2,462 research outputs found
Classification of Multidimensional Darboux Transformations: First Order and Continued Type
We analyze Darboux transformations in very general settings for
multidimensional linear partial differential operators. We consider all known
types of Darboux transformations, and present a new type. We obtain a full
classification of all operators that admit Wronskian type Darboux
transformations of first order and a complete description of all possible
first-order Darboux transformations. We introduce a large class of invertible
Darboux transformations of higher order, which we call Darboux transformations
of continued Type I. This generalizes the class of Darboux transformations of
Type I, which was previously introduced. There is also a modification of this
type of Darboux transformations, continued Wronskian type, which generalize
Wronskian type Darboux transformations
On the use of Mellin transform to a class of q-difference-differential equations
We explore the possibility of using the method of classical integral
transforms to solve a class of -difference-differential equations. The
Laplace and the Mellin transform of -derivatives are derived. The results
show that the Mellin transform of the -derivative resembles most closely the
corresponding expression in classical analysis, and it could therefore be
useful in solving certain -difference equations.Comment: 13 pages, LaTex, no figure
Duality, Biorthogonal Polynomials and Multi-Matrix Models
The statistical distribution of eigenvalues of pairs of coupled random
matrices can be expressed in terms of integral kernels having a generalized
Christoffel--Darboux form constructed from sequences of biorthogonal
polynomials. For measures involving exponentials of a pair of polynomials V_1,
V_2 in two different variables, these kernels may be expressed in terms of
finite dimensional ``windows'' spanned by finite subsequences having length
equal to the degree of one or the other of the polynomials V_1, V_2. The
vectors formed by such subsequences satisfy "dual pairs" of first order systems
of linear differential equations with polynomial coefficients, having rank
equal to one of the degrees of V_1 or V_2 and degree equal to the other. They
also satisfy recursion relations connecting the consecutive windows, and
deformation equations, determining how they change under variations in the
coefficients of the polynomials V_1 and V_2. Viewed as overdetermined systems
of linear difference-differential-deformation equations, these are shown to be
compatible, and hence to admit simultaneous fundamental systems of solutions.
The main result is the demonstration of a spectral duality property; namely,
that the spectral curves defined by the characteristic equations of the pair of
matrices defining the dual differential systems are equal upon interchange of
eigenvalue and polynomial parameters.Comment: Latex, 44 pages, 1 tabl
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