Inverse scattering for lasso graph

Abstract

The inverse problem for the magnetic Schrödinger operator on the lasso graph with different matching conditions at the vertex is investigated. It is proven that the Titchmarsh-Weyl function known for different values of the magnetic flux through the cycle determines the unique potential on the loop, provided the entries of the vertex scattering matrix S parametrizing matching conditions satisfy s12s23s31 ≠ s13s21s32. This is in contrast to numerous examples showing that the potential on the loop cannot be reconstructed from the boundary measurements.