Ionescu, Alexandru D.; Jerison, David On the absence of positive eigenvalues of Schrödinger operators with rough potentials. (English) Zbl 1055.35098 Geom. Funct. Anal. 13, No. 5, 1029-1081 (2003). Let \(V: \mathbb{R}^N\to\mathbb{R}\) denote a real-valued potential and \[ H= -\Delta+ V\tag{1} \] the associated Schrödinger operator. In this paper the authors consider the problem of proving the absence of positive eigenvalues of the operator \(H\) for a certain class of rough potential \(V\). To this end the authors for simplicity assume \(N\geq 3\) and \(V\in L^{N/2}(\mathbb{R}^N)\). They show that, absence of positive eigenvalues for (1) is a straight forward consequence of a Carleman inequality of the form \[ \| W_m u\|_{L^{p'}(\mathbb{R}^N)}\leq C\| W_n(\Delta+ 1)u\|_{L^p(\mathbb{R}^N)}, \] for a sequence \[ W_m(x)= \begin{cases} | x|^m &\text{for }| x|\leq R_m,\\ R^m_m\;&\text{for }| x|\geq R_m,\end{cases} \] with the property \(R_m\to \infty\) as \(m\to\infty\). Reviewer: Messoud A. Efendiev (Berlin) Cited in 1 ReviewCited in 47 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35J10 Schrödinger operator, Schrödinger equation 35P20 Asymptotic distributions of eigenvalues in context of PDEs Keywords:absence of positive eigenvalues; Carleman inequality; Schrödinger operator PDFBibTeX XMLCite \textit{A. D. Ionescu} and \textit{D. Jerison}, Geom. Funct. Anal. 13, No. 5, 1029--1081 (2003; Zbl 1055.35098) Full Text: DOI