Inverse problems for Sturm – Liouville operator. With potential functions from L2[0,π]

This paper deals with non-self-adjoint second-order differential operatorswith two constant delays. We consider four boundary value problems 𝐷𝑖,𝑘,𝑖=0,1,𝑘=1,2−𝑦′′(𝑥)+𝑞1(𝑥)𝑦(𝑥−𝜏1)+(−1)𝑖𝑞2(𝑥)𝑦(𝑥−𝜏2)=𝜆𝑦(𝑥),𝑥∈[0,𝜋]𝑦′(0)−ℎ𝑦(0)=0, 𝑦′(𝜋)+𝐻𝑘𝑦(𝜋)=0,where𝜋3≤𝜏2<𝜋2≤2𝜏2≤𝜏1<𝜋, ℎ,𝐻1,𝐻2∈ 𝑅\{0} and 𝜆 is a spectral parameter. We assumethat 𝑞1,𝑞2are real-valued potential functions from𝐿2[0,𝜋]such that 𝑞1(𝑥)=0,𝑥∈[0,𝜏1)and𝑞2(𝑥)=0,𝑥∈[0,𝜏2).The inverse spectral problem of recovering operators from their spectra hasbeen studied. We provethat delays 𝜏1,𝜏2and parameters ℎ,𝐻1,𝐻2are uniquely determined from thespectra. Then we prove that potentials are uniquely determined by Volterra linear integral equations.