On the absence of positive eigenvalues of Schrödinger operators with rough potentials (Q1423510)

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 \(m\to\infty\).