The adiabatic limit of Schrödinger operators on fibre bundles

Abstract

We consider Schrödinger operators $H=-{\Delta }_{g_{\epsilon }}+V$ on a fibre bundle $M\stackrel{\pi }{\to }B$ with compact fibres and a metric $g_{\epsilon }$ that blows up directions perpendicular to the fibres by a factor ${\epsilon }^{-1}\gg 1$. 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 ${\epsilon }^{N+1}$. 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 $H$.

References in corpus (3)

• Generalised Quantum Waveguides
• On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes
• Convergence of nodal sets in the adiabatic limit

Cited by in corpus (3)

• Spectral properties of relativistic quantum waveguides
• Ruled strips with asymptotically diverging twisting
• Quantum waveguides with magnetic fields