ar X iv : h ep - t h / 96 10 13 7 v 1 1 7 O ct 1 99 6 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 ∂P/∂t n = [(P n/p) + , P ], with p := ord P ≥ 2; (ii) find a matrix integral representation for the associated τ-function. First we construct an infinite dimensional space W = span C ψ 0 (z), ψ 1 (z),. .. of functions of z ∈ C invariant under the action of two operators, multiplication by z p and A c := z ∂/∂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 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 − ⊂ C defined by γ ± = R + e ±πi/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.