BOUND-STATES IN QUANTUM WAVE-GUIDES OF A SLOWLY DECAYING CURVATURE

Using two simple tricks, the results of our previous papers concerning bound states of Dirichlet Laplacians in curved tubes in R(nu), nu=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 nu=3) decays with respect to the arclength parameter s as Absolute value of s-1/2-epsilon. A sufficient condition is found and it is proved that it is necessary at the same time, provided the curvature (torsion) decays as Absolute value of s-1-epsilon and the tube is only mildly bent.