Spectral and scattering theory of fourth order differential operators

TL;DR

This paper develops scattering theory for fourth order differential operators with rapidly converging coefficients, including explicit formulas and classifications of zero-energy resonances, extending classical second order results.

Contribution

It introduces a comprehensive scattering framework for fourth order operators, including explicit formulas and a classification of zero-energy resonances, which is novel compared to existing second order theories.

Findings

Derived relations between scattering matrix and exponentially decaying modes.

Explicit formulas for scattering objects on the half-axis.

Classification of zero-energy resonances.

Abstract

An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the corresponding differential equation. In contrast to equations of second order "scattering" solutions contain exponentially decaying terms. A relation between the scattering matrix and a matrix of coefficients at exponentially decaying modes is found. In the second part of the paper the operator $D^4$ on the half-axis with different boundary conditions at the point zero is studied. Explicit formulas for basic objects of the scattering theory are found. In particular, a classification of different types of zero-energy resonances is given.