Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem

Details Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem

1999-07-08

arxiv.org/abs/math/9907053v1

Mathematics

Functional Analysis

Scientific paper

In this paper, we take the first step towards an extension of the nonlinear steepest descent method of Deift, Its and Zhou to the case of operator Riemann-Hilbert problems. In particular, we provide long range asymptotics for a Fredholm determinant arising in the computation of the probability of finding a string of n adjacent parallel spins up in the antiferromagnetic ground state of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in terms of the solution of an operator Riemann-Hilbert factorization problem.

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