Semirelativistic Hamiltonians and the auxiliary field method

Approximate analytical closed energy formulas for semirelativistic Hamiltonians of the form $\sigma \sqrt{{\mathit{p}}^2+m^2}+V(r)$ are obtained within the framework of the auxiliary field method. This method, which is equivalent to the envelope theory, has been recently proposed as a powerful tool to get approximate analytical solutions of the Schr"odinger equation. Various shapes for the potential $V(r)$ are investigated: power-law, funnel, square root, and Yukawa. A comparison with the exact results is discussed in detail.