A limiting absorption principle for Schrödinger operators with oscillating potentials. Part I: \(-\Delta +c \sin(b| x|^{\alpha})/| x|^{\beta}\) for certain c,\(\alpha\),\(\beta\)
This paper presents a limiting absorption principle for Schrödinger operators with an oscillating potential of the form \(p_ 0(x)=c \sin(b| x|^{\alpha})/| x|^{\beta}, \alpha,\beta>0\). The main step in obtaining a limiting principle for \(H(p_ 0)=-\Delta +p_ 0(x)\) is to prove an important inequality. As the potential \(p_ 0(x)\) is radially symmetric the Schrödinger operator for this potential is unitarily equivalent to a direct sum of ordinary second order differential operators. Then, the approach is to replace the potentials by more easily handled approximating potentials. The details of proofs are given.
scientific article; zbMATH DE number 3853520
scientific article