Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials

We study eigenvalues of the Dirac operator with canonical formL(p,q) ((u)(v)) = ((0)(-1)(1)(0))d/dt((u)(v))+((-p)(q) (q)(p)) ((u)(v)),where p and q are real functions. Under the assumption thatlim sup(x ->infinity) (x)root p(2)(x) + q(2)(x) < infinity,the essential spectrum of L-p,L-q is (-infinity, infinity). We prove that L-p,L-q has no eigen-values iflim sup(x ->infinity) (x)root Ip(2)(x) + q(2)(x) < 1/2.Given any A >= 1/2 and any lambda is an element of R, we construct functions p and q such that lim sup(x ->infinity)(x)root p(2)(x) + q(2)(x) = A and lambda is an eigenvalue of the corresponding Dirac operator L-p,L-q. We also construct functions p and q so that the corresponding Dirac operator L-p,L-q has any prescribed set (finitely or countably many) of eigenvalues.