems with a continuous spectrum is continued. In situations where it is applicable, Poincaré's theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant. In contrast, the theory of subdynamics as developed by the Brussels group permits the construction of a unique set of projectors , giving up the requirement of hermiticity which is replaced by “star-hermiticity”.

The theory of subdynamics is presented in a new self-contained way, starting from the commutation relation , where L_H is the Liouvillian. This presentation is far more direct, and avoids some of the lengthy discussions associated with previous presentations (based mainly on the resolvent of the Liouvillian).

Subdynamics appears to be of interest from many points of view. It generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed Hamiltonian. In contrast, degrees of freedom are lost when going to the spectral representation (e.g. in the Friedrichs model). Subdynamics permits us to solve the initial value problem associated with the Liouville equation retaining the “non-Markovian” contributions which appear in the standard presentation. Finally, it introduces a classification of large dynamical systems, classical or quantum, into integrable and nonintegrable ones. It is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems, such as the interaction of matter with light. The applications of this technique to these problems will be worked out in subsequent papers.     

A general rigorous quantum dynamics algorithm to calculate vibrational energy levels of pentaatomic molecules

An exact variational algorithm is presented for calculating vibrational energy levels of pentaatomic molecules without any dynamical approximation. The quantum mechanical Hamiltonian of the system is expressed in a set of orthogonal coordinates defined by four scattering vectors in the body-fixed frame. The eigenvalue problem is solved using a two-layer Lanczos iterative diagonalization method in a mixed grid/basis set. A direct product potential-optimized discrete variable representation (PO-DVR) basis is used for the radial coordinates while a non-direct product finite basis representation (FBR) is employed for the angular variables. The two-layer Lanczos method requires only the actions of the Hamiltonian operator on the Lanczos vectors, where the potential-vector products are accomplished via a pseudo-spectral transform technique. By using Jacobi, Radau and orthogonal satellite vectors, we have proposed 21 types of orthogonal coordinate systems so that the algorithm is capable of describing most five-atom systems with small and/or large amplitude vibrational motions. Finally, an universal program (PetroVib) has been developed. Its applications to the molecules , and the van der Waals cluster He₃Cl₂ are also discussed.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

Precise Coupling Terms in Adiabatic Quantum Evolution: The Generic Case

For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. Based on results from [BeTe₁] for special Hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. Our proof is based on a new norm which allows to rigorously implement Darboux' principle, a heuristic guideline widely used in asymptotic analysis. As shown in [BeTe₁], first order perturbation theory in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry's [Be] predictions on the universal form of adiabatic transition histories.     

An Ergodic Sampling Scheme for Constrained Hamiltonian Systems with Applications to Molecular Dynamics

This article addresses the problem of computing the Gibbs distribution of a Hamiltonian system that is subject to holonomic constraints. In doing so, we extend recent ideas of Cancès et al. (M2AN 41(2), 351–389, 2007) who could prove a Law of Large Numbers for unconstrained molecular systems with a separable Hamiltonian employing a discrete version of Hamilton’s principle. Studying ergodicity for constrained Hamiltonian systems, we specifically focus on the numerical discretization error: even if the continuous system is perfectly ergodic this property is typically not preserved by the numerical discretization. The discretization error is taken care of by means of a hybrid Monte-Carlo algorithm that allows for sampling bias-free expectation values with respect to the Gibbs measure independently of the (stable) step-size. We give a demonstration of the sampling algorithm by calculating the free energy profile of a small peptide.     

Spin Chain Models with Spectral Curves from M Theory

We construct the integrable model corresponding to the ?= 2 supersymmetric SU(N) gauge theory with matter in the antisymmetric representation, using the spectral curve found by Landsteiner and Lopez through M Theory. The model turns out to be the Hamiltonian reduction of a N+2 periodic spin chain model, which is Hamiltonian with respect to the universal symplectic form we had constructed earlier for general soliton equations in the Lax or Zakharov–Shabat representation. Received: 22 December 1999 / Accepted: 3 March 2000     

k-Cosymplectic Classical Field Theories: Tulczyjew and Skinner–Rusk Formulations

The k-cosymplectic Lagrangian and Hamiltonian formalisms of first-order classical field theories are reviewed and completed. In particular, they are stated for singular and almost-regular systems. Subsequently, several alternative formulations for k-cosymplectic first-order field theories are developed: First, generalizing the construction of Tulczyjew for mechanics, we give a new interpretation of the classical field equations. Second, the Lagrangian and Hamiltonian formalisms are unified by giving an extension of the Skinner–Rusk formulation on classical mechanics.