Stability of resonances for the Dirac operator

The Dirac operator on the semi-axis with a compactly supported potential is investigated. Let ( k n ) n ≥ 1 (k_n)_{n\geq 1} be the sequence of its resonances, taken with multiplicities and ordered so that | k n | |k_n| do not decrease as n n grows. It is proved that for any sequence ( r n ) n ≥ 1 ∈ ℓ 1 (r_n)_{n\geq 1} \in \ell ^1 such that the points k n + r n k_n + r_n remain in the lower half-plane for all n ≥ 1 n\geq 1 , the sequence ( k n + r n ) n ≥ 1 (k_n + r_n)_{n\geq 1} is also a sequence of resonances of a similar operator. Moreover, it is shown that the potential of the Dirac operator changes continuously under such perturbations.