The adiabatic limit of Schr\"odinger operators on fibre bundles

TL;DR

This paper analyzes the behavior of Schr"odinger operators on fibre bundles with collapsing metrics, showing how their spectral and dynamical properties can be approximated by effective operators on the fibres as the metric degenerates.

Contribution

It introduces a method to approximate the spectral and dynamical features of Schr"odinger operators on fibre bundles in the adiabatic limit using effective operators on eigenspaces.

Findings

Existence of invariant subspaces approximating the spectrum up to order ^{N+1}

Effective operators describe spectral and dynamical features in the adiabatic limit

Asymptotic expansions for Schr"odinger operators on fibre bundles

Abstract

We consider Schr\"odinger operators $H=-\Delta_{g_\varepsilon} + V$ on a fibre bundle $M\stackrel{\pi}{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $\lambda$ of the fibre-wise part of $H$, satisfying a local gap condition, and every $N\in \mathbb{N}$ there exists a subspace of $L^2(M)$ that is invariant under $H$ up to errors of order $\varepsilon^{N+1}$. The dynamical and spectral features of $H$ on this subspace can be described by an effective operator on the fibre-wise $\lambda$-eigenspace bundle $\mathcal{E}\to B$, giving detailed asymptotics for $H$.