Self-adjoint realizations of 2d-dimensional canonical systems and applications

TL;DR

This paper develops a framework for self-adjoint realizations of 2d-dimensional canonical systems using symplectic geometry, with applications to spectral analysis and stability of solutions in PDEs like NLS.

Contribution

It introduces a method to define self-adjoint boundary conditions via Lagrangian subspaces, linking symplectic geometry with spectral theory of canonical systems.

Findings

Self-adjoint boundary conditions characterized by Lagrangian matrices.

Application to spectral stability of traveling waves and NLS solitons.

Framework facilitates spectral analysis using Evans function and transfer matrix methods.

Abstract

This paper studies linear relations and their self-adjoint realizations arising from 2d-dimensional canonical systems, with a focus on how the symplectic structure interacts with boundary conditions. Understanding this interplay allows us to define self-adjoint realizations, which are crucial for analyzing the spectral properties of these systems. We prove that for each pair of Lagrangian boundary matrices {\Theta} and B satisfying appropriate orthonormality conditions, the restricted relation T{\Theta},B is self-adjoint. Our approach relies on the symplectic geometry of boundary spaces and the isotropic structure of Lagrangian subspaces. We also discuss extensions to semi-infinite intervals using asymptotic boundary conditions. In the second part of the paper, we show how this framework applies to spectral problems from partial differential equations, including the stability of traveling waves and the linearization of the focusing nonlinear Schrodinger equation around soliton profiles. In particular, the self-adjoint structure with respect to the H-weighted inner product ensures the spectral properties needed for stability analysis using Evans function and transfer matrix methods. Applications to integrable systems, such as the stability of NLS bright solitons, are also presented.