The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials
Thomas Hupfer, Hajo Leschke, Peter Müller, Simone Warzel
备注：
This paper will appear in "Communications in Mathematical Physics". It is a revised version of the second part of the first version of math-ph/0010013, which in its second version only contains the (revised) first part
The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{ö}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schrödinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.
We compute an explicit formula for the integrated density of states of the periodic Airy-Schr{ö}dinger operator on the real line. For this purpose, we study precisely the spectrum of the restriction of the periodic Airy-Schr{ö}dinger operator to a finite number of periods. We prove that all the eigenvalues of this restriction are in the spectral bands of the periodic Airy-Schr{ö}dinger operator and none of them are in its spectral gaps. We count exactly the number of eigenvalues in each of the spectral bands, this number being equal to the number of periods to which the periodic operator is re…
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We prove low frequency resolvent estimates and local energy decay for the Schr{ö}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schr{ö}dinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the …
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
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