Schrödinger operators and associated hyperbolic pencils (Q2426505)

The paper is concerned with two classes of Schrödinger operators: one-dimensional operators with operator-valued potentials \[ L (t) = -\frac{d^{2}}{d r^{2}} + t V (r), \quad r > 0, \] and the standard \[ H (t) = - \Delta + t V (t), \quad x \in \mathbb{R}^{3}, \] for the multidimensional case. In both cases, the coupling constant \(t \in \mathbb{R}\). For the family of operators \(L(t)\), the following result is proved. Assume that \(V(r)\) is a selfadjoint operator and \(\| V(r)\|\in L^{2}(\mathbb{R}_{+})\cap L^{\infty}(\mathbb{R}_{+})\). The corresponding result for the multidimensional case is the following. If \(V\in L^{\infty}(\mathbb{R}^{3})\) and \[ \int_{1}^{\infty} r| v(r)|^{2}\,dr<\infty, \] where \(v(r)= \sup_{| x| =r}| V(x)|\), then \(\sigma_{ac}(H(t))=\mathbb{R}_{+}\) for a.e.\ \(t\in\mathbb{R}_{+}\). The case \(| V(x)|<C(1+| x|^{2})^{- \gamma}\), \(\gamma>3/2\), is treated separately. In the proofs of the main results, the author uses spectral properties of some quadratic operator pencils.