Spectral properties of a confined nonlinear quantum oscillator in one and three dimensionsWe study the discrete Schrödinger operator H in Z d with the surface potential of theWe first consider the case where the components of the vector ␣ are rationally independent, i.e., the case of the quasi-periodic potential. We prove that the spectrum of H on the interval ͓Ϫd,d͔ ͑coinciding with the spectrum of the discrete Laplacian͒ is absolutely continuous. Then we show that generalized eigenfunctions, corresponding to this interval, have the form of volume ͑bulk͒ waves, which are oscillating and nondecreasing ͑or slow decreasing͒ in all variables. They are the sum of the incident plane wave and of an infinite number of reflected or transmitted plane waves, scattered by the subspace Z d 2 . These eigenfunctions are orthogonal, complete and verify a natural analog of the Lippmann-Schwinger equation. We discuss also the case where d 1 ϭd 2 ϭ1 and ␣ϭp/q is a rational number, i.e., a q-periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides the volume ͑Bloch͒ waves there are also the surface waves, whose amplitude decays exponentially as ͉x 1 ͉→ϱ. The part of the spectrum corresponding to the surface waves consists of a finite number of bands. For large q the bands outside of ͓Ϫd,d͔ are exponentially small in q, and converge in a natural sense to the pure point spectrum that was found ͓B. Khoruzhenko and L. Pastur, Phys. Rep. 288, 109-125 ͑1997͔͒ in the case of the Diophantine ␣'s.

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We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we obtain the Landau-Peierls formula in the low temperature and density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.

Abstract:In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated to the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω := eB/c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).MSC 2000: 82B10, 82B21, 81V99