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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.

A function$G$in a Bergman space$A$^{$p$}, 0<$p$<∞, in the unit disk$D$is called$A$^{$p$}-inner if |$G$|^{$p$}−1 annihilates all bounded harmonic functions in$D$. Extending a recent result by Hedenmalm for$p$=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ_{$D$}denotes the characteristic function of$D$and$G$is an arbitrary$A$^{$p$}-inner function, is continuous in$C$, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space$A$^{2}to a general$A$^{$p$}-setting.

No abstract uploaded!

No abstract uploaded!