| Summary: | We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence. © 2012 The Author(s). This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Received: 11 August 2012. Accepted: 24 August 2012. Published online: 5 October 2012. The authors wish to thank A. Laptev for stimulating their interest in this problem. U.S. NSF grants PHY-1068285 (R.L.F.) and PHY-1122309 (L.G.) and DFG grant GE 2369/1-1 (L.G.) are acknowledged. Communicated by A. Laptev. Published - art_3A10.1007_2Fs13373-012-0028-5.pdf Submitted - 1208.2327.pdf
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