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Methods Of Modern Mathematical Physics

Delone operators are Schrodinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given "single-site potential" centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.

Localisation for Delone operators via Bernoulli randomisation

We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for Boshernitzan's criterion under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e.\ frequency and phase) when the coupling is large.

https://ems.press/content/serial-article-files/27831

Spectral characteristics of Schrödinger operators generated by product systems

Random Hamiltonians with arbitrary point interactions in one dimension

We introduce and study the following model for random resonances: we take a collection $\Upsilon $ of point interactions $\Upsilon _j$ generated by a simple finite point process in $\mathbb R^3$ and consider the resonances of associated random Schrodinger Hamiltonians $H_\Upsilon = -\Delta + ``\sum \mathfrak {m}(\alpha ) \delta (x - \Upsilon _j)$”. These resonances are zeroes of a random exponential polynomial, and so form a point process $\Sigma (H_\Upsilon )$ in the complex plane $\mathbb C$. We show that the counting function for the set of random resonances $\Sigma (H_\Upsilon )$ in $\mathbb C$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $\Upsilon $, and obtain an explicit formula for the limiting distribution, as $m \rightarrow \infty $, of the leading parameter of the asymptotic chain of ‘most narrow’ resonances generated by a sequence of uniform binomial processes $\Upsilon ^m$ with m points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of m samples of a random point in the 3-D space and the related statistics of extreme values.

Asymptotics of random resonances generated by a point process of delta-interactions

Spectral statistics for Anderson models with sporadic potentials

We prove that the local eigenvalue statistics at energy E in the localization regime for Schrodinger operators with random point interactions on Rd, for d = 1, 2, 3, is a Poisson point process with the intensity measure given by the density of states at E times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrodinger operators in the continuum. The special structure of resolvent of Schrodinger operators with point interactions facilitates the proof of the Minami estimate for these models.

Eigenvalue statistics for Schrödinger operators with random point interactions on Rd, d = 1, 2, 3

We compute the density of states for the Cauchy distribution for a large class of random operators and show it is analytic in a strip about the real axis

Analyticity of density of states for the Cauchy distribution

In this paper we consider Schrodinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$\Gamma$-trimmed' random potential, namely a potential which vanishes outside a subset $\Gamma$ which is periodic with respect to a sub lattice We prove that (under appropriate assumptions) for strong disorder these operators have \emph{pure point spectrum } outside the set $\Sigma_{0}=\sigma(H_{0,\Gamma^{c}})$ where $H_{0,\Gamma^{c}} $ is the free (discrete) Laplacian on the complement $\Gamma^{c} $ of $\Gamma $ We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subset\Sigma_{0}$ Consequently, there is a mobility edge for such models We also consider the case $-M_{1}=M_{2}=\infty$, i~e~ $\Gamma $-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$ Again, we prove localisation outside $\Sigma_{0} $ by showing exponential decay of the Green function $G_{E+i\eta}(x,y) $ uniformly in $\eta>0 $ For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+i\eta} $ is \emph{not} (uniformly) in $\ell^{1}$ as $\eta$ approaches $0$ This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here

Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrodinger operators with random point interactions on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrodinger operators in the continuum. The special structure of resolvent of Schrodinger operators with point interactions facilitates the proof of the Minami estimate for these models.

M. Krishna

Papers

Spectral statistics for Anderson models with sporadic potentials

Eigenvalue statistics for Schrödinger operators with random point interactions on Rd, d = 1, 2, 3

Analyticity of density of states for the Cauchy distribution

Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

Eigenvalue statistics for Schr\"odinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$