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• W3100579904 abstract "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 Pge 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 zinBC 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 Hankel function (for p=2); our representation of the generalized Hankel 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^- subsetBC defined by gamma^pm=BR_+E^{pmpiI/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." @default.
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• W3100579904 date "1996-09-01" @default.
• W3100579904 modified "2023-10-14" @default.
• W3100579904 title "A matrix integral solution to [P, Q]=P and matrix laplace transforms" @default.
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• W3100579904 doi "https://doi.org/10.1007/bf02101187" @default.
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