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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 \{ψ_0(z),ψ_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 ψ_0 is a certain function generalizing the classical Hänkel function (for p=2); our representation of the generalized Hänkel function as a double Laplace transform of a simple function, which was unknown even for the p=2 case, enables us to represent the τ-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 γ:= γ^+ + γ^- \subset\BC defined by γ^\pm=\BR_+\E^{\pmπ\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.