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On the absence of positive eigenvalues of Schrödinger operators with rough potentials. (English) Zbl 1055.35098
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 \(m\to\infty\).

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
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