On the index instability for some nonlocal elliptic problems

TL;DR

This paper investigates how nonlocal perturbations with non-vanishing coefficients can alter the Fredholm index of operators in nonlocal elliptic problems, challenging previous assumptions about index stability.

Contribution

It provides the first examples demonstrating that the index can change under certain nonlocal perturbations with non-zero coefficients at conjugation points.

Findings

Index may change under nonlocal perturbations with non-vanishing coefficients

Constructed explicit examples showing index instability

Challenges previous results assuming index invariance

Abstract

The Fredholm index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions.