Examples of discrete Schrödinger operators with pure point spectrum
The discrete Schrödinger operator \((Hu)_ i=\epsilon \sum_{| j| =1}u_{i+j}+q_ iu_ i, i\in Z^ m\) on \(\ell^ 2(Z^ m)\), \(m\geq 1\), with small coupling \(\epsilon\) is considered. Examples of limit periodic potentials \((q_ i)\) are constructed, for which H has pure point spectrum with a complete set of exponentially localized eigenstates. The spectrum can be an interval or a Cantor set. H is considered as a perturbation of the multiplicative operator \((Qu)_ i=q_ iu_ i\). If Q satisfies certain small divisor conditions, then another diagonal operator \(\hat Q\) of order \(\epsilon^ 2\) and an isomorphism V of \(\ell^ 2(Z^ m)\) via KAM-iteration are constructed such that \(V^{-1}(H+\hat Q)V=Q\). Thus, actually an inverse spectral problem is solved. These results are special cases of more general results about perturbations of multiplication operators. They include also a number of other known examples of discrete Schrödinger operators with pure point spectrum.
perturbations of multiplication operators
scientific article; zbMATH DE number 3844264