Must the Spectrum of a Random Schrödinger Operator Contain an Interval? (original) (raw)

The Spectrum of Schrödinger Operators with Poisson Type Random Potential

Annales Henri Poincaré, 2006

We consider the Schrödinger operator with Poisson type random potential, and derive the spectrum which is deterministic almost surely. Apart from some exceptional cases, the spectrum is equal to [0, ∞) if the single-site potential is nonnegative, and is equal to R if the negative part of it does not vanish with positive probability, which is consistent with the naive observation. To prove that, we use the theory of admissible potential and the Weyl asymptotics.

Absolutely continuous spectrum in random Schrödinger operators on the Bethe strip

2012

The Bethe Strip of width m is the cartesian product B × {1,. .. , m}, where B is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians H λ = 1 2 ∆ ⊗ 1 + 1 ⊗ A + λV on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, V is a random matrix potential, and λ is the disorder parameter. Given any closed interval I ⊂ (− √ K + amax, √ K + a min), where a min and amax are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.

Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

2011

The Bethe Strip of width mmm is the cartesian product Btimes1,...,m\B\times\{1,...,m\}Btimes1,...,m, where B\BB is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians ;Hlambda=frac12Deltaotimes1+1otimesA+lambdaVv\;H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A + \lambda \Vv;Hlambda=frac12Deltaotimes1+1otimesA+lambdaVv on a Bethe strip with connectivity Kgeq2K \geq 2Kgeq2, where AAA is an mtimesmm\times mmtimesm symmetric matrix, Vv\VvVv is a random matrix potential, and lambda\lambdalambda is the disorder parameter. Given any closed interval Isubset(−sqrtK+amathrmmax,sqrtK+amathrmmin)I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})Isubset(−sqrtK+amathrmmax,sqrtK+amathrmmin), where amathrmmina_{\mathrm{min}}amathrmmin and amathrmmaxa_{\mathrm{max}}amathrmmax are the smallest and largest eigenvalues of the matrix AAA, we prove that for lambda\lambdalambda small the random Schr\"odinger operator ;Hlambda\;H_\lambda;Hlambda has purely absolutely continuous spectrum in III with probability one and its integrated density of states is continuously differentiable on the interval III.

Schr�dinger operators with an electric field and random or deterministic potentials

Communications in Mathematical Physics, 1983

We prove that the Schrδdinger operator H= --~ + V(x)+F-x has ax purely absolutely continuous spectrum for arbitrary constant external field F, for a large class of potentials this result applies to many periodic, almost periodic and random potentials and in particular to random wells of independent depth for which we prove that when F = 0, the spectrum is almost surely pure point with exponentially decaying eigenfunctions.

The spectrum minimum for random Schrödinger operators with indefinite sign potentials

Journal of Mathematical Physics, 2006

This paper sets out to study the spectral minimum for an operator belonging to the family of random Schrödinger operators of the form Hλ,ω=−Δ+Wper+λVω, where we suppose that Vω is of Anderson type and the single site is assumed to be with an indefinite sign. Under some assumptions we prove that there exists λ0>0 such that for any λ∊[0,λ0], the minimum of the spectrum of Hλ,ω is obtained by a given realization of the random variables.

One-dimensional Schr�dinger operators with random decaying potentials

Communications in Mathematical Physics, 1988

We investigate the spectrum of the following random Schrόdinger operators: 72 where F(X t (ω)) is a Markovian potential studied by the Russian school [8]. We completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order of a(t) grows. This is an extension to a continuous case of the result due to Delyon-Simon-Souillard [6], who deal with the lattice case.