Nodal sets of thin curved layers

Abstract

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the neighbourhood tends to zero, it is known that spectral properties of the Laplacian are approximated well by an effective Schrödinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures. By applying techniques of elliptic partial differential equations, we strengthen the known perturbation results to get a convergence of eigenfunctions in Hölder spaces. This enables us in particular to conclude that every nodal set has a nonempty intersection with the boundary of the tubular neighbourhood.

Convergence of nodal sets in the adiabatic limit

“…is of order ε 2 . This is also true for typical examples with ∂M = 0, such as those discussed in [4,8,12], where actually ∆ F log ρ = O(ε 2 ) holds point-wise. All of our results will be valid also in more general situations if the following condition holds.…”

“…This unfortunately gives no meaningful estimate if dim B > 3. In the recent article [12] the authors were able to obtain some results without restriction on the base dimension. Their technique involves proving convergence ϕ → π * ψφ 0 in Sobolev spaces of arbitrary order.…”

A lower bound to the spectral threshold in curved quantum layers

“…for any fixed u ∈ (−a, a) and all x ∈ Σ. The last inequality can be established for non-zero u's by writing (15) and follows more easily for u = 0.…”

Location of hot spots in thin curved strips

“…Since the extremal points of ψ 0 n are explicitly known, convergence (14) is sufficient to conclude with a weaker version of (4) and (5), namely with just closures of S (n) 0 (δ) and S (n) n−1 (δ) instead of the more restricted setsS (n) 0 andS (n) n−1 on the right-hand side of the formulae. Here the main idea is to locate the stationary points with help of (14). To exclude the possibility of existence of extremal points of ψ ε n in the boundary narrow regions S (n) 0 (δ) and S (n) n−1 (δ) as well (which is particularly crucial for the proof of the hot spots conjecture), we still try to continue with the ideas of elliptic regularity theory as above, but now we are only able to show the bound (15) ψ ε n − ψ 0 n C 2,γ (Π) ≤ C γ with any γ ∈ (0, 1), where C γ is an ε-independent constant (possibly different from that of (14)).…”

“…Remark 1.5. Since this paper is primarily motivated by the hot spots conjecture, we focus on the corresponding consequences of (14) and (15) summarised in Theorem 1.2. Notice, however, that the established convergence results for eigenfunctions can be interested in different contexts, too.…”