## Journal of Geometry and Symmetry in Physics

### On Spectral Theory of Lax Operators on Symmetric Spaces: Vanishing Versus Constant Boundary Conditions

Vladimir S. Gerdjikov

#### Abstract

We outline several specific issues concerning the theory of multicomponent nonlinear Schrödinger equations with vanishing and constant boundary conditions. We start with the spectral properties of the Lax operator $L$ for vanishing boundary conditions. We introduce the fundamental analytic solutions (FAS) and demonstrate their importance for relating the scattering problem to a Riemann-Hilbert problem, and for the construction of the resolvent of $L$. Then we generalize this procedure to constant boundary conditions case. We start with the structure of the class of allowed potentials $\mathcal{M}$ and give a recipe of how FAS can be constructed on each of the leafs of the relevant Riemannian surface. This allows us to relate the scattering problem to a Riemann-Hilbert problem posed on a Riemannian surface. Next we use these FAS to construct the resolvent of $L$ and study its spectral properties. We also introduce the minimal set of scattering data on the continuous spectrum of $L$ which generically has varying multiplicity. The general construction is illustrated by three representative examples related to A.III, C.II and D.III symmetric spaces. Finally we consider regularized Wronskian relations which allow us to analyze the mapping between the potential of $L$ and the scattering data.

#### Article information

Source
J. Geom. Symmetry Phys., Volume 15 (2009), 1-41.

Dates
First available in Project Euclid: 24 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1495591216

Digital Object Identifier
doi:10.7546/jgsp-15-2009-1-41

Mathematical Reviews number (MathSciNet)
MR2572534

Zentralblatt MATH identifier
1180.37105

#### Citation

Gerdjikov, Vladimir S. On Spectral Theory of Lax Operators on Symmetric Spaces: Vanishing Versus Constant Boundary Conditions. J. Geom. Symmetry Phys. 15 (2009), 1--41. doi:10.7546/jgsp-15-2009-1-41. https://projecteuclid.org/euclid.jgsp/1495591216