On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder

Abstract

Consider the equation − Δ u + V u = 0 -\Delta u + Vu = 0 in the half-cylinder [ 0 , ∞ ) × ( 0 , 2 π ) d [0, \infty ) \times (0,2\pi )^d with periodic boundary conditions. Assume that the potential V V is bounded. The possible rate of decay at infinity for a nontrivial solution is studied. It is shown that the fastest rate of decay is e − c x e^{-cx} for d = 1 d=1 or 2 2 and e − c x 4 / 3 e^{-cx^{4/3}} for d ≥ 3 d\ge 3 ; here x x is the axial variable.