Symplectic Operators Research Articles

Local symplectic operators and structures related to them

Matrices with entries being differential operators, that endow the phase space of an evolution system with a (pre)symplectic structure are considered. Special types of such structures are explicitly described. Links with integrability, geometry of loop spaces, and Bäcklund transformations are traces.

Singular operators on boson fields as forms on spaces of entire functions on Hilbert space

Invariant scales of entire analytic functions on Hilbert space are introduced and applied. Singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the Hilbert space with its dual. The Weyl (or, exponentiated boson field) operators act smoothly and irreducibly on corresponding spaces of entire functions. Arbitrary symplectic operators on a single-particle Hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators.

On the bi-Hamiltonian structure, strong and hereditary operator of symmetries for a new nonlinear system

The bi-Hamiltonian structure, strong, and hereditary operator for symmetries for a new nonlinear integrable system have been deduced by the procedure of Fourier analysis and small amplitude expansion. The recursion operator for the gradient of conserved quantities has been written down. An important feature of this approach is that, it is possible to obtain several solutions of the equation determining the symplectic operator. But unfortunately one cannot construct the strong (hereditary) operator from all such pairs of solution. Only for some special solutions it was possible to do so. In these cases the strong and hereditary property can be proven explicitly. In the following, examples of operators belonging to both of these classes will be cited.

On a different approach to the bi-Hamiltonian structure of higher-order water-wave equations

Conservation laws and the bi-Hamiltonian structure of higher-order water-wave equations are obtained by a method different from the usual approach of symmetry analysis. The method is based on the technique of Fourier analysis and small amplitude expansion. Some comments are made about the symmetries of the equation. The recursion operator Lambda is then constructed from the two symplectic operators and it is explicitly verified that it is both a strong operator and a hereditary one.

On differential operators that generate Hamiltonian structures

Differential operators can serve in a one-dimensional Hamiltonian theory of evolution equations either as symplectic operators or as Hamiltonian ones. Restrictions on the coefficients brought by each of these possibilities are considered. A complete characterization of all symplectic operators whose coefficients are linear functions of the dependent variable and its derivatives is presented.

Invariant Lagrangian Subspaces

It is proved that on Hilbert spaces with strong symplectic form, every symplectic operator $I + C$ with $C$ compact has an invariant Lagrangian subspace.

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL 2(R 3) and the Einstein’s relativistic equation in the spaceL 2(R 3, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL 2(R 3) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL 2(R 3).

On the twisted convolution product and the Weyl transformation of tempered distributions

It is well known that the Weyl transformation in a phase space R 21, transforms the elements of L ( R 21) in trace class operators and the elements of L 2( R 21) in the Hilbert-Schmidt operators of the Hilbert space L 2( R 1); this fact leads to a general method of quantization suggested by E. Wigner and J.E. Moyal and developed by M. Flato, A. Lichnerowicz, C. Fronsdal, D. Sternheimer and F. Bayen for an arbitrary symplectic manifold, known under the name of star-product method. In this context, it is important to study the Weyl transforms of the tempered distributions on the phase space and that of the star-exponentials which gave the spectrum in this process of quantization. We analyze here the relations between the star-product, the twisted convolution product and the Weyl transformation of tempered distributions. We introduce symplectic differential operators which permit us to study the structure of the space O 1 λ λ ≠ 0, (similar to the space O 1 C) of the left (twisted) convolution operators of L ( R 21) which permit to define the twisted convolution product in the space L( R 21), and the structures of the admissible symbols for the Weyl transformation (i.e. the domain of the Weyl transformation). We prove that the bounded operators of L 2( R 1) are exactly the Weyl transforms of the bounded (twisted) convolution operators of L 2( R 21). We give an expression of the integral formula of the star product in terms of twisted convolution products which is valid in the most general case. The unitary representations of the Heisenberg group play an important role here.

Infinite-dimensional metagonal and metaplectic groups. I. General concepts and metagonal groups

We study central S1 -extensions of the groups of orthogonal and symplectic operators on a Hilbert space with Hilbert-Schmidt antilinear part. We investigate in detail the corresponding 2-cocyles on their Lie algebras. The passage to the limit of the corresponding finite-dimensional groups is described and it is shown that both extensions are nontrivial even in the topological sense. Among the applications is a unified construction of the basic modulus for Kac-Moody algebras.

Some covariant representations of massless boson fields

Considers the c*-algebra of the canonical commutation relations (CCR), acted on by a group G of one-particle symmetry transformations V. A symplectic operator T defines a representation pi T= pi F omicron T where pi F is the Fock representation. The automorphisms of the CCR algebra that are induced by G are shown to be continuously implemented in pi T if and only if A-V(g)AV*(g) is a continuous Hilbert-Schmidt 1-cocycle of G; here, A is related to T by T=exp A, A being a suitable bounded, antilinear self adjoint operator. Some new examples of fully Poincare-covariant representations of massless fields in 1+1 dimensions are constructed.

Irreducible U(3) tensor interactions in the symplectic collective model

The potential V( β, γ) of the Bohr-Mottelson and symplectic collective models is expressed as a linear combination of U(3) irreducible tensor operators in the symplectic enveloping algebra. This many-body collective potential is then projected onto the symplectic two-body tensor operators. The projected two-body potential is shown to give results similar to the many-body potential in 20Ne. Hence, in the symplectic shell model, one has obtained a collective model with two-body forces.

On the structure of symplectic operators and hereditary symmetries

On the structure of symplectic operators and hereditary symmetries

A simple model of the integrable Hamiltonian equation

A method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Bäcklund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrödinger equation and the so-called Harry Dym equation turn out to be particular examples.