EXISTENCE OF RESONANCES FOR DIFFERENTIAL OPERATORS

EXISTENCE OF RESONANCES FOR DIFFERENTIAL OPERATORS

Let H be a Schrodinger operator in $L^2$(R) H =(equation omitted) ＋ V(x), with supp V ⊂ [-R, R]. A number $z_{0}$ / in the lower half-plane is called a resonance for H if for all $\phi$ with compact support 〈$\phi$, $(H - z)^{-l}$ $\phi$〉 has an analytic continuation from the upper half-plane to a part of the lower half-plane with a pole at z = $z_{0}$ . Thus a resonance is a sort of generalization of an eigenvalue. For Im k > 0, ($H - k^2$)$^{-1}$ is an integral operator with kernel, given by Green's function(omitted)

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