112 research outputs found
Introduction to middle convolution for differential equations with irregular singularities
We introduce middle convolution for systems of linear differential equations
with irregular singular points, and we presend a tentative definition of the
index of rigidity for them. Under some assumption, we show a list of terminal
patterns of irreducible systems of linear differential equations by iterated
application of middle convolution when the index is positive or zero.Comment: 24 page
Integral transformation and Darboux transformation
We review Darboux-Crum transformation of Heun's differential equation. By
rewriting an integral transformation of Heun's differential equation into a
form of elliptic functions, we see that the integral representation is a
generalization of Darboux-Crum transformation. We also consider conservation of
monodromy with respect to the transformations.Comment: 7 pages, based on the talk presented at International Workshop on
Nonlinear and Modern Mathematical Physics, China, 19th July 200
Integral transformation of Heun's equation and some applications
It is known that the Fuchsian differential equation which produces the sixth
Painlev\'e equation corresponds to the Fuchsian differential equation with
different parameters via Euler's integral transformation, and Heun's equation
also corresponds to Heun's equation with different parameters, again via
Euler's integral transformation. In this paper we study the correspondences in
detail. After investigating correspondences with respect to monodromy, it is
demonstrated that the existence of polynomial-type solutions corresponds to
apparency (non-branching) of a singularity. For the elliptical representation
of Heun's equation, correspondence with respect to monodromy implies
isospectral symmetry. We apply the symmetry to finite-gap potentials and
express the monodromy of Heun's equation with parameters which have not yet
been studied.Comment: 43 page
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