arXiv subjects

A. Kiselev

Publications and source records attributed to A. Kiselev.

23 records · Page 2Linked to original sources

arXiv2001

Dynamical upper bounds on wavepacket spreading

We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian--the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

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arXiv2001

Imbedded Singular Continuous Spectrum for Schrödinger Operators

We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.

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arXiv2001

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|q(x)| \leq C(1+|x|)^{-{1/4}-ε};$ in the smoothness direction, a sufficient condition in Hölder classes is $q \in C^{{1/2}+ε}(\reals)$. On the other hand, we show that there exist potentials which both satisfy $|q(x)| \leq C(1+|x|)^{-1/4}$ and belong to $C^{1/2}(\reals)$ for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

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arXiv2001

Stability of Singular Spectral Types under Decaying Perturbations

We look at invariance of a.e. boundary condition spectral behavior under perturbations, $W$, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported $W$'s to suitable short-range $W$. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.

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arXiv2001

Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials

We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in $L^p(\reals)$, $p<2$. If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.

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