Lifschitz singularities for periodic operators plus random potentials

Mezincescu, G. A.

doi:10.1007/bf01017565

Cited by 29 publications

(47 citation statements)

References 16 publications

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“…As there are ( j) d smaller cubes in λ L and the probability that a fixed one of them contributes an eigenvalue below α 4-1" 2 is given by (4.6) we obtain: In the general case we use the ideas from [22] in order to get the analog of (4.5). To this end consider boundary conditions of the type Η%(ω) which are defined in [22] by the help of a periodic strictly positive generalized eigenfunction h of H per .…”

Section: Jq+-hmentioning

confidence: 99%

“…To this end consider boundary conditions of the type Η%(ω) which are defined in [22] by the help of a periodic strictly positive generalized eigenfunction h of H per . In [22] it is described how to modify the proof of [17] in order to show that these operators still satisfy (4.6) above. Moreover, from Proposition 1 in [22] it follows that (4.7) is valid.…”

Section: Jq+-hmentioning

confidence: 99%

“…In [22] it is described how to modify the proof of [17] in order to show that these operators still satisfy (4.6) above. Moreover, from Proposition 1 in [22] it follows that (4.7) is valid. The final observation necessary to adopt the above proof to the general case is the fact that, as for the Neumann boundary conditions, we have which is apparent from the special choice of χ in [22].…”

Section: Jq+-hmentioning

confidence: 99%

See 2 more Smart Citations

Localization for random perturbations of periodic Schrödinger operators

Kirsch¹,

Stollmann²,

Stolz³

1998

Random Operators and Stochastic Equations

125

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Abstract-We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R near the band edges.General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schr dinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging.The paper aims at high accessibility in providing details for all the main steps in the proof.

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Section: Jq+-hmentioning

confidence: 99%

Section: Jq+-hmentioning

confidence: 99%

Section: Jq+-hmentioning

confidence: 99%

See 1 more Smart Citation

Localization for random perturbations of periodic Schrödinger operators

Kirsch¹,

Stollmann²,

Stolz³

1998

Random Operators and Stochastic Equations

125

View full text Add to dashboard Cite

show abstract

Section: Resultsmentioning

confidence: 99%

“…The asymptotic behavior of the IDS N 0 (E) as E ↓ E − has been investigated in [28] - [29] in the case d = 1, and in [19] in the case d ≥ 1 and E − = inf σ(−∆ + ω − W ). Note that the proofs of the results of [28], [29], and [19], essentially rely on the non-degeneracy of E − . Later, the Lifshitz tails for the operator −∆ + V ω near the edge E − were investigated in [15] under the assumptions that d ≥ 1, E − > inf σ(−∆ + ω − W ), and that E − is non-degenerate edge in the spectrum of −∆ + ω − W ; due to the last assumption these results are conditional.…”

Section: Resultsmentioning

confidence: 99%

Lifshitz Tails in Constant Magnetic Fields

Klopp

Raikov

2006

Commun. Math. Phys.

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We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] in the case of a vanishing magnetic field. 2000

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Lifshitz tails for Anderson models with sign‐indefinite single‐site potentials

Shen

2015

Mathematische Nachrichten

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We study the spectral minimum and Lifshitz tails for continuum random Schrödinger operators of the formwhere V 0 is the periodic potential, {ω i } i∈Z d are i.i.d random variables and u is the sign-indefinite impurity potential. Recently, this model has been proven to exhibit Lifshitz tails near the bottom of the spectrum under the small support assumption of u and the reflection symmetry assumption of V 0 and u. We here drop the reflection symmetry assumption of V 0 and u. We first give characterizations of the bottom of the spectrum. Then, we show the existence of Lifshitz tails in the regime where the characterization of the bottom of the spectrum is explicit. In particular, this regime covers the reflection symmetry case.

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Lifschitz singularities for periodic operators plus random potentials

Cited by 29 publications

References 16 publications

Localization for random perturbations of periodic Schrödinger operators

Localization for random perturbations of periodic Schrödinger operators

Lifshitz Tails in Constant Magnetic Fields

Lifshitz tails for Anderson models with sign‐indefinite single‐site potentials

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