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Register a new userA short guide to pure point diffraction in cut-and-project sets
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We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochners theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples.
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We show that real model sets with real internal spaces are determined, up to translation and changes of density zero by their two- and three-point correlations. We also show that there exist pairs of real (even one dimensional) aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose two- and three-point correlations are identical but which are not related by either translation or inversion of their windows. All these examples are pure point diffractive. Placed in the context of ergodic uniformly discrete point processes, the result is that real point processes of model sets based on real internal windows are determined by their second and third moments.
We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function argument with help of parallel coordinates outside the cut-locus of the curve, we establish the existence of discrete eigenvalues. This is a special variant of a recent result of Exner in a non-smooth case and via a different technique which does not require non-positive constraining potentials.
We consider resonant tunneling between disorder localized states in a potential energy displaying perfect correlations over large distances. The phenomenon described here may be of relevance to models exhibiting many-body localization. Furthermore, in the context of single particle operators, our examples demonstrate that exponential resolvent localization does not imply exponential dynamical localization for random Schrodinger operators with correlated potentials.
A $p$-adic Schr{o}dinger-type operator $D^{alpha}+V_Y$ is studied. $D^{alpha}$ ($alpha>0$) is the operator of fractional differentiation and $V_Y=sum_{i,j=1}^nb_{ij}<delta_{x_j}, cdot>delta_{x_i}$ $(b_{ij}inmathbb{C})$ is a singular potential containing the Dirac delta functions $delta_{x}$ concentrated on points ${x_1,...,x_n}$ of the field of $p$-adic numbers $mathbb{Q}_p$. It is shown that such a problem is well-posed for $alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for $alpha>1$. In the latter case, the spectral analysis of $eta$-self-adjoint operator realizations of $D^{alpha}+V_Y$ in $L_2(mathbb{Q}_p)$ is carried out.
The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared to numerical values from the literature.
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