Analyticity properties of eigenfunctions and scattering matrix |
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Authors: | Erik Balslev |
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Institution: | (1) Denmark and Institute for Advanced Study, University of Aarhus, 08540 Princeton, NJ, USA |
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Abstract: | For potentialsV=V(x)=O(|x|–2– ) for |x|![rarr](/content/k8j1r761x2g1439m/xxlarge8594.gif) ,x![isin](/content/k8j1r761x2g1439m/xxlarge8712.gif) 3 we prove that if theS-matrix of (– , – +V) has an analytic extension
to a regionO in the lower half-plane, then the family of generalized eigenfunctions of – +V has an analytic extension
toO such that
for |Imk|<b. Consequently, the resolvent (– +V–z
2)–1 has an analytic continuation from + to {k O Imk|<b} as an operator
from
b
={f=e
–b|x|
g|g L
2( 3)} to –b
. Based on this, we define for potentialsW=o(e
–2b|x|) resonances of (– +V, – +V+W) as poles of
and identify these resonances with poles of the analytically continuedS-matrix of (– +V, – +V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1) |
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Keywords: | |
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