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An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 51 (8 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
The integrated density of states for random Schrödinger operators
 in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is di ..."
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Cited by 20 (1 self)
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Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current
LOCALIZATION NEAR FLUCTUATION BOUNDARIES VIA FRACTIONAL MOMENTS AND APPLICATIONS
, 2006
"... We present a new, short, selfcontained proof of localization properties of multidimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2], but does not require the ra ..."
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Cited by 7 (5 self)
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We present a new, short, selfcontained proof of localization properties of multidimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2], but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.
4. Regularity of the Integrated Density of States References
"... Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is di ..."
Abstract
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Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems