By Krishnamurti Jiddu
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Extra info for Action And Relationship
DMS-0635607. The first and second authors were also partially supported by the Agence Nationale de la Recherche under the grant ANR -09-JCJC-0099-01. Thanks also to Edward Ott for his permission to include Fig. 3 in our paper. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. From Open Quantum Systems to Open Quantum Maps 47 References 1.
9], the equality [Re , Re ] = 0 is equivalent to Gg((∇ Q X Q)(Y ), Z ) = 0, where G means the cyclic sum over X, Y, Z and ∇ is the Levi-Civita connection of g. To prove the latter identity we use the fact that the Levi-Civita connection ∇ and the Chern connection D + of (g, J+ ) are related by 1 g(∇ X Y, Z ) = g(D +X Y, Z ) − d + F + (X, J+ Y, J+ Z ). 2 Set P = J+ J− + J− J+ . Then, by (6) we have 2g((∇ X Q)(Y ), Z ) = 2g((∇ X QY, Z ) + 2g(∇ X Y, Q Z ) = 2g((D +X Q)(Y ), Z ) − d + F + (X, J+ QY, J+ Z ) − d + F + (X, J+ Y, J+ Q Z ) = d + F + (X, PY, Z ) + d + F + (X, Y, P Z ) + 2d + F + (X, J− Y, J+ Z ) + 2d + F + (X, J+ Y, J− Z ).
Now, the lower order terms in G 0 may change the norms only by factors (1 + O(Mh log(1/ h))), so the same norm estimates hold if we replace G 0 by its principal symbol G 0,0 in the definition of the new norms. 2. Take G 0 (x , xn , ξ , ξn ) = g 0 (x , ξ ), G 0 ∈ Cc∞ (X ) satisfying G 0 = G 0 ◦ κ −1 in , and G = Mh log(1/ h) G 0 , g = Mh log(1/ h) G 0 . 9) hold in the spaces HG ( ), Hg ( ). 2. Globally defined operators and finite rank weighted spaces. In this section we transform our microlocal Grushin problem into a globally defined one.