  The spinless Salpeter equation may be considered either as a standard
approximation to the Bethe--Salpeter formalism, designed for the description of
bound states within a relativistic quantum field theory, or as the most simple,
to a certain extent relativistic generalization of the costumary
nonrelativistic Schr\"odinger formalism. Because of the presence of the rather
difficult-to-handle square-root operator of the relativistic kinetic energy in
the corresponding Hamiltonian, very frequently the corresponding (discrete)
spectrum of energy eigenvalues cannot be determined analytically. Therefore, we
show how to calculate, by some clever choice of basis vectors in the Hilbert
space of solutions, for the rather large class of power-law potentials, at
least (sometimes excellent!) upper bounds on these energy eigenvalues, for the
lowest-lying levels this even analytically.
