Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result

摘要

In a finite-dimensional Euclidian space we consider a connected metric graphwith the following property: each two cycles can have at most one common point.Such graphs are called A-graphs. On noncompact A-graph we consider a scatteringproblem for Sturm-Liouville differential operator with standard matchingconditions in the internal vertices. Transport, spectral and scatteringproblems for differential operators on graphs appear frequently in mathematics,natural sciences and engineering. In particular, direct and inverse problemsfor such operators are used to construct and study models in mechanics,nano-electronics, quantum computing and waveguides. The most complete resultson (both direct and inverse) spectral problems were achieved in the case ofSturm-Liouville operators on compact graphs, in the noncompact case there areno similar general results. In this paper, we establish some properties of thespectral characteristics and investigate the inverse problem of recovering theoperator from the scattering data. A uniqueness theorem for such inverseproblem is proved.