  A spectral analysis is done on the $L$ operator of the Lax pair for the
Benjamin-Ono equation. Simplicity and finiteness of the discrete spectrum are
established as are needed for the Fokas and Ablowitz inverse scattering
transform scheme. A crucial step in the simplicity proof is the discovery of a
new identity connecting the $L^2$ norm of the eigenvector to its inner product
with the scattering potential. The proof for finiteness is an extension of the
ideas involved in the Birman-Schwinger bound for Schr\"odinger operators.
