In this paper, I describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. I describe also all the limits with minimal support (the scars).
We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on XZ. If A is a pseudo-differential operator on X of degree <3, we prove that B=P^* A
P is a pseudo-differential operator on Z and calculate the principal symbol of B.
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
r on an homogeneous tree. We developp this scattering theory using the classical recipes for Schrodinger operators in Euclidian spaces.
In this note, we present a natural proof of a recent and surprising result of Gregory Berkolaiko (arXiv 1110.5373) interpreting the Courant nodal defect of a Schrodinger operator on a finite graph as a Morse index associated to the deformations of th
e operator by switching on a magnetic field. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.
We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3).
in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schrodinger operator with magnetic field. We show how to recover their formula by using the method
s developped by the geometers in the seventies for the heat expansions.
In this paper we give an exact relation between the Greens function in a scattering problem for a wave equation and the correlation of scattered plane waves. This general relation was proved in a special case by Sanchez-Sesma and al.
