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By Roger Godement

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In the rest of this section, we construct an h-dependent sequence of eigenfunctions of H , all with energy Eo, concentrating on 7 as ti -+ 0 in the sense explained in section 1. Wave Packets Localised on Closed Classical Trajectories 29 First, we briefly recall the definition of coherent states. We define where K is the matrix It is then well known that where we introduced the notation I w I= n cw;. 76) i= 1 The coherent state I q , p > being optimally localized around the phase space point ( q , p ) , it is natural to construct a state which is a superposition of coherent states localized on points of the trajectory 7.

From estimate (39) and the fact that Au, = 0(1), we obtain a boundary estimate for r = &E, z E X ( En ) ( A c e \ &c) : L ( x , c / c w Ivu,12 = O(E2N1-2). 4 such that B ( z , 2 r ) C Ac, (so r = O ( E )as above). These results yield Step 4 We now relate estimate (39) on u, and the L2-estimate (41)on V u , to uo. Let x E C r ( B ( O , 2 ) )be a smooth cut-off function such that x 2 O,xIB(O,1) = 1. /CoE))uc(z), where CO > 0 is chosen such that T ( E n ) dC c B ( 0 , y ) . We have that Z, E H 2 ( C ) n H,'(C) and hence - ACE, = E l ( ~ ) i i+, re, (42) which follows by a simple calculation.

For a constant C > 0, define Ace G B(0,C E )f l C ( E ) . )"14, (35) which is proved, for example, in [2]. This allows us to derive the bound I I U c l l L 2 ( A ~ ,= ) 0(E"/2> (36) which, for n > 2, is stronger than the bound which follows from the Poincarh inequality. We first use the Harnack inequality in the interior of Ace. For CO> 0, we define Lower Bounds on Eigenfunctions and the First Eigenvalue Gap 45 Estimate (34) and a version of the Harnack inequality due Jerison [7] (see also [2]) yield To extend (37) from Bcoc to Ace we have to use the boundary Harnack inequality developed for non-negative solutions to parabolic equations of the form Lu = Au - Otu = 0, where A is an elliptic operator (see [3] for a discussion).

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