26 research outputs found
The absolute position of a resonance peak
It is common practice in scattering theory to correlate between the position
of a resonance peak in the cross section and the real part of a complex energy
of a pole of the scattering amplitude. In this work we show that the resonance
peak position appears at the absolute value of the pole's complex energy rather
than its real part. We further demonstrate that a local theory of resonances
can still be used even in cases previously thought impossible
Category of nonlinear evolution equations, algebraic structure, and r-matrix
This paper deals with the category of nonlinear evolution equations (NLEEs)
associated with the spectral problem and provides an approach for constructing
their algebraic structure and -matrix. First we introduce the category of
NLEEs, which composes of various positive order and negative order hierarchies
of NLEEs both integrable and non-integrable. The whole category of NLEEs
possesses a generalized Lax representation. Next, we present two different Lie
algebraic structures of the Lax operator, one of them is universal in the
category,i.e. independent of the hierarchy, while the other one is nonuniversal
in the hierarchy, i.e. dependent on the underlying hierarchy. Moreover, we find
that two kinds of adjoint maps are -matrices under the algebraic structures.
In particular, the Virasoro algebraic structures without central extension of
isospectral and non-isospectral Lax operators can be viewed as reductions of
our algebraic structure. Finally, we give several concrete examples to
illustrate our methods. Particularly, the Burgers category is linearized when
the generator, which generates the category, is chosen to be independent of the
potential function. Furthermore, an isospectral negative order hierarchy in the
Burger's category is solved with its general solution. Additionally, in the KdV
category we find an interesting fact: the Harry-Dym hierarchy is contained in
this category as well as the well-known Harry-Dym equation is included in a
positive order KdV hierarchy.Comment: 24 pages, 0 figure
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
Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
We define a theory of Galilean gravity in 2+1 dimensions with cosmological
constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke
group, extending our previous study of classical and quantum gravity in 2+1
dimensions in the Galilean limit. We exhibit an r-matrix which is compatible
with our Chern-Simons action (in a sense to be defined) and show that the
associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the
classical double of the extended Heisenberg algebra. We deduce that, in the
quantisation of the theory according to the combinatorial quantisation
programme, much of the quantum theory is determined by the quantum double of
the extended q-deformed Heisenberg algebra.Comment: 22 page