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We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrodinger operator with an attractive $delta$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $phi$ of the $n$-th eigenfunction of the Schrodinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the
We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can b
The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its as
We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operato