The adiabatic limit of Schrödinger operators on fibre bundles

Abstract

We consider Schrödinger operators H = −∆ gε + V on a fibre bundle M π → B with compact fibres and a metric g ε that blows up directions perpendicular to the fibres by a factor ε −1 1. We show that for an eigenvalue λ of the fibre-wise part of H, satisfying a local gap condition, and every N ∈ N there exists a subspace of L 2 (M ) that is invariant under H up to errors of order ε 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 E → B, giving detailed asymptotics for H.

The Adiabatic Limit of the Connection Laplacian

“…Starting from this point, we will improve the approximation and find projections P ε such that P ε H is invariant under H up to errors of order ε N (for any given N ∈ N). This generalises the work of the second author with Teufel [LT17], where such an approximation was derived for the scalar case E = M × C, ∇ E = d. The generalisation of these results to vector bundles requires an in-depth discussion of the analytical setup. We discuss the structure of the fibre bundle E Π E − − → B, whose fibre E x is the vector bundle E| Mx ∼ = F, in Section 2 and show that it inherits a specific form of bounded geometry from E → M and M → B in Proposition 2.8.…”

“…B.7]). Via the identification H ∼ = L 2 (H F ) (see [Lam14,Corollary B.6]), the operator P 0 defines a bounded operator on H, whose image P 0 H is isomorphic to L 2 (P), the L 2 -sections of the finite-rank vector bundle π P : P → B.…”

“…These corollaries depend only on the general structure of Theorem 1.1 and not on the details of the problem, such as the choice of H = L 2 (E). We thus refer to [LT17] for complete proofs and focus here on the construction of P ε .…”

Ruled Strips with Asymptotically Diverging Twisting

“…Analogous results are also known for tubes embedded in a Riemannian manifold A instead of the Euclidean space: see [11,12,19,10,8,18]. In the simplest non-trivial situation where Σ is a curve in a two-dimensional surface A and the cross-section ω is a symmetric interval, any non-trivial curvature of Σ and/or non-negative Gauss curvature of A, both vanishing at infinity in an appropriate sense, lead to the existence of discrete spectra (see [11]), while Hardy-type inequalities hold if Σ is a geodesic and A is non-positively curved (see [12,10]).…”

“…with the function f ∞ defined in (18). Notice that the one-dimensional operator −|θ ′ (s)| −1 ∂ s |θ ′ (s)| −1 ∂ s in L 2 R, |θ ′ (s)| ds , understood as the Friedrichs extension of the operator initially defined on the domain C ∞ 0 (R), is unitarily equivalent to the standard Laplacian −∂ 2 u in L 2 (R) with the usual domain W 2,2 (R); indeed, the unitarily equivalence is accomplished by the change of variables u = s 0 |θ ′ (σ)| dσ.…”