Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSchrodinger operators generated by substitutions
40
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Schrodinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically verifiable sufficient condition for their spectrum to be singular continuous and supported on a Cantor set of zero Lebesgue measure. Applications to specific examples are discussed.
rate research
Read More
Let $mathcal{L}=-Delta+V$ be a Schr{o}dinger operator, where the nonnegative potential $V$ belongs to the reverse H{o}lder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, ${e^{-tmathcal{L}^{alpha}}}_{t>0},$ associated with $mathcal{L}$. As an application, we obtain the $BMO^{gamma}_{mathcal{L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $mathcal{L}$ via $T1$ theorem, respectively.
The present work aims at obtaining estimates for transformation operators for one-dimensional perturbed radial Schrodinger operators. It provides more details and suitable extensions to already existing results, that are needed in other recent contributions dealing with these kinds of operators.
We consider a family ${mathcal{H}^varepsilon}_{varepsilon>0}$ of $varepsilonmathbb{Z}^n$-periodic Schrodinger operators with $delta$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $minmathbb{N}$ surfaces. We show that in the limit when $varepsilonto 0$ and the interactions strengths are appropriately scaled, $mathcal{H}^varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.
We consider a Schrodinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have the new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of Im part of eigenvalues plus singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.
In this work we consider an example of a linear time-degenerate Schrodinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates as the linear solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
