Conditions are given to ensure the existence and finiteness of the number of eigenvalues below the essential spectrum of self-adjoint realizations H of the operator \((D^ 2+m^ 2)^{1/2}+q(x)\) in \(L^ 2({\mathbb{R}}^ n)\) \((D=-i\partial /\partial x\), \(m>0\) a constant, q real valued). If \(n=1\), the results are as follows: 1) Assume \(\int_{E}q(x)dx<0\) and q(x)\(\leq 0\) a.e. outside E for some measurable \(E\subset {\mathbb{R}}\). If the essential spectrum of H is contained in [m,\(\infty)\), then H has eigenvalues below m. 2) Let \(B=\int_{{\mathbb{R}}}\max \{-q(x),0\}dx<\infty\). Then there is a self-adjoint realization H which has in (-\(\infty,m)\) at most \(CB^ 2\) eigenvalues, where C is a constant independent of q.