Schrödinger operators with singular potentials from the space of multiplicators
Schrödinger operators \(-\Delta + Q\) are considered where \(Q\) is a multiplication operator by a generalized function \(q\). The following problems are studied. 1. What singular functions \(q\) do admit a well-defined operator \(-\Delta + q(x)\)? This leads to the question when a generalized function \(q\) is a multiplicator from \(H^1(R^n)\) to \(H^{-1}(R^n)\), i.e. when the multiplication by this function is a bounded operator from \(H^1\) to \(H^{-1}\). 2. If the operator \(-\Delta + q\) with a given singular potential \(q\) is well-defined, is it possible to approximate it by operators with smooth potentials so that the spectra of the approximating operators are close to the spectrum of \(-\Delta + q\)? The main results of the paper give answers to the questions mentioned. Also a generalization to the polyharmonic operator \((-\Delta)^n\) is given.
scientific article; zbMATH DE number 1517555