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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 sections we apply these tools to prove some results on the count of zeros of the eigenfunctions of quantum graphs.
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
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the
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
Given two Hilbert spaces, $mathcal{H}$ and $mathcal{K}$, we introduce an abstract unitary operator $U$ on $mathcal{H}$ and its discriminant $T$ on $mathcal{K}$ induced by a coisometry from $mathcal{H}$ to $mathcal{K}$ and a unitary involution on $mat
We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function argument with hel