Inverse scattering problems on a noncompact star graph. (31st August 2018)

• Record Type: Journal Article
• Authors: Xu, Xiao-Chuan; Yang, Chuan-Fu
• Is Part Of: Inverse problems. Volume 34:Number 11(2018:Nov.)
• Publication Date: 2018-08-31
• Subjects: Schrodinger operator -- inverse scattering problem -- quantum graph -- mixed data -- reconstruction algorithm; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aadb1f
• Languages: English
• ISSNs: 0266-5611

Abstract

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.