Abstract |
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[eng] In this paper we solve the following problems: (i) find two differential
operators P and Q satisfying [P,Q]=P, where P flows according to the KP
hierarchy \partial P/\partial t_n = [(P^{n/p})_+,P], with p := \ord P\ge 2;
(ii) find a matrix integral representation for the associated $\t
au$-function. First we construct an infinite dimensional space {\cal
W}=\Span_\BC \{\psi_0(z),\psi_1(z),... \} of functions of z\in\BC invariant
under the action of two operators, multiplication by z^p and A_c:= z
\partial/\partial z - z + c. This requirement is satisfied, for arbitrary p,
if \psi_0 is a certain function generalizing the classical H\"ankel function
(for p=2); our representation of the generalized H\"ankel function as a
double Laplace transform of a simple function, which was unknown even for the
p=2 case, enables us to represent the \tau-function associated with the KP
time evolution of the space \cal W as a ``double matrix Laplace transform''
in two different ways. One representation involves an integration over the
space of matrices whose spectrum belongs to a wedge-shaped contour \gamma :=
\gamma^+ + \gamma^- \subset\BC defined by \gamma^\pm=\BR_+\E^{\pm\pi\I/p}.
The new integrals above relate to the matrix Laplace transforms, in contrast
with the matrix Fourier transforms, which generalize the Kontsevich integrals
and solve the operator equation [P,Q]=1.
Comment: 27 pages, LaTeX, 1 figure in PostScript |