We consider the three-dimensional Schrödinger operators H 0 and H ± where H 0 = (i∇+A) 2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H ± = H 0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski's representation of the spectral shift function (SSF) for the pair of operators H ± , H 0 is well defined for energies E = 2qb, q ∈ Z + . We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z + . Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Résumé. On considère les opérateurs de Schrödinger tridimensionnelsAlors, la représentation obtenue par A. Pushnitski de la fonction du décalage spectral pour les opérateurs H ± , H 0 est bien définie pour desénergies E = 2qb, q ∈ Z + . Onétudie le comportement du représentant associé de la classe d'équivalence déterminée par la fonction du décalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z + . En réduisant l'analyseà l'investigation de l'asymptotique des valeurs propres d'une famille d'opérateurs de Toeplitz compacts, onétablit une relation entre le type des singularités de la fonction du décalage spectral aux niveaux de Landau et la vitesse de la décroissance de Và l'infini.

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 niteness 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 rst time that commutator methods are applied in the context of rotation algebras, for the study of their generators.

Let U be a unitary operator defined on a infinite-dimensional separable complex Hilbert space H. Assume there exists a self-adjoint operator A on H such thatfor some positive constant c and compact operator K. Then, assuming the commutators U * AU − A and [A, U * AU ] admit a bounded extension over H, we prove the spectrum of the operator U has no singular continuous component and only a finite number of eigenvalues of finite multiplicity. We give a localized version of this result and apply it to study the spectrum of the Floquet operator of periodic time-dependent kicked quantum systems.

Left atrial wall haematoma is a very uncommon entity, associated mainly to cardiac surgery, interventional procedures, or trauma. Spontaneous cases are supposed to be associated with left atrial wall pathology. We present a case of a 53-year-old male who was admitted for prolonged chest pain, with transthoracic and transesophagic echocardiography documentation of a left atrial mass in close proximity to a mitral annular calcification. Tissue characterization with cardiac magnetic resonance suggested the aetiology of the mass, which was confirmed histologically.

Explicit lower bounds are given for the size of the imaginary parts of resonances for Schrόdinger operators with non-trapping or trapping potentials, and for the Dirichlet Laplacian in the exterior of a star-shaped obstacle, both acting in three dimensions,