Hamiltoniens àN corps avec champs magnétiques très singuliers du type ‘courte portée’

We study the spectral properties of a large class ofN-body Hamiltonians, in the Agmon formalism, considering magnetic fields with strong singularities and of short-range type. More precisely, we obtain the essential spectrum (a problem that has been solved in a more general setting by Iftimie by a different method), we control the point spectrum embeded in the continuous spectrum, we prove the absence of singularly continuous spectrum and a Limiting Absorption Principle. Our method consists in performing a gauge transformation (obtained by a procedure similar to that used by Boutet-de Monvel-Berthier and Purice that eliminates the magnetic field by adding some perturbation to the electric potential. The perturbation is a first-order differential operator, generally of the long-range type. At this stage we can use the theory without the magnetic field given by Amreinet al.

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Résumé Nous étudions le spectre d'une classe trés générale d'hamiltoniens à N corps dans le formalisme d'Agmon avec des champs magnétiques très singuliers. Il s'agit notamment de calculer le spectre essentiel (probléme deja résolu par Iftimie dans des conditions plus générales, par une autre méthode), de contrôler le spectre ponctuel plongé dans le spectre continu, d'établir l'absence du spectre singulier continu et de prouver un principe d'absorption limite. Par un changement de jauge (dont la construction est basée sur une idée de Boutet de Monvel-Berthier et Purice) on se ramène au cas d'un hamiltonien sans champ magnétique, mais avec les potentiels électriques modifiés, les perturbations étant des opérateurs différentiels du premier ordre, en général du type longue portée. À ce moment là on peut utiliser la théorie sans champ magnétique de Amreinet al.

     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

Microlocal analysis and phase-space decompositions

A general class of phase-space decompositions f(x)=a(x,p) dp of functions f defined in ⁿ is presented. In the latter, a(x,p) depends on values of the Fourier transform F of f in a region around p whose width tends to zero as |x| increases and it decays exponentially, for each p, in all directions in x-space outside the microsupport at p of F, with a rate of exponential fall-off linked to analyticity properties of F in local tubes (in complex space) around p. A possible application in quantum-field theory is mentioned.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.     

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Distinguished Hamiltonian theorem for homogeneous symplectic manifolds

A diffeomorphism of a finite-dimensional flat symplectic manifold which is canonoid with respect to all linear and quadratic Hamiltonians, preserves the symplectic structure up to a factor: so runs the quadratic Hamiltonian theorem. Here we show that the same conclusion holds for much smaller sufficiency subsets of quadratic Hamiltonians, and the theorem may thus be extended to homogeneous infinite-dimensional symplectic manifolds. In this way, we identify the distinguished Hamiltonians for the Kähler manifold of equivalent quantizations of a Hilbertizable symplectic space.     

Vortex configurations in regular arrays of pinning sites

The vortex states interacting with a triangular lattice of pinning sites in a two-dimensional (2D) superconducting sample have been investigated by using a molecular dynamics approach. The Nordborg–Vinokur potential is used to model the interaction between the vortices and the pinning sites. We have found several ordered vortex configurations, such as pentagons, hexagons, and shells depending on two critical parameters of the system, namely pinning radius and vorticity. Our results are in good agreement with the results of Bitter decoration experiments performed on type-II superconductors with blind hole and pillar arrays.     

Ostrogradski's theorem for higher-order singular Lagrangians

A demonstration is given of the equivalence of Euler-Lagrange and Hamilton-Dirac equations for constrained systems derived from singular Lagrangians of higher order in the derivatives.     

Integrable three-body systems with distinct two-body forces

Translationally invariant one-dimensional three-body systems with mutually different pair potentials are derived that possess a third constant of motion, both classically and quantum-mechanically; a Lax pair is given, and all (even) regular solutions of the corresponding functional equation are obtained.     

Lagrange–Poincaré field equations

The Lagrange–Poincaré equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin–Noether Theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, and molecular strands illustrate various aspects of the theory.     

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R^p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.     

Multisymplectic Structures and Discretizations for One-way Wave Equations

Multisymplectic structures for one-way wave equations are presented in this letter. Based on the multisymplectic formulation, we obtain the corresponding multisymplectic discretizations. The structure-preserving property of a finite difference scheme for the first-order one-way wave equation is proved. Implications and applications of this result are explored.      

Equilibrium theory in 2D Riemann manifold for heterogeneous biomembranes with arbitrary variational modes

Based on the first and second gradient operators and their integral theorems in 2D Riemann manifold, the equilibrium differential equations and geometrically constraint equations for heterogeneous biomembranes with arbitrary variation modes are developed. Through the combination of these equations, the equilibrium theory for heterogeneous biomembranes is established in 2D Riemann manifold. From the equilibrium theory, various interesting information is revealed: Different from homogeneous biomembranes, heterogeneous one posses new equations within the membrane’s tangential planes, i.e. the in-plane equilibrium differential equations, the in-plane boundary conditions and the in-plane geometrically constraint equations. Different from the equilibrium theory in Euclidean space, the one in 2D Riemann manifold displays strict constraints between the physical coefficients and characteristic geometric parameters of biomembranes.     

Symplectic manifolds with Lagrangian fibration

A structure theorem is presented for certain kinds of symplectic manifold with a Lagrangian fibration. As a corollary, the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of symplectic manifold considered in the theorem for which in addition, a certain cohomology class vanishes. These results and techniques are then applied to two situations in classical mechanics where symplectic manifolds foliated by Lagrangian submanifolds arise, namely, the Legendre transformation and Hamilton-Jacobi theory.     

Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

We give a Cwikel–Lieb–Rozenblum type bound on the number of bound states of Schrödinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.