Serge Richard

Spectral and scattering theory for the Aharonov-Bohm operators

We review the spectral and the scattering theory for the Aharonov–Bohm model on ℝ2. New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high energy and at low energy of the scattering operator are computed.

Levinson's theorem for Schrödinger operators with point interaction: a topological approach

In this note Levinson theorems for Schrodinger operators in R n with one point interaction at 0 are derived using the concept of winding numbers. These results are based on new expressions for the associated wave operators.

Some Improvements in the Method of the Weakly Conjugate Operator

We present some improvements in the method of the weakly conjugate operator, one variant of the Mourre theory. When applied to certain two-body Schrödinger operators, this leads to a limiting absorption principle that is uniform on the positive real axis.

Commutator methods for unitary operators

We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generators.

Spectral Analysis for Adjacency Operators on Graphs

Abstract.We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there is at most an eigenvalue located at the origin. Among other examples, the one-dimensional XY model of solid-state physics is covered. The proofs rely on commutators methods.

ON THE WAVE OPERATORS AND LEVINSON'S THEOREM FOR POTENTIAL SCATTERING IN ℝ3

The paper is a presentation of recent investigations on potential scattering in ℝ3. We advocate a new formula for the wave operators and deduce the various outcomes that follow from this formula. A topological version of Levinsons theorem is proposed by interpreting it as an index theorem.

Weber–Schafheitlin-type integrals with exponent 1

Explicit formulae for Weber–Schafheitlin-type integrals with exponent 1 are derived. The results of these integrals are distributions on ℝ+.

The topological meaning of Levinson's theorem, half-bound states included

We propose to interpret Levinsons theorem as an index theorem. This exhibits its topological nature. It furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds.

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

The paper is devoted to operators given formally by the expression