Let V {element_of} L{sub c}{sup {infinity}} (R{sup n}) be a real-valued potential, n {ge} 3 odd. The scattering matrix, Sv(A), corresponding to V extends to be a meromorphic function in C. Our normalization is that P, the physical half plane, is the open lower half plane, C{sup -}. Thus, S{sub v}({lambda}) only has a finite number of poles in P and they correspond to the bound states of the Hamiltonian {Delta} + V ({Delta} is the positive Laplacian). In Zworski has shown that the number of scattering poles n(r) in a disc of radius r is bounded by C(r + 1){sup n}, and in he has given similar lower bounds for n(r) for certain radial potentials. The question of the existence of pure imaginary scattering poles was investigated by Lax and Phillips for both obstacle and potential scattering. In the case of obstacle scattering they showed that the number N(s) of pure imaginary scattering poles of absolute value less than s satisfies C{sub 1} s{sup 2} 0, and there is an analogous upper bound if the obstacle is star-shaped. They also proved that much of the machinery can bemore » translated into the setting of potential scattering by non-negative potentials. This reduces the problem of obtaining lower bounds for N(s) to finding the corresponding bounds when V is tile characteristic function of a ball. In this paper we make the simple observation that their proof can be modified to accommodate non-positive potentials, and we prove that if V or -V is bounded below by a positive multiple of the characteristic function of a ball, then for some C, C{prime} > 0.« less