Construction of KdV flow I. $\tau$-Function via Weyl function

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.