A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t_n=[(P^n/p)₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{₀(z,₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz^p andA_c:=z/z–z+c. This requirement is satisfied, for arbitraryp, if₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW 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^-+^- defined by^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.