  This paper is concerned with the discrete spectra of Schroedinger operators H
= -Delta + V, where V(r) is an attractive potential in N spatial dimensions.
Two principal results are reported for the bottom of the spectrum of H in each
angular-momentum subspace H_{ell}: (i) an optimized lower bound when the
potential is a sum of terms V(r) = V^{(1)}(r) + V^{(2)}(r), and the bottoms of
the spectra of -Delta + V^{(1)}(r) and -Delta + V^{(2)}(r) in H_{ell} are
known, and (ii) a generalized comparison theorem which predicts spectral
ordering when the graphs of the comparison potentials V^{(1)}(r) and V^{(2)}(r)
intersect in a controlled way. Pure power-law potentials are studied and an
application of the results to the Coulomb-plus-linear potential V(r) = -a/r +
br is presented in detail: for this problem an earlier formula for energy
bounds is sharpened and generalized to N dimensions.
