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

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Abstract

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 ϖP/ϖt n =[(P n/p)+,P], withp:=ordP≥2; (ii) find a matrix a integral representation for the associated τ-function. First we construct an infinite dimensional spaceW= spanℂ{ψ 0(z,ψ 1(z,...)} of functions ofzεℂ invariant under the action of two operators, multiplication byz p andA c :=zϖ/ϖz−z+c. This requirement is satisfied, for arbitraryp, ifψ 0 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 τ-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 contourγ≔γ -+γ - ⊂ℂ defined byγ ± = ℝ+e±πi/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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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, 02254, Waltham, Mass, USA

    M. Adler & P. van Moerbeke

  2. ITEP, Moscow, Russia

    A. Morozov

  3. Department of Mathematics, Kyoto University, 606-01, Kyoto, Japan

    T. Shiota

  4. Department of Mathematics, Universite de Louvain, 1348, Louvain-la-Neuve, Belgium

    P. van Moerbeke

Authors
  1. M. Adler
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  2. A. Morozov
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  3. T. Shiota
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  4. P. van Moerbeke
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Corresponding author

Correspondence to P. van Moerbeke.

Additional information

Communicated by R.H. Dijkgraaf

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.

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Adler, M., Morozov, A., Shiota, T. et al. A matrix integral solution to [P, Q]=P and matrix laplace transforms. Commun.Math. Phys. 180, 233–263 (1996). https://doi.org/10.1007/BF02101187

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  • DOI: https://doi.org/10.1007/BF02101187

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