Solvability Conditions for Some non Fredholm Operators

TL;DR

This paper establishes solvability conditions for certain elliptic equations involving non-Fredholm operators using spectral and scattering theory, despite the lack of Fredholm property.

Contribution

It introduces solvability criteria for elliptic equations with non-Fredholm operators through spectral and scattering theory methods.

Findings

Solvability conditions are expressed via orthogonality to solutions of the adjoint equation.

The approach applies spectral theory to non-Fredholm elliptic operators.

The results extend classical solvability criteria beyond Fredholm operators.

Abstract

We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrodinger type operators. Though the Fredholm property is not satisfied, the solvability conditions are formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation.