333 research outputs found
Isospectral flows on a class of finite-dimensional Jacobi matrices
We present a new matrix-valued isospectral ordinary differential equation
that asymptotically block-diagonalizes zero-diagonal Jacobi
matrices employed as its initial condition. This o.d.e.\ features a right-hand
side with a nested commutator of matrices, and structurally resembles the
double-bracket o.d.e.\ studied by R.W.\ Brockett in 1991. We prove that its
solutions converge asymptotically, that the limit is block-diagonal, and above
all, that the limit matrix is defined uniquely as follows: For even, a
block-diagonal matrix containing blocks, such that the
super-diagonal entries are sorted by strictly increasing absolute value.
Furthermore, the off-diagonal entries in these blocks have the same
sign as the respective entries in the matrix employed as initial condition. For
odd, there is one additional block containing a zero that is
the top left entry of the limit matrix. The results presented here extend some
early work of Kac and van Moerbeke.Comment: 19 pages, 3 figures, conjecture from previous version is added as
assertion (iv) of the main theorem including a proof; other major change
Isospectral Flow and Liouville-Arnold Integration in Loop Algebras
A number of examples of Hamiltonian systems that are integrable by classical
means are cast within the framework of isospectral flows in loop algebras.
These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger
systems and the sine-Gordon equation. Each system has an associated invariant
spectral curve and may be integrated via the Liouville-Arnold technique. The
linearizing map is the Abel map to the associated Jacobi variety, which is
deduced through separation of variables in hyperellipsoidal coordinates. More
generally, a family of moment maps is derived, identifying certain finite
dimensional symplectic manifolds with rational coadjoint orbits of loop
algebras. Integrable Hamiltonians are obtained by restriction of elements of
the ring of spectral invariants to the image of these moment maps. The
isospectral property follows from the Adler-Kostant-Symes theorem, and gives
rise to invariant spectral curves. {\it Spectral Darboux coordinates} are
introduced on rational coadjoint orbits, generalizing the hyperellipsoidal
coordinates to higher rank cases. Applying the Liouville-Arnold integration
technique, the Liouville generating function is expressed in completely
separated form as an abelian integral, implying the Abel map linearization in
the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth
Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199
Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
In these notes we review and further explore the Lie enveloping algebra of a
post-Lie algebra. From a Hopf algebra point of view, one of the central
results, which will be recalled in detail, is the existence of a second Hopf
algebra structure. By comparing group-like elements in suitable completions of
these two Hopf algebras, we derive a particular map which we dub post-Lie
Magnus expansion. These results are then considered in the case of
Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined
in terms of solutions of modified classical Yang-Baxter equation. In this
context, we prove a factorization theorem for group-like elements. An explicit
exponential solution of the corresponding Lie bracket flow is presented, which
is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl
Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients
We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and
Killip-Simon from asymptotically constant orthogonal polynomials on the real
line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC.
The key tool is a characterization of the isospectral torus that is well
adapted to the study of perturbations.Comment: 64 pages, to appear in Ann. of Mat
Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras
Darboux coordinates are constructed on rational coadjoint orbits of the
positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by
the values of the spectral parameters at the divisors corresponding to
eigenvector line bundles over the associated spectral curves, defined within a
given matrix representation. A Liouville generating function is obtained in
completely separated form and shown, through the Liouville-Arnold integration
method, to lead to the Abel map linearization of all Hamiltonian flows induced
by the spectral invariants. Serre duality is used to define a natural
symplectic structure on the space of line bundles of suitable degree over a
permissible class of spectral curves, and this is shown to be equivalent to the
Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general
construction is given for or , with
reductions to orbits of subalgebras determined as invariant fixed point sets
under involutive automorphisms. The case is shown to reproduce
the classical integration methods for finite dimensional systems defined on
quadrics, as well as the quasi-periodic solutions of the cubically nonlinear
Schr\"odinger equation. For , the method is applied to the
computation of quasi-periodic solutions of the two component coupled nonlinear
Schr\"odinger equation.Comment: 61 pg
The Atiyah--Hitchin bracket and the open Toda lattice
The dynamics of finite nonperiodic Toda lattice is an isospectral deformation
of the finite three--diagonal Jacobi matrix. It is known since the work of
Stieltjes that such matrices are in one--to--one correspondence with their Weyl
functions. These are rational functions mapping the upper half--plane into
itself. We consider representations of the Weyl functions as a quotient of two
polynomials and exponential representation. We establish a connection between
these representations and recently developed algebraic--geometrical approach to
the inverse problem for Jacobi matrix. The space of rational functions has
natural Poisson structure discovered by Atiyah and Hitchin. We show that an
invariance of the AH structure under linear--fractional transformations leads
to two systems of canonical coordinates and two families of commuting
Hamiltonians. We establish a relation of one of these systems with Jacobi
elliptic coordinates.Comment: 26 pages, 2 figure
Post-Lie Algebras and Isospectral Flows
In this paper we explore the Lie enveloping algebra of a post-Lie algebra
derived from a classical -matrix. An explicit exponential solution of the
corresponding Lie bracket flow is presented. It is based on the solution of a
post-Lie Magnus-type differential equation
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