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In the semi-classical regime we study the resonances of the operator$P$_{$t$}=$h$^{2}Δ+$V$+$t$·δ$V$in some small neighborhood of the first spectral band of$P$_{0}. Here$V$is a periodic potential, δ$V$a compactly supported potential and$t$a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent of$P$_{$t$}, and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues when$t$crosses a certain threshold.

Let Ω be a bounded domain in$C$^{$n$}. This paper deals with the study of the behavior of the pluricomplex Green function$g$_{Ω}($z, w$) when the pole$w$tends to a boundary point$w$_{0}of Ω. We find conditions on Ω which ensure that lim_{$w→wo$}$g$_{$Ω$}$(z, w)$=0, uniformly with respect to$z$on compact subsets of $$\bar \Omega \backslash \{ w_0 \} $$ . Our main result is Theorem 5; it gives a sufficient condition for the above property to hold, formulated in terms of the existence of a plurisubharmonic peak function for Ω at$w$_{0}which satisfies a certain growth condition.

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