Witten’s Proof of the Morse Inequalities

Part of the Theoretical,Mathematical Physics book series (TMP)


Thus far, we have describend the study of Schrödinger operators for their own sake. In ths chapter and the next, we will discuss some rather striking application of the Schrödinger operators to analysis on manifolds. In a remarkable paper, Witten(370) showed that one can obtain the strong Morse inequalities from the semiclassical anlysis of the eigenvalues of some appropriately chosen Schrödinger operatorson a compact manifold M. The semiclassical eigenvalues theorems are discussed in Sect. 11.1, and Witten`s choice of operators in Sect. 11.4. The Morse inequalities are stated in Sect.11.2 and proven in Sect.11.5. Some backround from Hodge theory is described in Sect. 11.3.


Information Technology Quantum Information Compact Manifold Morse Inequality Remarkable Paper 
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© Springer-Verlag Berlin Heidelberg 2008

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