Location of the nodal set for thin curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are in this limit approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrödinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigenvalues, the uniform convergence of eigenfunctions and locate the nodal set of the Dirichlet Laplacian in the tube near nodal points of the one-dimensional Schrödinger operator. As a consequence, we prove the “nodal-line conjecture” for a class of non-convex and possibly multiply connected domains. The results are based on a perturbation theory developed for Schrödinger-type operators in a straight tube of diminishing cross-section. This also enables us to obtain similar results in the case where the cross-section is allowed to vary along the reference curve, provided we impose certain restrictions on the deviation from the constant cross-section case. MSC2000: Primary 35J25; Secondary 35B25, 35B38.