arXiv subjects
Publications and source records attributed to Shinichi Kotani.
A Toda flow is constructed starting from a certain class of unbounded initial conditions including sequences growing with power order of less than 1. Unbounded ergodic sequences are allowed, and especially \b{eta}-ensembles matrix models in random matrix theory can be an initial data and they yiled invariant measures for the flow.
C. Remling obtained a theorem on limit set of the shift operation on a space of functions on R when the associated 1-D half line Schrödinger operators have absolutely continuous component in their spectrum. The purpose of the paper is to define a KdV flow on a certain class of functions containing algebro-geometric functions and to extend this result to the KdV flow.
A Toda flow is constructed on a space of bounded initial data through Sato-Segal-Wilson theory. The flow is described by the Weyl functions of the underlying Jacobi operators. This is a continuation of the previous work on the KdV flow.
A KdV flow is constructed on a space whose structure is described in terms of the spectrum of the underlying Schrödinger operators. The space includes the conventional decaying functions and ergodic ones. Especially any smooth almost periodic function can be initial data for the KdV equation.
Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators on Hardy space of the unit disc. This paper gives another representation of the tau-functions by the Weyl functions for 1d Schrödinger operators with real valued potentials, which will make it possible to extend the class of initial data for the KdV equation to more general one.
We consider the 1d Schrödinger operators with random decaying potentials where the spectrum is pure point(sub-critical case). We show that the point process composed of the rescaled eivenvalues, together with those zero points of the corresponding eigenfunctions, converges to the Poisson process.
We study the level statistics of one-dimensional Schrödinger operator with random potential decaying like $x^{-α}$ at infinity. We consider the point process $ξ_L$ consisting of the rescaled eigenvalues and show that : (i)(ac spectrum case) for $α> \frac 12$, $ξ_L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii)(critical case) for $α= \frac 12$, $ξ_L$ converges to the limit of the circular $β$-ensemble.
In the case of Krein's strings with spectral functions of polynomial growth a necessary and sufficient condition for the Krein's correspondence to be continuous is given.