Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

Abstract

We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of Livsic (Soobshch Akad Nauk Gruzin SSSR 91(2):281–284, 1978). Our work may be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm–Liouville differentiable equations on the half axis (0,∞) with singularity at 0. We also develop a rich and interesting theory of vessels with deep connections to the notion of the τ function, arising in non linear differential equations (LDE), and to the Galois differential theory for LDEs.