Reviews in Mathematical Physics

Using the whurl relation of the first two authors, we define a new discrete solitonic system, which we call the box-basket-ball system, generalizing the box-ball system of Takahashi and Satsuma. In box-basket-ball systems, balls may be put either into boxes or into baskets. While boxes stay fixed, both balls and baskets get moved during time evolution. Balls and baskets behave as fermionic and bosonic particles, respectively. We classify the solitons of this system, and study their scattering.

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

We present new results on the block-diagonalization of operators with spectral gaps, based on a method of Langer and Tretter, and apply them to Dirac operators on three-dimensional Euclidean space with unbounded potentials. For the Coulomb potential, we achieve an exact diagonalization up to nuclear charge Z = 124 (which covers all chemical elements) and prove the convergence of an approximate block-diagonalization up to Z = 62, thus considerably improving the upper bounds Z = 93 and Z = 51, respectively, established by Siedentop and Stockmeyer.

This paper is a part of an ongoing study on the diamagnetic behavior of a 3-dimensional quantum gas of non-interacting charged particles subjected to an external uniform magnetic field together with a random electric potential. We prove the existence of an almost-sure non-random thermodynamic limit for the grand-canonical pressure, magnetization and zero-field orbital magnetic susceptibility. We also give an explicit formulation of these thermodynamic limits. Our results cover a wide class of physically relevant random potentials which model not only crystalline disordered solids, but also amorphous solids.