ﻻ يوجد ملخص باللغة العربية
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 operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $phi - (n-1)$.
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 fro
An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graphs non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graphs first Betti number $beta$. We study the
We prove that after an arbitrarily small adjustment of edge lengths, the spectrum of a compact quantum graph with $delta$-type vertex conditions can be simple. We also show that the eigenfunctions, with the exception of those living entirely on a loo
The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e., conditions under which it holds sup_{t in R} | (psi(t),H(t) psi(t)) |<infty w
We describe some basic tools in the spectral theory of Schrodinger operator on metric graphs (also known as quantum graph) by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In the later sectio