Francoise Truc

Essential self-adjointness for combinatorial Schrodinger operators II- Metrically non complete graphs

We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.

Remarks on the spectrum of the Neumann problem with magnetic field in the half-space

We consider a Schrodinger operator with a constant magnetic field in a one-half three-dimensional space, with Neumann-type boundary conditions. It is known from the works by Lu–Pan and Helffer–Morame that the lower bound of its spectrum is less than b, the intensity of the magnetic field, provided that the magnetic field is not normal to the boundary. We prove that the spectrum under b is a finite set of eigenvalues (each of infinite multiplicity). In the case when the angle between the magnetic field and the boundary is small, we give a sharp asymptotic expansion of the number of these eigenvalues.

Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators

Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.

Scattering theory for graphs isomorphic to a regular tree at infinity

We describe the spectral theory of the adjacency operator of a graph which is isomorphic to a regular tree at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on a regular tree. We develop this scattering theory using the classical recipes for Schrodinger operators in Euclidian spaces.

Topological Resonances on Quantum Graphs

In this paper, we try to put the results of Smilansky et al. on “Topological resonances” on a mathematical basis. A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exist compactly supported eigenfunctions. We give several estimates on the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case of trees is also discussed.

Eigenvalue bounds for radial magnetic bottles on the disk

We consider a Schrodinger operator H with a non-vanishing radial magnetic field B=dA and Dirichlet boundary conditions on the unit disk. We assume growth conditions on B near the boundary which guarantee in particular the compactness of the resolvent of this operator. Under some assumptions on an additional radial potential V the operator H + V has a discrete negative spectrum and we obtain an upper bound on the number of negative eigenvalues. As a consequence we get an upperbound of the number of eigenvalues of H smaller than any positive value, which involves the minimum of B and the square of the L^2 -norm of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of the magnetic field through the disk of radius r centerde in the origin.

Magnetic bottles on geometrically finite hyperbolic surfaces

We consider a magnetic Laplacian on a geometrically finite hyperbolic surface, when the corresponding magnetic field is infinite at the boundary at infinity. We prove that the counting function of the eigenvalues has a particular asymptotic behaviour when the surface has an infinite area.

THE MAGNETIC LAPLACIAN ACTING ON DISCRETE CUSPS

We study several toy-models of cups-like weighted graphs. We prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.

Eigenvalue Asymptotics for Magnetic Fields and Degenerate Potentials

We present various asymptotic estimates of the counting function of eigenvalues for Schrodinger operators in the case where the Weyl formula does not apply. The situations treated seem to establish a similarity between magnetic bottles (magnetic fields growing at infinity) and degenerate potentials, and this impression is reinforced by an explicit study in classical mechanics, where the classical Hamiltonian induced by an axially symmetric magnetic bottle can be seen as a perturbation of the Hamiltonian derived from an operator with a degenerate potential.