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arXiv · 1402.0382

The adiabatic limit of Schrödinger operators on fibre bundles

Jonas Lampart·Stefan Teufel
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Abstract

We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ 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 $λ$ 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 $λ$-eigenspace bundle $\mathcal{E}\to B$, giving detailed asymptotics for $H$.

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Jonas Lampart, Stefan Teufel. 2016-03-22. The adiabatic limit of Schrödinger operators on fibre bundles. https://doi.org/10.1007/s00208-016-1421-2

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