Internal Lifschitz singularities of disordered finite-difference Schr�dinger operators

Mezincescu, G. A.

doi:10.1007/bf01464286

Cited by 17 publications

(17 citation statements)

References 13 publications

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“…We refer to this phenomenon as "internal Lifshitz tails." Internal Lifshitz tails have been proven for the Anderson model by Mezincescu [139] and Simon [169]. Their proofs apparently cannot be translated to the continuum case.…”

Section: 22mentioning

confidence: 99%

The integrated density of states for random Schrödinger operators

Kirsch¹,

Metzger²

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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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 research.

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Section: 22mentioning

confidence: 99%

The integrated density of states for random Schrödinger operators

Kirsch¹,

Metzger²

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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“…This leads to the fact (see Okura [22], that if B ( r ) is the ball of radius r, and Ad is the first eigenvalue of -+A in the ball of radius 1. Such results are known as Lifschitz tail results (see Lifschitz [20]), and also arise for other models in Kirsch-Martinelli [19], Mezincescu [21].…”

mentioning

confidence: 89%

Lifschitz Tail and Wiener Sausage on hyperbolic space

Sznitman

1989

Comm Pure Appl Math

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IntroductionConsider H,,, the hyperbolic space of dimension n 2 2; the hyperbolic Laplacian A H on smooth compactly supported functions defines an essentially selfadjoint operator on L2( V ) , if dV is the hyperbolic volume. The bottom of the spectrum of -$ A H is known to be i ( n -1)2. Here we suppose that we are given a random distribution of points on H,,, by means of a Poisson point process with intensity vdV, and we suppose that each point is surrounded by a hyperbolic ball of radius p . We are interested in the following type of question:Let A be a small positive number; what is the number of eigenvalues less than Q(n -1)2 + X for the operator -$A, in a ball B , centered around a fixed point 0, of radius N , with Dirichlet boundary condition on JB,, as well as on each of the random balls intersecting B,, when N is large enough? More precisely, we consider the random measure on [i(n -1)2, co), where lBNl is the volume of B,, and Xi, , are the eigenvalues of -$ A H in B, with the above described boundary conditions. In the first section we show that this sequence of random measures almost surely converges vaguely to a deterministic measure m(dX) on [i(n -1)2, co), called the density of states. We study the behavior of this measure near the edge $ ( n -1)2, by means of its Laplace transform /Fe-"'m(dX), for large values of t , and a Tauberian theorem.Such a Laplace transform in the case of d-dimensional Euclidean space and the usual Laplacian, can be expressed in terms of the Wiener sausage WP (neighborhood of radius p) of a Euclidean Brownian bridge by the formula ( m is now a measure of [0, co)):The large time behavior of this quantity was studied in the work of Donsker-

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“…In particular, this method can be applied to show that the asymptotics (1) holds for the Anderson model on ℓ 2 (Z d ) and its continuum counterpart, the alloy type model on L 2 (R d ). This strategy of proof was pursued in the 1980s in several papers [52,80,98,55,81,82,99] by Kirsch, Martinelli, Simon and Mezincescu. These two models will serve as a point of reference in the present note, since they are closely related to percolation Hamiltonians.…”

Section: Lifshitz Asymptoticsmentioning

confidence: 99%

“…The conjugation with U relates the upper spectral edge to the lower one, see e.g. [98,81,54]. For any subgraph…”

Section: Percolation Hamiltonians On Cayley Graphsmentioning

confidence: 99%

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Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

Autunović

Veselić

Methods of Spectral Analysis in Mathematical Physics

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Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum.The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem.

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Internal Lifschitz singularities of disordered finite-difference Schr�dinger operators

Cited by 17 publications

References 13 publications

The integrated density of states for random Schrödinger operators

The integrated density of states for random Schrödinger operators

Lifschitz Tail and Wiener Sausage on hyperbolic space

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

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