Inverse scattering problems on a noncompact star graph. (31st August 2018) - Xu, Xiao-Chuan; Yang, Chuan-Fu
Abstract: In this work, we study the Schrödinger operator on the noncompact star graph with p 1 infinite rays and p   −   p 1 line segments ( ). (1) In the case p   −   p 1   =  1, it was known that the scattering data uniquely recover all potentials. Here we show that for the unique determination of all potentials: (i) if potentials on m infinite rays super-exponentially decrease, then the corresponding mN normalization constants can be missing; (ii) if potentials on m infinite rays are known a priori, then the corresponding m refraction coefficients s jj ( k ) and mN normalization constants can be missing; (iii) if q j on some infinite ray is known a priori for for some a   >  0, then the corresponding j th normalization constants can be missing. (2) In the case, it is shown that the scattering data and p   −   p 1   −  1 additional reflection coefficients uniquely determine all potentials. Moreover, we provide the reconstruction algorithms for the corresponding uniqueness theorems.