arXiv subjects

Riko Ukena

Publications and source records attributed to Riko Ukena.

4 recordsLinked to original sources

arXiv2023

Finite section method for aperiodic Schrödinger operators

We consider 1D discrete Schrödinger operators with aperiodic potentials given by a Sturmian word, which is a natural generalisation of the Fibonacci Hamiltonian. Via a standard approximation by periodic potentials, we establish Hausdorff convergence of the corresponding spectra for the Schrödinger operators on the axis as well as for their compressions to the half-axis. Based on the half-axis results, we study the finite section method, which is another operator approximation, now by compressions to finite but growing intervals, that is often used to solve operator equations approximately. We find that, also for this purpose, the aperiodic case can be studied via its periodic approximants. Our results on the finite section method of the aperiodic operator are illustrated by confirming a result on the finite sections of the special case of the Fibonacci Hamiltonian.

math.SP↗

arXiv2023

Spectral approximation of generalized Schrödinger operators via approximation of subwords

We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D.

math.SP↗

arXiv2022

Half-line compressions and finite sections of discrete Schrödinger operators with integer-valued potentials

We study 1D discrete Schrödinger operators $H$ with integer-valued potential and show that, $(i)$, invertibility (in fact, even just Fredholmness) of $H$ always implies invertibility of its half-line compression $H_+$ (zero Dirichlet boundary condition, i.e. matrix truncation). In particular, the Dirichlet eigenvalues avoid zero -- and all other integers. We use this result to conclude that, $(ii)$, the finite section method (approximate inversion via finite and growing matrix truncations) is applicable to $H$ as soon as $H$ is invertible. The same holds for $H_+$.

math.FA↗

arXiv2022

Finite Sections of Periodic Schrödinger Operators

We study discrete Schrödinger operators $H$ with periodic potentials as they are typically used to approximate aperiodic Schrödinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates $H$ by growing finite square submatrices $H_n$. For integer-valued potentials, we show that the finite section method is applicable as soon as $H$ is invertible. This statement remains true for $\{0, λ\}$-valued potentials with fixed rational $λ$ and period less than nine as well as for arbitrary real-valued potentials of period two.

math.SP↗