On eigenfunction expansions of first-order symmetric systems and ordinary differential operators of an odd order

TL;DR

This paper develops a spectral theory framework for first-order symmetric systems with unequal deficiency indices, introducing boundary conditions, an m-function, and eigenfunction expansions, extending Titchmarsh-Weyl theory to odd-order differential operators.

Contribution

It generalizes eigenfunction expansion theory for non-Hamiltonian symmetric systems with unequal deficiency indices, including boundary conditions, m-functions, and spectral parametrization.

Findings

Defined boundary conditions analogous to separated self-adjoint conditions.

Constructed an m-function as an analog of Titchmarsh-Weyl coefficient.

Characterized spectrum via spectral functions and parametrized all spectral functions.

Abstract

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b\rangle $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ satisfy $n_+(\Tmi)< n_-(\Tmi)$. We define $\l$-depending boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate an exit space self-adjoint extension $\wt T$ of $\Tmi$ and the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. By using $m$-function we obtain the eigenfunction expansion with the spectral function $\Si(\cd)$ of the minimally possible dimension and characterize the case when spectrum of $\wt T$ is defined by $\Si(\cd)$. Moreover, we parametrize all spectral functions in terms of a Nevanlinna type boundary parameter. Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh-Weyl theory of such operators.