Inverse spectral problems for the Sturm–Liouville operator on a d-star graph

Abstract

In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d 2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that(1) if the potential function q j (x) on a fixed edge e j is prescribed on the interval [ π 2 , π ], then the reciprocal of d of the spectrum suffices to determine q j (x) on the whole interval [0, π ];(2) the 2 over d of the spectrum suffices to determine q j (x) on a fixed edge e j .

Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with mixed boundary conditions

“…We derive the asymptotic formulas for the eigenvalues of this operator, and then solve the partial inverse problems which consist in constructing the potential on one edge by a part of the spectrum when the potentials are known on all the other edges. Some aspects of partial inverse problems on graphs were studied in [2,3,15,18,19,21]. Such problems generalize the Hochstadt-Lieberman problem on a finite interval [9].…”

Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for $${\varvec{L}}_\mathbf{1}$$ L 1 -potentials and an Ambartsumian Theorem

“…Therefore it seems attractive to start investigations assuming that the metric graph and the vertex conditions are fixed. The simplest graph to be considered is the star graph formed by a finite number of compact edges, and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [14,30,[33][34][35]. The case of general trees has also been studied and we have a rather good understanding of the problem [2,8,9,31,37].…”

Inverse problem for a quadratic pencil of Sturm–Liouville operator

“…In recent years, inverse spectral problems for Sturm-Liouville operator are mostly solved in Refs. [2,3,6,8,[13][14][15][16]18,19].…”