Spectral asymptotics for the Schr\"odinger operator on the line with spreading and oscillating potentials

TL;DR

This paper analyzes the spectral behavior of multiscale Schr"odinger operators with oscillating and decaying potentials, using normal form techniques to simplify the problem across different scaling regimes.

Contribution

It introduces a method to reduce oscillating Schr"odinger operators to effective non-oscillating Hamiltonians through normal form transformations.

Findings

Spectral asymptotics depend on the scaling regime.

Normal form transformation effectively simplifies oscillating potentials.

Results provide insights into the spectral properties of multiscale Schr"odinger operators.

Abstract

This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal form filtrating the oscillations, a reduction to a non-oscillating effective Hamiltonian is performed.