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.

No abstract

We consider an inverse spectral problem for Sturm-Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M -matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M -matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M -matrix. This enables us to recover, from the M -matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

This paper generalises the work done in Currie and Love 2010 , where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affine λ-dependent boundary conditions at the end points, where λ is the eigenparameter. We now consider general λ-dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the secondorder difference equation constant but possibly increasing or decreasing the dependence on λ of the boundary conditions at each step. In addition, we show that the transformed boundary value problem either gains or loses an eigenvalue, or the number of eigenvalues remains the same as we step up or down the hierarchy.

We consider a system formulation for Sturm-Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm-Liouville problem on a graph from its M-matrix.