Inverse nodal problems for Sturm–Liouville equations on graphs

Abstract

We consider inverse nodal problems on graphs. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph. Based on this, the uniqueness of the potential for given nodal data is proved and we give a construction of q as a limit, in , of a sequence of functions whose nth term is dependent only on the nth eigenvalue and its associated nodal data.

Inverse nodal problem for a Sturm-Liouville operator with discontinuous coefficient

“…From then on, their results have been generalized to various problems. Inverse nodal problems for Sturm-Liouville operators without discontinuities have been studied in the several papers ( [8], [10], [12], [13], [14], [19], [21], [22] and [24]). The …rst result on inverse nodal problems for the Sturm-Liouville operators with a discontinuity condition was obtained by Shieh and Yurko [20].…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…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. Recently, some authors have reconstructed the potential function for generalizations of the Sturm-Liouville problem from the nodal points (for example, refer to [3,5,7,10,11,14,19,24,25,26,30,33,34,35,37,41,42]). …”

“…Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [20,21,22,31,39] and inverse [4,23,32,38,40,41,42], etc. Nowadays there are only a number of papers devoted to inverse nodal problems for differential operators on graphs (for example, refer to [8,11,41,42]). …”

Reconstruction of the Dirac Operator From Nodal Data

“…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. Recently, some authors have reconstructed the potential function for generalizations of the Sturm-Liouville problem from the nodal points (for example, refer to [1][2][3][4][5][7][8][9][10][11]15,[17][18][19]21,24]). …”