We derive a generalization of the virial theorem in terms of the canonically conjugate pair of variables. Then, we apply it to the Salpeter equation and to the reductions of the Salpeter equation. It is shown that the linear mass form and the quadratic mass form of the reductions of the Salpeter equation will be the same in the nonrelativistic limit but different in the ultrarelativistic limit. Therefore, different reductions are appropriate for different bound systems.

In this paper, we discuss the bound-state problem for the spinless Salpeter equation with the Yukawa potential. Due to the nonlocal term of the Hamiltonian encountered, we use the eigenfunction for the ground state of the hydrogen atom as a trial function and employ the variational method to solve the spinless Salpeter equation. We derive the upper bounds on the eigenvalues to obtain the bound state inequality. The constraint on the interaction strength α is given, (2.42m–1.32μ)μ/(2.37m2–mμ) ≤ α < 8/(3π). And the maximum of the screening parameter of the Yukawa potential μ is obtained, μmax = 1.14 m.