On the Convergence of the Fractional Relativistic Schrodinger Operator

In this paper, we deal with the convergence of the fractional relativistic Schrodinger operator (-Δ+m2)sass→1-. Intuitively, this operator converges to (- Δ + m2) but the proof of this result is not so simple and it is based on a careful analysis of the involved modified Bessel function Kν . The convergence of the operator makes natural the following question: do the solutions of the problem (-Δ+m2)sw=f(w)inRN converge, in a suitable sense, to a solution u of the problem (-Δ+m2)v=f(v)inRNass→1-? We will show that the answer is affirmative under reasonable hypotheses. The proof of the convergence of solutions is obtained by combining an immersion result for Bessel potential spaces with the uniform convergence on compact sets. To the best of our knowledge, the analogous result for the fractional Laplacian operator (- Δ) s has not been established yet.