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The aim of this article is twofold. First we establish sharper lower bounds for the sums of eigenvalues of      ( −  Δ  ) 1 2 | D , (-\Delta )^{\frac {1}{2}}|_{D},       the Klein-Gordon operator restricted to a bounded domain      D ⊂  R d , D\subset {\mathbb R}^d,       than the bounds obtained in works by E. Harrell; S. Yıldırım Yolcu; and G. Wei, H. Sun, and L. Zeng. Then we study upper bounds for the sums of negative powers of the eigenvalues of      ( −  Δ  ) 1 2 | D . (-\Delta )^{\frac {1}{2}}|_{D}.

Refined bounds for the eigenvalues of the Klein-Gordon operator