# Spectrum of the Iwatsuka Hamiltonian at thresholds

## TL;DR

This paper analyzes the spectral properties of the Iwatsuka Hamiltonian, focusing on the behavior of band functions at thresholds, and examines the effects of perturbations on the spectral density and shift function.

## Contribution

It provides asymptotic analysis of band functions and derivatives, estimates on current and localization near thresholds, and studies the spectral shift function under perturbations.

## Key findings

- Asymptotic behavior of band functions and derivatives at infinity.
- Estimates on current and localization of states near spectral thresholds.
- Asymptotic behavior of the spectral shift function at thresholds.

## Abstract

We consider the bi-dimensional Schr\"odinger operator with unidirectionally constant magnetic field, $H_0$, sometimes known as the "Iwatsuka Hamiltonian". This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given \emph{threshold} in the spectrum of $H_0$. In addition, for a non-negative electric perturbation $V$ we study the spectral density of $H_0\pm V$ by considering the Spectral Shift Function associated to the operator pair $(H_0\pm V,H_0)$. We describe the continuity and boundedness properties of the spectral shift function, and we compute the asymptotic behavior at the thresholds, which are the only points where it can grows to infinity.