ESTIMATES OF PERIODIC POTENTIALS IN TERMS OF GAP LENGTHS

Define the Hill operator T = -d(2)/dx(2) + q(x) in L-2(R) and suppose q is an element of L-2(0, 1) is a 1-periodic real potential, integral( 0)(1) q(x)dx = 0. We prove the estimate parallel to q parallel to less than or equal to 2 parallel to gamma parallel to(1 + parallel to gamm a parallel to(1/3)), where parallel to gamma parallel to(2) = Sigma(n greater than or equal to 1) \gamma\ greater than or equal to 0, n grea ter than or equal to 1, is the gap length of T.