New matrix model solutions to the Kac-Schwarz problem

We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying $[{\cal K}_1,W] = 1$ always leads to Kontsevich type models, in the case of $[{\cal K}_1,W] = W$ the corresponding KP $\tau$-functions are represented as more sophisticated matrix integrals.Comment: 19 pages, latex, no figures, contribution to the proceedings of the 29th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, German

A Matrix Integral Solution to [P,Q]=P and Matrix Laplace Transforms

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 PostScrip