1996/08/30 by Kodama, Yuji, Ye, Jian
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.solv-int/9609001
We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996), 321-339]. These Toda lattices consist of 2N-1 different systems with hamiltonians H = (1/2) ∑k=1N yk2 + ∑k=1N-1 sksk+1 exp(xk-xk+1), where si=± 1. We compactify the manifolds by adding infinities according to the Toda flows which blow up in finite time except the case with all sisi+1=1. The resulting manifolds are shown to be nonorientable for N>2, and the symmetric group is the semi-direct product of (\ZZ2)N-1 and the permutation group SN. These properties identify themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction, we give a formula on the total numbers of zeroes for a system of exponential polynomials generated as Hankel determinant.