Half-inverse problem for the Dirac operator

Inverse spectral problems for Dirac operators on a star graph with mixed boundary conditions

“…Horváth 12,13 further gave the uniqueness theorem of recovering the potential from finitely many spectra and part of the potential (on less than or more than half of the interval) and an Ambarzumian‐type theorem for Dirac operators. Recently, Yang and Liu 14 gave an reconstruction algorithm of recovering the potential as well as one boundary condition from one spectrum, half the potential and another boundary condition. However, limited work has been done in inverse problems for Dirac operators on graphs, which initially served as a model of a simple scattering system by Bulla and Trenkler 15 .…”

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

“…In particular, global solvability of Inverse Problem 1.4 yields the results of Martinyuk and Pivovarchik for the Hochstadt–Lieberman problem and global solvability for partial inverse problems on star‐shaped graphs (see Section ). One can also obtain necessary and sufficient conditions for solvability of the half‐inverse problem for the Dirac operator from Yang and Liu, by applying our approach to the Dirac system with arbitrary entire functions in the boundary condition. Our results on local solvability and stability generalize their analogues for the inverse transmission eigenvalue problem (see Bondarenko and Buterin) and improve the results of Bondarenko for the partial inverse problem on a graph.We also mention that Inverse Problem 1.3 can be treated as the problem of Horváth, which consists in recovering the potential q ( x ) from a set of eigenvalues corresponding to different boundary conditions However, as far as we know, Horváth and Kiss studied only uniqueness and stability issues (see Horváth).…”

“…In particular, global solvability of Inverse Problem 1.4 yields the results of Martinyuk and Pivovarchik for the Hochstadt–Lieberman problem and global solvability for partial inverse problems on star‐shaped graphs (see Section ). One can also obtain necessary and sufficient conditions for solvability of the half‐inverse problem for the Dirac operator from Yang and Liu, by applying our approach to the Dirac system with arbitrary entire functions in the boundary condition. Our results on local solvability and stability generalize their analogues for the inverse transmission eigenvalue problem (see Bondarenko and Buterin) and improve the results of Bondarenko for the partial inverse problem on a graph.…”