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Examples for the Infinite Dimensional Morse Lemma

Buchner, Michael and Marsden, Jerrold E. and Schechter, Stephen (1983) Examples for the Infinite Dimensional Morse Lemma. SIAM Journal on Mathematical Analysis, 14 (6). pp. 1045-1055. ISSN 0036-1410. doi:10.1137/0514084.

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Examples are presented which show how to use the Morse lemma in specific infinite dimensional examples and what can go wrong if various hypotheses are dropped. One of the examples shows that the version of the Morse lemma using singularity theory can hold, yet the hypotheses of the Morse–Palais and Morse–Tromba lemmas fail. Another example shows how to obtain a concrete normal form in infinite dimensions using the splitting lemma and hypotheses related to those in the Morse–Tromba lemma. An example of Dancer is given which shows that for the validity of the Morse lemma in Hilbert space, some hypotheses on the higher order terms must be made in addition to smoothness, if the quadratic term is only weakly nondegenerate. A general conjecture along these lines is made.

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Additional Information:©1983 Society for Industrial and Applied Mathematics Received by the editors June 28, 1982, and in revised form October 29, 1982.
Issue or Number:6
Record Number:CaltechAUTHORS:BUCsiamjma83
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4633
Deposited By: Archive Administrator
Deposited On:30 Aug 2006
Last Modified:08 Nov 2021 20:18

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