A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces
The author develops a Morse theory for strongly indefinite functionals \(f\in{\mathcal{C}}^2(H)\) relative to a closed linear subspace \(E\) of the Hilbert space \(H\). If \(f\) has the form \(f(x)={1\over 2}\langle Lx,x\rangle-b(x)\) with \(L\) a self-adjoint invertible linear operator on \(H\), then \(E\) is the positive eigenspace of \(L\) which may be infinite-dimensional and infinite-codimensional. It is assumed that \(f\) satisfies the Palais-Smale condition and has only nondegenerate critical points. The Morse theory can be applied to \(f\) if \(b\) is weakly continuous, \(\nabla b\) is Lipschitz and completely continuous, \(D^2b(x)\) is compact for every \(x\in H\), and \(b(x)\leq C(1+\| x\|^p)\) for some \(C>0\), some \(p\in(0,2)\). The conditions on \(b\) can be weakened but not omitted. The Morse theory is based on a new cohomology theory \(H^*_E\) defined on the category of pairs \((X,A)\) with \(A\subset X\subset H\). The author considers two classes of morphisms \(\Phi\): \((X,A)\to(Y,B)\), namely \(E\)-morphisms and positive \(E\)-morphisms. An \(E\)-morphism has the form \(\Phi=L+K\) with \(L\in GL(H)\), \(LE=E\) and \(K\) satisfies certain compactness conditions. \(\Phi\) is positive if \(L\) is a positive operator. \(H^*_E\) has coefficients in an arbitrary ring \({\mathcal A}\) if only positive \(E\)-morphisms are considered; otherwise \({\mathcal A}=\mathbb{F}_2\). Appropriate versions of the Eilenberg-Steenrod axioms hold. The dimension axiom reads as follows: If \(S\subset E^+\) is the unit sphere then \(H^q_E(S)=0\) for \(q\neq-1\) and \(H^{-1}_E(S)\cong {\mathcal A}\). Using \(H^*_E\) and defining Morse indices relative to \(E\), the Morse relations take a familiar form. The paper is clearly and carefully written. It does not contain applications.
scientific article; zbMATH DE number 1130753