Maximally Superintegrable Gaudin Magnet: A Unified Approach

A classical integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Poisson algebra, while a quantum integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Jordan–Lie algebra of Hermitian operators. We propose a method for obtaining large Abelian subalgebras inside the tensor product of free tensor algebras, and we show that there exist canonical morphisms from these algebras to Poisson algebras and Jordan–Lie algebras of operators. We can thus prove the integrability of some particular Hamiltonian systems simultaneously at both the classical and the quantum level. We propose a particular case of the rational Gaudin magnet as an example.     

Noncommutative Integrability from Classical to Quantum Mechanics

We propose in this work a definition of integrable quantum system, which is based upon the correspondence with the concept of noncommutative integrability for classical mechanical systems. We then determine sufficient conditions under which, given an integrable classical system, it is possible to construct an integrable quantum system by means of a quantization procedure based on the symmetrized product of operators. As a first example of application of such an approach, we will consider the possible cases of noncommutative integrability for systems with rotational symmetry in an n-dimensional Euclidean configuration space.     

Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions

We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

On a Lagrangian Reduction and a Deformation of Completely Integrable Systems

We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm \(H^1\) in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered, and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them, we found two important equations, the Camassa–Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation.     

Integrability of geodesic flows and isospectrality of Riemannian manifolds

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds M and M′ which are isospectral for the Laplace operator on functions and such that M has completely integrable geodesic flow in the sense of Liouville, while M′ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by two maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in M, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for M, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both M and M′ satisfy the so-called Clean Intersection Hypothesis. The author was partially supported by DFG Sonderforschungsbereich 647.     

Integrability and singularity structure of coupled nonlinear Schrödinger equations

Multimode propagation of electromagnetic waves in optical fibre is often described by coupled nonlinear Schrödinger (NLS) equations. To understand the integrability properties of such coupled NLS systems, we extend the Painlevé singularity structure analysis of two coupled systems to three coupled systems and identify four integrable sets of parameters. We bilinearize these cases to obtain soliton solutions. The results are extended to N-coupled systems, completing the earlier analysis of Sahadevan, Tamizhmani and Lakshmanan.     

Propagating wave patterns for the ‘resonant’ Davey–Stewartson system

The resonant nonlinear Schrödinger (RNLS) equation exhibits the usual cubic nonlinearity present in the classical nonlinear Schrödinger (NLS) equation together with an additional nonlinear term involving the modulus of the wave envelope. It arises in the context of the propagation of long magneto-acoustic waves in cold, collisionless plasma and in capillarity theory. Here, a natural (2 + 1) (2 spatial and 1 temporal)-dimensional version of the RNLS equation is introduced, termed the ‘resonant’ Davey–Stewartson system. The multi-linear variable separation approach is used to generate a class of exact solutions, which will describe propagating, doubly periodic wave patterns.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

On integrability and chaos in discrete systems

The scalar nonlinear Schrödinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of “effective” chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.     

Algebraic integrability for the Schrödinger equation and finite reflection groups

Algebraic integrability of ann-dimensional Schrödinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.In memory of M. K. PolivanovMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 253–275, February, 1993.     

Integrability of induction cocycles for Kac-Moody groups

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L^p for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.Prépublication de l’Institut Fourier nº 637 (2004); e-mail: http://www-fourier.ujf-grenoble.fr/prepublicatons.html     

Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics

We construct well-known integrable equations with their Lax pairs from the corresponding linear equations using our nonlinearization scheme. Using negative powers in the spectral flow to deform the time Lax operator, we find a class of perturbations that unlike the usual perturbations, which spoil the system integrability, exhibit a twofold integrable hierarchy, including those for the KdV, modified KdV, sine-Gordon, nonlinear Schrödinger (NLS), and derivative NLS equations. We discover hidden possibilities of using the perturbed hierarchy of the NLS equations to amplify and control optical solitons propagating through a fiber in a doped nonlinear resonant medium.     

Projective Completions of Jordan Pairs,Part II: Manifold Structures and Symmetric Spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields , and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.Mathematics Subject Classiffications (2000). primary: 17C36, 46H70, 17C65; secondary: 17C30, 17C90     

Sparse grid spaces for the numerical solution of the electronic Schrödinger equation

This article complements the author’s recent work [Numer. Math. 98, 731–759 (2004)] on the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. It has been shown there that the solutions of this equation are surprisingly smooth and possess square integrable mixed weak derivatives of order up to N+1 with N the number of electrons across the singularities of the interaction potentials, and it has been claimed that this result can help to break the complexity barriers in computational quantum mechanics using correspondingly antisymmetrized sparse grid trial functions. A construction of this kind that can be interpreted as a sparse grid sampling theorem is sketched here.     

On global integrability of BMO functions on general domains

Extending results of Staples and Smith-Stegenga, we characterize measurable subsets of a given domainD ⊃R ⁿ on which BMO(D) functions areL ^p integrable or exponentially integrable. In particular, we characterize uniform domains by the integrability of BMO functions. We also remark on the boundedness of domains satisfying a certain integrability condition for the quasihyperbolic metric.     

Integrability aspects of NLS–MB system with variable dispersion and nonlinear effects

In this paper, we analyze the integrability aspects of the NLS–MB system with variable dispersion and nonlinear effects. We obtain the constraints for which the above system becomes integrable by using the Painlevé singularity analysis. Obtained results are in agreement with the known results.     

A note on degenerate corank-one singularities of integrable Hamiltonian systems

We prove that, in a neighborhood of a corank-1 singularity of an analytic integrable Hamiltonian system with n degrees of freedom, there is a locally-free analytic symplectic \Bbb T^n-1 {\Bbb T}^{n-1} -action which preserves the moment map, under some mild conditions. This result allows one to classify generic degenerate corank-one singularities of integrable Hamiltonian systems. It can also be applied to the study of (non)integrability of perturbations of integrable systems.     

Integrability of dual coactions on Fell bundle C*-algebras

We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied by Ruy Exel, Ralf Meyer and Marc Rieffel. Our goal is to study the close relationship between integrable group coactions and Fell bundles. As a main result, we prove that dual coactions on C*-algebras of Fell bundles are integrable, generalizing results by Ruy Exel for abelian groups.     

A characterization of strongly measurable Kurzweil–Henstock integrable functions and weakly continuous operators

We give necessary and sufficient conditions for the Kurzweil–Henstock integrability of functions given by , where x_n belong to a Banach space and the sets (E_n)_n are measurable and pairwise disjoint. Also weakly completely continuous operators between Banach spaces are characterized by means of scalarly Kurzweil–Henstock integrable functions.