The Hardy estimates in $\mathbb R^n$ and absence of positive eigenvalues for Schrodinger operators with complex potentials
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 33-34
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Using the following estimit
$$
\int_{\mathbb R^n}|x|^{2p+2}|\Delta\varphi+\varphi|^2\,dx\geqslant C(p)\int_{\mathbb R^n}|x|^{2p}|\varphi|^2\,dx
$$
with $C(p)\to\infty$ as $p\to\infty$, we prove the absence of $L_2$-solution of
$$
\Delta\varphi+v\varphi=\varphi
$$
with $|v(x)|\leqslant C(1+|x|)^{-1-\varepsilon}$.
@article{ZNSL_1984_138_a2,
author = {A. F. Vakulenko},
title = {The {Hardy} estimates in $\mathbb R^n$ and absence of positive eigenvalues for {Schrodinger} operators with complex potentials},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {33--34},
publisher = {mathdoc},
volume = {138},
year = {1984},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a2/}
}
TY - JOUR AU - A. F. Vakulenko TI - The Hardy estimates in $\mathbb R^n$ and absence of positive eigenvalues for Schrodinger operators with complex potentials JO - Zapiski Nauchnykh Seminarov POMI PY - 1984 SP - 33 EP - 34 VL - 138 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a2/ LA - ru ID - ZNSL_1984_138_a2 ER -
%0 Journal Article %A A. F. Vakulenko %T The Hardy estimates in $\mathbb R^n$ and absence of positive eigenvalues for Schrodinger operators with complex potentials %J Zapiski Nauchnykh Seminarov POMI %D 1984 %P 33-34 %V 138 %I mathdoc %U http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a2/ %G ru %F ZNSL_1984_138_a2
A. F. Vakulenko. The Hardy estimates in $\mathbb R^n$ and absence of positive eigenvalues for Schrodinger operators with complex potentials. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 16, Tome 138 (1984), pp. 33-34. http://geodesic.mathdoc.fr/item/ZNSL_1984_138_a2/