Minimal spectral functions of an ordinary differential operator

TL;DR

This paper develops a framework for describing minimal spectral functions of differential operators using boundary triplets and introduces an $m$-function concept that generalizes classical spectral functions, applicable to operators with operator-valued coefficients.

Contribution

It introduces a new approach to characterize all minimal spectral functions of differential operators with operator-valued coefficients, extending classical theory to more general boundary conditions and deficiency indices.

Findings

Describes all minimal spectral functions of the boundary problem.

Introduces the $m$-function concept generalizing the Titchmarsh-Weyl function.

Improves estimates of spectral multiplicity for certain selfadjoint extensions.

Abstract

Let $l[y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b> \;(b\leq \infty)$ and let $L_0$ be the corresponding minimal operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation $l[y]-\l y=f\;(f\in L_2[0,b>)$ and the Nevanlinna $\l$-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the $m$-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e., all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices $n_\pm(L_0)$ and not decomposing boundary conditions) the known estimate of the spectral multiplicity of the (exit space) selfadjoint extension $\wt A\supset L_0$. The results of the paper are obtained for expressions $l[y]$ with operator valued coefficients and arbitrary (equal or unequal) deficiency indices $n_\pm(L_0)$.