Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction
type: Item
instance of: scholarly article
title: Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction
zbMATH Open document ID: 1181.81054
DOI: 10.1134/S1061920808030063
published in: Russian Journal of Mathematical Physics
publication date: 2009-11-10T00:00:00Z
review text: \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.
reviewed by: Miyeon Kwon
Mathematics Subject Classification ID: 81Q35
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47N50
zbMATH DE Number: 5628036
zbMATH Keywords: Neumann Schrödinger operator
zbMATH Keywords: Krein formula
zbMATH Keywords: Kirchhoff condition
author: Boris S. Pavlov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5628036