Finite dimensional Sturm Liouville vessels and their tau functions

Theory of Vessels was started by M. Livšic in late 70’s as a part of more general theory, developed for n dimensional systems, defined by non self-adjoint commuting operators. In the Phd thesis of the author this theory is developed for the case n = 2 and there arise overdetermined time invariant systems and corresponding Vessels. A key idea of the construction is that transfer function of the system intertwines solutions of Linear Differential Equations (LDEs) with a spectral parameter. In this manner there are constructed solutions of Sturm Liouville differential equations d 2 dx2 y(x) − q(x)y(x) = λy(x) with the spectral parameter λ and coefficient q(x), called potential. On the one hand this work can be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm Liouville differential equations on a half axis (0,∞) with singularity at 0. On the other hand, there is developed a rich and interesting theory of Vessels which has connections to the notion of τ function, arising in non linear differential equations and to the Galois differential theory for linear ODEs. The transfer function of a Vessel plays a key role in this research work. From a realization formula for the transfer function one can construct ,,tau” function τ , which is a determinant of a self-adjoint matrix function, corresponding to the SL Vessel, and can prove that there is a differential ring R∗ generated by τ ′ τ , e ∫ , to which all the relevant objects belong. Further, using R∗ one can evaluate the Picard-Vessiot ring of the output LDE and so connections to the Differential Galois theory are obtained. It seems that finite dimensional SL Vessels as the most convenient environment to handle the ,,deformation theory” of Sturm Liouville differential equations. In order to precisely understand this deformation, there is studied the dependence of Vessels on spectral parameters.