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On Positive Solutions of Critical Schrödinger Operators in Two Dimensions

✍ Scribed by F. Gesztesy; Z. Zhao

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
696 KB
Volume
127
Category
Article
ISSN
0022-1236

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✦ Synopsis

Using a probabilistic approach based on the Feynman-Kac formalism and the spectral radius of the shuttle operator, we prove that two-dimensional Schrödinger operators (H=-\Delta+V) with short-range potentials (V) satisfying (V(x)=\left.\right|_{\mid x \rightarrow \infty} 0\left(|x|^{-2}(\ln (|x|))^{-2-\varepsilon}\right)) for some (\varepsilon>0) are critical if and only if (H \psi=0) has a positive bounded distributional solution (\psi). It is shown that (apart from logarithmic refinements) our decay assumptions on (V(x)) as (|x| \rightarrow \infty) are the best possible. This yields a complete solution of a problem posed by Simon and extends an earlier result of Murata. 1995 Academic Press, Inc

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