Reconstruction of the Dirac Operator From Nodal Data

Abstract

Inverse nodal problems consist in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, we deal with the inverse nodal problem of reconstructing the Dirac operator on a finite interval. We prove that a dense subset of nodal points uniquely determine the parameters of the boundary conditions, the mass of a particle and the potential function of the Dirac system. We also provide a constructive procedure for the solution of the inverse nodal problem. Mathematics Subject Classification (2000). Primary 34A55; Secondary 34L05, 34L40, 45C05.

Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter

“…However, it is outside the scope of this paper, the interested reader may consult [3,12,24] for recent developments. We note that the obtained results are natural generalizations of the well-known results on inverse nodal problems for the Dirac operator which were studied in [27,28].…”

“…Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see [1,2,15,[20][21][22]29] and the references therein). The inverse nodal problem, first posed and solved by McLaughlin [13,23], is the problem of constructing operators from given nodes (zeros) of their eigenfunctions (refer to [3][4][5]12,14,17,24,[26][27][28]). From the physical point of view this corresponds to finding, e.g., the density of a string or a beam from the zero-amplitude positions of their eigenvibrations.…”

Inverse Nodal Problems for Dirac-Type Integro-Differential System with Boundary Conditions Polynomially Dependent on the Spectral Parameter

“…Inverse nodal problems have been addressed by various researchers in several papers for different operators [4], [5], [6], [7], [8], [9], [10], [11] and [12]. The inverse nodal problems for Dirac operators with various boundary conditions have been studied and shown that the dense subsets of nodal points which are the first components of the eigenfunctions determines the coefficients of discussed operator by Yang C-F, Huang Z-Y [13]; Yang C-F, Pivovarchik VN [14] and Guo Y, Wei Y [15].…”

Inverse nodal problems for Dirac-type integro-differential operators

“…Inverse nodal problems for Sturm-Liouville or diffusion operators have been studied in the several papers ( [1], [2], [3], [6], [14], [15], [16], [19] and [20]). The inverse nodal problems for Dirac operators with various boundary conditions have been solved in [8], [18] and [21]. In their works, it was shown that the zeros of the first components of the eigenfunctions determines the coefficients of operator.…”