Finite-dimensional Sturm–Liouville vessels and their tau functions Andrey Melnikov Drexel University, USA September 4, 2017 Abstract We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of M. Livˇ sic [Liv1]. This 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 τ function, arising in non linear differential equations and to the Galois differential theory for LDEs. Contents 1 Introduction 2 2 Overdetermined time invariant 2D systems 4 2.1 Conservative vessel [MVc] ...................... 4 2.2 The transfer function of a vessel .................. 8 3 Sturm–Liouville vessels 11 3.1 Elementary input vessels ....................... 11 3.1.1 Definition of a vessel with elementary input ........ 11 3.1.2 The τ function of an elementary input vessel ....... 14 3.1.3 Anti-adjoint spectral values ................. 16 3.2 Vessels as B¨acklund transformations. Crum transformations . . . 17 3.3 The differential ring R ∗ associated to an elementary input SL vessel ................................. 19 3.4 General Sturm–Liouville vessels ................... 29 4 Conclusions and remarks 35 1