The Adiabatic Limit of the Connection Laplacian

• Published in: The Journal of geometric analysis Vol. 29; no. 3; pp. 2644 - 2673
• Main Authors: Haag, Stefan, Lampart, Jonas
• Format: Journal Article
• Language: English
• Published: New York Springer US 15-07-2019; Springer Nature B.V; Springer
• Subjects: Abstract Harmonic Analysis; Adiabatic flow; Boundary conditions; Convex and Discrete Geometry; Differential Geometry; Dirichlet problem; Dynamical Systems and Ergodic Theory; Eigenvalues; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematical Physics; Mathematics; Mathematics and Statistics; Operators; Spectral Theory; Bounded geometry; Adiabatic limit; Quantum waveguide; Laplacian on vector bundles
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-018-0087-2

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
ε
-
1
≫
1
. Under a gap condition on the fibrewise eigenvalues, we prove the 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
L
2
(
E
)
.