Mathematische Annalen

, Volume 367, Issue 3–4, pp 1647–1683 | Cite as

The adiabatic limit of Schrödinger operators on fibre bundles

Article

Abstract

We consider Schrödinger operators \(H=-\Delta _{g_\varepsilon } + V\) on a fibre bundle \(M\mathop {\rightarrow }\limits ^{\pi }B\) 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 \(\mathcal {E}\rightarrow B\), giving detailed asymptotics for H.

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.PSL Research University and CEREMADE (UMR CNRS 7534)Université de Paris-DauphineParis Cedex 16France
  2. 2.Fachbereich MathematikUniversität TübingenTübingenGermany

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