Quasi self-adjoint extensions of a class of formally non-self-adjoint discrete Hamiltonian systems

TL;DR

This paper characterizes quasi self-adjoint extensions of certain non-self-adjoint discrete Hamiltonian systems, establishing a bijection with self-adjoint extensions and providing a complete classification in the limit point case.

Contribution

It introduces a novel framework for characterizing quasi self-adjoint extensions and relates them to self-adjoint extensions via a bijective projection.

Findings

Established a bijective projection between quasi self-adjoint and self-adjoint extensions.

Provided a complete classification of quasi self-adjoint extensions in the limit point case.

Derived properties of solutions and minimal linear relations for non-self-adjoint systems.

Abstract

This paper is concerned with the characterizations of quasi self-adjoint extensions of a class of formally non-self-adjoint discrete Hamiltonian systems. Some properties of the solutions and the characterization of the minimal linear relations of the non-self-adjoint systems are obtained. A bijective projection between all the quasi self-adjoint extensions of non-self-adjoint systems and all the self-adjoint extensions of the self-adjoint systems generated by the non-self-adjoint Hamiltonian systems is established in the general case. When the system is in the limit point case and $\I=[a,\infty)$, a complete characterization of all the quasi self-adjoint extensions is obtained by a subspace $Q\subset \C^{2n}$ with $\dim Q=n$ in terms of boundary conditions.