# Finite section method for aperiodic Schrödinger operators

@article{Gabel2021FiniteSM, title={Finite section method for aperiodic Schr{\"o}dinger operators}, author={Fabian Gabel and Dennis Gallaun and Julian Grossmann and Riko Ukena}, journal={ArXiv}, year={2021}, volume={abs/2104.00711} }

We consider discrete Schrödinger operators with aperiodic potentials given by a Sturmian word, which is a natural generalisation of the FibonacciHamiltonian. We introduce the finite section method, which is often used to solve operator equations approximately, and apply it first to periodic Schrödinger operators. It turns out that the applicability of the method is always guaranteed for integer-valued potentials. By using periodic approximations, we find a necessary and sufficient condition for…

## One Citation

Minimal Families of Limit Operators

- Mathematics
- 2021

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both…

## References

SHOWING 1-10 OF 61 REFERENCES

Spectral Approximation for Quasiperiodic Jacobi Operators

- Mathematics
- 2014

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be…

Spectral properties of Schrödinger operators arising in the study of quasicrystals

- Mathematics, Physics
- 2015

We survey results that have been obtained for self-adjoint operators, and especially Schrodinger operators, associated with mathematical models of quasicrystals. After presenting general results that…

Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials

- Physics, MathematicsJournal of Spectral Theory
- 2019

We show that the spectrum of a discrete two-dimensional periodic Schrodinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least…

The spectrum of a quasiperiodic Schrödinger operator

- Mathematics
- 1987

AbstractThe spectrum σ(H) of the tight binding Fibonacci Hamiltonian (Hmn=δm,n+1+δm+1,n+δm,nμv(n),v(n)=
$$\chi _{[ - \omega ^3 ,\omega ^2 [} $$
((n−1)ω), 1/ω is the golden number) is shown to…

Half-line Schrödinger operators with no bound states

- Physics, Mathematics
- 2003

acting in L2([0, c~)) with the boundary condition r For convenience, we require that the potential, V, be uniformly locally square integrable. We write l~(L 2) for the Banach space of such functions.…

Strictly Ergodic Subshifts and Associated Operators

- Mathematics
- 2005

We consider ergodic families of Schr\"odinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely…

Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure

- Mathematics, Physics
- 1999

We consider discrete one-dimensional Schr\"odinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely…

Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential

- Mathematics
- 1996

Treated in this paper are one-dimensional discrete Schr dinger operators with a quasiperiodic potential, which are derived from the model proposed by Kohmoto, Kadanoff and Tang in 1983. The aim of…

Spectral analysis of tridiagonal Fibonacci Hamiltonians

- Mathematics, Physics
- 2011

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution…

Note on Spectra of Non-Selfadjoint Operators Over Dynamical Systems

- Mathematics, PhysicsProceedings of the Edinburgh Mathematical Society
- 2018

Abstract We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical…