A matrix integral solution to [P,Q]=P and matrix Laplace transforms
(1996) Communications in Mathematical Physics — Vol. 180, n° 1, p. 233-263 (1996)
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- 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 derivative P/partial derivative t(n)=[(P-n/p)(+), P], with p:= ord P greater than or equal to 2; (ii) find a matrix integral representation for the associated tau-function. First we construct an infinite dimensional space W = = span(C){psi(0)(z), psi(1)(z),...} of functions of z is an element of C invariant under the action of two operators, multiplication by z(p) and A(c):= z partial derivative/partial derivative 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 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 of C defined by gamma(+/-) = R(+)e(+/-pi 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.
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Adler, M., Morozov, A., Shiota, T., & Van Moerbeke, P. (1996). A matrix integral solution to [P,Q]=P and matrix Laplace transforms. Communications in Mathematical Physics, 180(1), 233-263. https://doi.org/10.1007/BF02101187 (Original work published 1996)