The adiabatic limit of the connection Laplacian

Stefan Haag (1) , Jonas Lampart (2, 3)

1 Fachbereich Mathematik, Eberhard Karls Universität Tübingen
2 ICB - Laboratoire Interdisciplinaire Carnot de Bourgogne
3 CNRS - Centre National de la Recherche Scientifique

We study the behaviour of Laplace-type operators H on a complex vector bundle E → M in the adiabatic limit of the base space. This space is a fibre bundle M → B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/ε. Under a gap condition on the fibre-wise eigenvalues we prove existence of effective operators that provide asymptotics to any order in ε for H (with Dirichlet boundary conditions), on an appropriate almost-invariant subspace of L²(E).

Mots clés: Bounded geometry; Quantum waveguide; Adiabatic limit; Laplacian on vector bundles

Domaines: Physique mathématique [math-ph]; Géométrie différentielle [math.DG]; Théorie spectrale [math.SP]

Stefan Haag, Jonas Lampart. The adiabatic limit of the connection Laplacian. The Journal of Geometric Analysis, 2018, 29 (3), ⟨10.1007/s12220-018-0087-2⟩. ⟨hal-01527928v2⟩