Finite Section Method

Discrete Schrödinger operators are used to describe physical systems on lattices and, therefore, play an important role in theoretical solid-state physics. For a fixed $p \in [1,\infty]$, consider the Schrödinger operator $H \colon \ell^p(\mathbb{Z}) \to \ell^p(\mathbb{Z})$ given by

$$ (H x_n)_n = x_{n + 1} + x_{n - 1} + v(n) x_n, n \in \mathbb{Z}, $$

and its one-sided counterpart $H_+ \colon \ell^p(\mathbb{N}) \to \ell^p(\mathbb{N})$ given by

$$ (H_+ x)_n = x_{n + 1} + x_{n - 1} + v(n) x_n, \quad n \in \mathbb{N}, \quad x_0 = 0;. $$

Based on Definitions , one can associate $H$ and $H_+$ with infinite tridiagonal matrices $ A = (a_{ij})_{i,j \in \mathbb{Z}}$ and $A_+ = (a_{i,j})_{i,j \in \mathbb{N}}$ .

Looking at the corresponding infinite linear system of equations

$$ A x = b \quad\text{and}\quad A_+ y = c $$

it is interesting to know if the solutions $x$ and $y$ to theses systems can be computed approximately by solving the large but finite linear systems

$$ A_m x^{(m)} = b^{(m)} \quad\text{and}\quad (A_+)_m y^{(m)} = c^{(m)} $$

and letting $m \to \infty$. This is the main idea of the Finite Section Method (FSM). In order to assure the applicability of the above procedure, one investigates further properties of the operator $A$, the sequence $(A_n)$ and its one-sided counterparts. In particular, Fredholm Theory, spectral theory and the concept of limit operators play a central role in this investigation.

This research project deals with the investigation of the applicability of the FSM to problems surging from aperiodic discrete Schrödinger Operators. A famous example for theses operators is the so called Fibonacci-Hamiltonian, where the potential $v$ is given as

$$ v(n) := \chi_{[1 - \alpha, 1)}(n \alpha \operatorname{mod} 1);, \quad n \in \mathbb{Z}. $$

For this particular example, the central objects of investigation are periodic approximations $(A_m)$. It is crucial to assure that the spectrum of these approximations eventually avoids the point $0$ for larger numbers of $m$.