Virtual levels, virtual states, and the limiting absorption principle for higher order differential operators in 1D

TL;DR

This paper investigates the resolvent estimates and virtual states of higher order differential operators in one dimension, extending the analysis of the limiting absorption principle to third-order and higher operators using Jost solutions.

Contribution

It introduces a method for analyzing virtual states and resolvent estimates for higher order 1D differential operators, generalizing previous second-order results.

Findings

Established resolvent estimates for third-order operators

Characterized virtual states for higher order differential operators

Extended the limiting absorption principle to higher order cases

Abstract

We consider the resolvent estimates and properties of virtual states of the higher order derivatives in one dimension, focusing on Schroedinger-type operators of degree $N=3$ (the approach applies to higher orders). The derivation is based on the construction of the Jost solution for higher order differential operators and on restricting the resolvent onto subspaces of finite codimension.