Reduced transfer operators for singular difference equations

TL;DR

This paper develops reduced transfer operators for singular difference equations, enabling the extension of oscillation theory to systems with non-invertible off-diagonal blocks by ensuring unitarity and monotonicity properties.

Contribution

It introduces a method to construct reduced transfer operators for singular tridiagonal block Jacobi operators, overcoming invertibility limitations.

Findings

Reduced transfer operators maintain Krein space unitarity.

Established monotonicity in the energy variable.

Extended oscillation theory to singular systems.

Abstract

For tridiagonal block Jacobi operators, the standard transfer operator techniques only work if the off-diagonal entries are invertible. Under suitable assumptions on the range and kernel of these off-diagonal operators which assure a homogeneous minimal coupling between the blocks, it is shown how to construct reduced transfer operators that have the usual Krein space unitarity property and also a crucial monotonicity in the energy variable. This allows to extend the results of oscillation theory to such systems.