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 consider an electron with an anomalous magnetic moment g > 2 confined to a plane and interacting with a nonzero magnetic field B perpendicular to the plane. We show that if B has compact support and the magnetic flux in the natural units is F ≥ 0, the corresponding Pauli Hamiltonian has at least 1 + [F ] bound states, without making any assumptions about the field profile. Furthermore, in the zero-flux case there is a pair of bound states with opposite spin orientations. Using a Birman-Schwinger technique, we extend the last claim to a weak rotationally symmetric field with B(r) = O(r −2−δ ) correcting thus a recent result. Finally, we show that under mild regularity assumptions the existence can be proved for non-symmetric fields with tails as well.

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive δ interaction supported by a periodic curve in R d , d = 2, 3.