Bound states in quantum waveguides of a slowly decaying curvature

Abstract

Using two simple tricks, the results of our previous papers concerning bound states of Dirichlet Laplacians in curved tubes in Rν, ν=2, 3 are improved. If the tube is smooth and thin enough, the bound states are shown to exist if the tube curvature (and torsion for ν=3) decays with respect to the arclength parameter s as ‖s‖−1/2−ε. A sufficient condition is found and it is proved that it is necessary at the same time, provided the curvature (torsion) decays as ‖s‖−1−ε and the tube is only mildly bent.