Convergence of nodal sets in the adiabatic limit

Abstract

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $\pi{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the base is given by $\varepsilon \ll 1$. We assume $M$ to be compact and allow for fibres $F$ with boundary, under the condition that the ground state eigenvalue of the Dirichlet-Laplacian on $F_x$ is independent of the base point. We prove for $\mathrm{dim} B \leq 3$ that the nodal set of the Dirichlet-eigenfunction $\varphi$ converges to the pre-image under $\pi$ of the nodal set of a function $\psi$ on $B$ that is determined as the solution to an effective equation. In particular this implies that the nodal set meets the boundary for $\varepsilon$ small enough and shows that many known results on this question, obtained for some types of domains, also hold on a large class of manifolds with boundary. For the special case of a closed manifold $M$ fibred over the circle $B=S^1$ we obtain finer estimates and prove that every connected component of the nodal set of $\varphi$ is smoothly isotopic to the typical fibre of $\pi{:}\, M\to S^1$.Comment: revised version, Annals of Global Analysis and Geometry, 201

The adiabatic limit of Schrödinger operators on fibre bundles

“…In Section 4.2 we will also derive some results on the approximation of eigenfunctions. These are relevant for the application to nodal sets [40] where it is shown that, for certain low lying eigenvalues, the behaviour of the nodal set is essentially determined by the nodal set of the eigenfunction of H a with the corresponding eigenvalue. A large portion of the literature on the adiabatic limit of Schrödinger operators is concerned with quantum waveguides.…”

“…Here we will only consider connected fibres and an operator H 1 of a special form, that is relevant to the applications in [27] and [40]. By small energies we mean energies whose distance to…”

Nodal sets of thin curved layers

“…[12]). The present work and [25] are also independent from the point of view of the technical handling of the limit ε → 0. While Lampart starts with an adiabatic perturbation theory developed for fibre bundles in [26] and finally ends up with H 1 -estimates, we rather rely on the norm-resolvent convergence established in [24] and proceed to higher-order Sobolev spaces.…”

“…When finishing this paper a related work of Lampart [25] appeared. On the one hand, it is concerned with the much more general geometrical setting of the Laplacian on a diminishing fibre bundle of arbitrary dimension.…”

“…On the one hand, it is concerned with the much more general geometrical setting of the Laplacian on a diminishing fibre bundle of arbitrary dimension. On the other hand, the base space of the fibre is assumed to be compact and of dimension at most 3 in [25], which would correspond to the restriction to compact Σ and d ≤ 4 in our setting. It is remarkable that we do not need these constraints to prove Conjecture 1, but our fibre is always one-dimensional (although there seem not to be any technical limitations with extending our methods to higher-dimensional cross-sections, cf.…”