dorsal/arxiv
View SchemaIntegrable Fredholm Operators and Dual Isomonodromic Deformations
| Authors | J. Harnad, Alexander R. Its |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9706002 |
| URL | https://arxiv.org/abs/solv-int/9706002 |
| DOI | 10.1007/s002200200614 |
| Journal | Commun.Math.Phys.226:497-530,2002 |
Abstract
The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.
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"abstract": "The Fredholm determinants of a special class of integral operators K\nsupported on the union of m curve segments in the complex plane are shown to be\nthe tau-functions of an isomonodromic family of meromorphic covariant\nderivative operators D_l. These have regular singular points at the 2m\nendpoints of the curve segments and a singular point of Poincare index 1 at\ninfinity. The rank r of the vector bundle over the Riemann sphere on which they\nact equals the number of distinct terms in the exponential sums entering in the\nnumerator of the integral kernels. The deformation equations may be viewed as\nnonautonomous Hamiltonian systems on an auxiliary symplectic vector space M,\nwhose Poisson quotient, under a parametric family of Hamiltonian group actions,\nis identified with a Poisson submanifold of the loop algebra Lgl_R(r) with\nrespect to the rational R-matrix structure. The matrix Riemann-Hilbert problem\nmethod is used to identify the auxiliary space M with the data defining the\nintegral kernel of the resolvent operator at the endpoints of the curve\nsegments. A second associated isomonodromic family of covariant derivative\noperators D_z is derived, having rank n=2m, and r finite regular singular\npoints at the values of the exponents defining the kernel of K. This family is\nsimilarly embedded into the algebra Lgl_R(n) through a dual parametric family\nof Poisson quotients of M. The operators D_z are shown to be analogously\nassociated to the integral operator obtained from K through a Fourier-Laplace\ntransform.",
"arxiv_id": "solv-int/9706002",
"authors": [
"J. Harnad",
"Alexander R. Its"
],
"categories": [
"solv-int",
"cond-mat",
"hep-th",
"math-ph",
"math.MP",
"nlin.SI"
],
"doi": "10.1007/s002200200614",
"journal_ref": "Commun.Math.Phys.226:497-530,2002",
"title": "Integrable Fredholm Operators and Dual Isomonodromic Deformations",
"url": "https://arxiv.org/abs/solv-int/9706002"
},
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