On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials
Let \(H_D\), \(h_D\) respectively denote the continuous and discrete one-dimensional Schrödinger operators: \[ H_D\psi(x)= -{d^2\psi\over dx^2}+ q(x)\psi(x)\quad\text{on }L^2([0,\infty))\quad\text{with }\psi(0)= 0 \] and \[ h_D\psi(n)= \begin{cases} \psi(n+ 1)+ \psi(n- 1)+ q(n)\psi(n),\quad & n=1,2,\dots,\\ \psi(1)+ q(0)\psi(0),\quad & n= 0.\end{cases} \] It is proved that if \(q\in L^2\) then the absolutely continuous spectrum of \(H_D\) is essentially supported by \([0,\infty)\), and, if \(q\in l^2\), the absolutely continuous spectrum of \(h_D\) is essentially supported by \([-2,2]\). By essentially supported is meant that every subset of positive Lebesgue measure has positive measure with respect to some spectral measure. The results are optimal in the sense that it is known that there exist potentials decaying to zero at infinity and belonging to \(\bigcap_{2< p\leq\infty} L^p\left(\bigcap_{2< p\leq\infty}l^p\right)\) such that \(H_D(h_D)\) has purely singular continuous spectrum; [see \textit{A. Kiselev}, \textit{Y. Last} and \textit{B. Simon}, Commun. Math. Phys. 194, No. 1, 1-45 (1998; Zbl 0912.34074)]. The main ingredients of the proofs are estimates for the transmission coefficients.
scientific article; zbMATH DE number 1334926