Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction
\textit{P. Kuchment} and \textit{H. Zeng} showed in [J. Math. Anal. Appl. 258, No. 2, 671--700 (2001; Zbl 0982.35076)], using the probabilistic and variational approaches, that the essential spectral properties of the Neumann Laplacian on the 2D junction with shrinking leads are inherited by the limiting 1D Laplacian on the corresponding quantum graph with Kirchhoff-type boundary conditions at the vertex. This paper presents similar results for spectral properties of the Neumann Schrödinger operator on a thin 2D star-shaped junction with semi-infinite straight leads. First it drives an approximate formula for the scattering matrix of the junction in terms of the Neumann-to-Dirichlet mapping and then gives an observation on the compensation of singularities inherited from the unperturbed Hamiltonian in the corresponding Krein formula. The compensation of singularities is used to construct the corresponding solvable models of the junction in the form of star-shaped 1D quantum graphs and to drive the boundary conditions at the vertex. Lastly the Neumann Schrödinger results are compared with the previous results by Kuchment and Zeng.
scientific article; zbMATH DE number 5628036
scientific article