Abstract

this paper we work in the semi-classical setting of "black box scattering" introduced by Sjostrand and the second author in [10] and then extended by Sjostrand in [9]. In the modification of the argument of Stefanov and Vodev we replace the Phragm'en-Lindelof principle by a local application of the maximum principle adapted to the semi-classical setting. The global bound (1) is replaced by a local bound given in Lemma 1 below and coming essentially from the recent work of Sjostrand on the local trace formula for resonances [9]. It is interesting to note that the global trace formula for resonances (established in successive generality by Bardos-Guillot-Ralston, Melrose, Sjostrand-Zworski and S'a Barreto-Zworski -- see [9],[15] and references given there) can be proved using the minimum modulus theorem as in the proof of (1) -- see [3],[15], while for the more generally valid local formula of [9], the local estimates of the type used here are needed. Finally we remark that the results of Stefanov and Vodev, [11],[12], and the more recent results of Popov and Vodev [7] are also concerned with situations in which one cannot construct quasi-modes in the standard sense. By proceeding as in their papers and using the methods of this paper one could generalize their results to even dimensions and to suitable non-compactly supported perturbations

Citations