Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

Abstract

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 equationswith 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 , eq , 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.

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation

“…The main reason to consider NLS vessel is Theorem 5, whose proof we present in full detail, and also it appeared in different settings in [5,[11][12][13][14]. It turns out that we can see an NLS vessel as a Bäcklund transformation of the trivial NLS equation to a more complicated one.…”

“…Finally we present constructions of the Solitons in Section 5.4. Much of the results presented here were originated in the work of the author and collaborators [5][6][7][8]. As we pointed out, as this paper was being processed for the publication, the result of solving the KdV equation with analytic initial condition was accepted for publication in the Journal of Math.…”

Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. On solution of the Korteweg-de Vries equation in the first quadrant

“…In a series of papers of the author [Mel09,Mela,Melc,Melb] and collaborators [AM09,AM12] there was developed a similar approach to the inverse scattering of many LDEs (like SL, NLS, Canonical systems) using a theory of vessels. In the most general setting in the theory of vessels, the spectral function ρ(λ) is translated into a complex-valued matrix function, belonging to a special class of functions, parallel to the "scattering matrix" s(λ) appearing in the work on Inverse scattering of Faddeyev [Fad74]:…”

Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

“…The time-dependent and/or non-commutative cases were also considered by H. Gauchman [17][18][19][20][21] in a very general setting of connections on Hilbert bundles on smooth manifold, with an application to Lie groups and their Lie algebras, by M. S. Livšic [31], and by D. Alpay, A. Melnikov and the second author [3,36]; see also [34,44] and [33,Ch. 2].…”