Asymptotic behavior of solutions of \(-\Delta v+qv=\lambda v\) and the distance of \(\lambda\) to the essential spectrum
The main results of the paper are upper bounds for the distance of \(\lambda\in {\mathbb{R}}\) to the essential spectrum of an operator \(\overline{-\Delta +q},\) depending on growth or decay properties of distributional solutions v for the equation \(-\Delta v+qv=\lambda v\). From these bounds it is derived that if \(q(x) \geq -o(| x|^ 2)\) outside a ball, then the existence of a bounded solution guarantees \(\lambda \in \sigma (\overline{-\Delta +q}),\) and every eigenfunction to a discrete eigenvalue decays exponentially. Furthermore, an example of an eigenvalue in the essential spectrum and lower bounds for \(non\)-L\({}_ 2\)-solutions are given. \(q(x) \sim -O(| x|^ 2)\) appears as a borderline case, and the question is raised if, in that case, there may be a \(\lambda\) outside the spectrum of \(\overline{-\Delta +q}\) with a bounded solution v of \(-\Delta v+qv=\lambda v.\)
scientific article; zbMATH DE number 4008962