Inverse problems for Dirac operators with constant delay: uniqueness, characterization, uniform stability

Abstract

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $L_2$-potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary and sufficient for the unique determination of the potentials. Moreover, necessary and sufficient conditions for the solvability of both inverse problems are obtained. For the inverse problem involving the complete spectra, we establish also uniform stability in each ball. For this purpose, we use recent results on uniform stability of sine-type functions with asymptotically separated zeros.