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Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schrödinger operator with complex potential. (Russian. English summary) Zbl 0557.35023

Summary: Using the following estimate \[ \int_{{\mathbb{R}}^ n}| x|^{2p+2}\quad | (\Delta \phi +\phi)|^ 2\quad dx\geq C(p)\int_{{\mathbb{R}}^ n}| x|^{2p}\quad | \phi |^ 2\quad dx \] with C(p)\(\to \infty\) as \(p\to \infty\), we prove the absence of an \(L_ 2\)-solution of \(1/2-\Delta \phi +\vartheta\phi=\phi\) with \(|\vartheta(x)| \leq C(1+| x|)^{-1-\epsilon}\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
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