Spectral flow and bifurcation of critical points

C. R. Acad. Sci. Paris, t. 325, Analyse fonctionnelle/Funcfioional

Spectral Patrick

flow

S&ie

I, p. 743-747, Analysis

and bifurcatio:n

M. FITZPATRICK,

Jacob0

P. F. : Department of’ Mathematics, E-mail: [email protected] J. P. : Dipartimento di Matematica, L. R. : Departamento

1997

de Matematica,

of critical

PEJSACHOWICZ

Univrrsity

and Lazaro

of Md.., College Park,

Politecniro Llniversidad

di Torino,

Torino,

points RECHT

MD, 20742.

U.S.A.

10129, Italy

dr Simon Bolivar,

Caracas, Venezuela.

To each path of strongly indefinite sellfadjoint Fredholm operators with invertible ends there is associated an integer called spectral @w. We develop a new approach to spectral flow which permits us to prove that for a one-parameter family of strongly indefinite functionals, there is bifurcation of critical points from a trivial branchif the

Abstract.

spectralflow of the path of Hessiansalong the branchis non-zero. Flat

R&urn&

Version

spectral

et bifurcation

de points

critiques

A toute courbe d’optkuteurs de Fredholm auto-adjoints fortement indkjinis ~5extre’mitks inversibles, on ussocie un entier appele’ flat spectral. On dkveloppe une nouvelle upproche du flot spectral qui permet de montrer que pour une ,famille ci un parumktre de fonctionnelles fortement ind$nies, il y a bifurcation de points critiques d partir d’une branche rriviale pourvu que le flat spectrul de la courbe de hessiens le long de la brunche triviale ne soit pas nul.

franqaise

abrkge’e

Pour les familles B un paramktre de fonctionne.lles essentiellementpositives il y a bifurcation de points critiques g partir d’une branche triviale si les indices de Morse des points critiques non-dCgCnCr&aux extrkmitCs de la branche sont diffkents (voir [3] et [9]). Pour les fonctionnelles fortement indkfinies, ce critkre n’a aucun senscar les indices de Morse sont infinis. 11est cependant possible d’assigner A une courbe d’optkateurs de Fredholm auto-adjoints fortement indkfinis, 2 extrCmitCs inversibles, un entier appeltJiot spectral. Le flot spectral depend de la classed’homotopie de la courbe et pas seulementdes extrkmitts. En particulier, une courbe fermCepeut avoir un flot spectral non nul. A notre connaissance, cette notion apparait pour la premiere fois dans [I], en rapport avec l’asymktrie spectrale, et s’est depuis retrouvke dans de nombreux domaines (voir [7] et [5]). MotivCs par la methode utiliske dans [6] pour montrer la bifurcation des zQos d’une courbe d’opkrateurs de Fredholm, nous dkveloppons une nouvelle approche du flot spectral, basCesur une paramktrix cogrkdiente. Cette approche est Note pr&entCe

par Louis

NIRENBERG.

0764-4442197103250743 0 Academic des SciencesIElsevier, Paris

743

P. M.

Fitzpatrick,

J. Pejsachowicz

and

1. Recht

adaptee a la demonstration du fait que pour 11:s courbes de fonctionnelles fortement indefinies. il y a bifurcation de points critiques a partir d’une branche triviale pourvu que le flot spectral de la courbe des hessiens le long de la branche triviale ne soit pas nul. &ant don& un espace de Hilbert reel separable H, @s(H) dtsigne l’ensemble des operateurs auto-adjoints de L(H) qui sont de Fredholm. L’espace @s(H) a trois composantes, qui consistent en les operateurs essentiellement positifs (resp. negatifs), dont le spectre essentiel est contenu dans le demi-axe reel positif (resp. negatif), et le complement que nous appelons ensemble des operateurs fortement indejnis et que nous designons par Q;(H). Le flot spectral d’une courbe dans @i(H) a extremites inversibles est introduit en deux temps. Tout d’abord, pour une symetrie donnee J de H (un operateur J E @i(H) tel que J2 =Id), le flot spectral d’une courbe de perturbations compactes de J est introduit comme etant la limite du flot spectral de l’approximation de Galerkin de la courbe ; ensuite, le flot spectral est identifie a l’indice de Morse relatif aux extremitts de cette courbe (voir [S] et [2]). Nous montrons alors que toute courbe dans @i(H) avec extremites inversibles peut &tre transformee de facon cogrediente en une courbe de perturbations compactes de 3. Le flot spectral de la courbe transformee ne depend ni du choix de la symetrie ni de la tranformation cogrediente utilide. Cette construction du flot spectral par les transformations cogrtdientes nous permet de demontrer le theoreme suivant. THBORBME. - Soient U un voisinage parametre de fonctionnelles fortement 0 soit un point critique de $J(X, .) et {Lx}xEt ait des extremites inversibles contient un point critique non trivial

1. Introduction

de I x (0) duns R, x H et I+!) : indejnies de classe C2. Pour tout soit Lx le hessien de $(X, .) en 0. et un jlot spectral non nul. Alors, de $(X, .).

U + R une famille a un X dans I, supposons que Supposons que la courbe tout voisinage de I x (0)

’

For a one-parameter family of essentially-positive functionals there is bifurcation of critical points from a trivial branch if the Morse indices of the non-degenerate critical points at the ends of the branch, are different (see [3] and [9]). For strongly-indefinite functionals this criterion has no meaning since the Morse indices are infinite. It is possible however to assign to a path of strongly-indefinite selfadjoint Fredholm operators, with invertible ends, an integer called spectraljlow. The spectral flow depends on the homotopy class of the path, not just on the end-points. In particular, a closed path can have non-zero spectral flow. To the best of our knowledge, this notion appeared for the first time in [ 11, in connection with the spectral asymmetry, and has since reappeared in many other areas (see [7] and [5]). Motivated by the method used in [6] to prove bifurcation of zeroes of a path of Fredholm maps, we develop a new approach to spectral flow, based on a cogredient parametrix. It is tailored to show that for paths of strongly-indefinite functionals, non-zero spectral flow of the path of Hessians along a trivial branch forces bifurcation of critical points. Complete proofs, computations of spectral flow and applications will appear elsewhere 2. 2. Spectral

flow

for paths

of compact

symmetric

perturbations

of 3

The Morse index, M(L), of a symmetric linear operator L on a finite-dimensional inner-product space, is the dimension of the direct sum of the eigenspaces of L corresponding to the negative eigenvalues, and its signature, sig L, is the difference M(--L) - M(L). For an interval I = [a, b], of such operators with invertible ends, sf (L, I), is defined to the spectral flow of a path {Lx}xEr be M(L,) - M(Lb) = [sig(Lb) - sig(L,)]/2. 744

Spectral

flow

and

bifurcation

of critical

points

Let Q in L(H) be an orthogonal projector such that both H + =: im (Q) and H- F im (Id - Q) have infinite dimension and define J E Q - (Id - Q). Then ker(ld - 3) = Hf and ker(ld + J’) = H-, so the selfadjoint operator ,7 is a symmetry (3’ = Id) whose spectrum, which coincides with its essential spectrum, equals {*l}. Choose orthonormal Schauder bases (et}k=l,...,oo and {e<}k=l....,oa for H+ and H-, respectively. For each natural number n, let H, be the span of {et / 1 < k < n}, P, be the orthogonal projector of H onto the space H,, and for an operator L in L(H), define the n-th Galerkin approximation of L to be the restriction of P, o L to H,. The spectral flow, sf (L,I), of a path {LX }xEl of compact symmetric perturbations of 3, with invertible ends, is defined to be the spectral flow of sufficiently high Galerkin approximations of the path. Since sig & = 0 and P,J.= JP, for all n, the uniform convergence on compact sets of {P,} to Id and the continuity of the signature of isomorphisms imply that the spectral flow is a properly defined homotopy invariant: For a family H: [(I: l] x I -+ @g(H) of compact symmetric perturbations of 3 such that each path H(t, .) has invertible ends, the spectral flow sf (H(t, .), 1) is constant.

3. A relative

Morse

index

A pair (P, Q) of orthogonal projectors in I-(H) is called Fredholm if im P n im (Id - Q) and im Q n im (Id - P) are finite dimensional, and the index of such a pair is defined by: ind(P,Q)

~dim(imPnim(ld

-Q))

-dim(imQnim(Id

-p)).

Observe that a pair of orthogonal projectors whose difference is compact is Fredholm. This notion was investigated, among others, in [8] 3 and [4], and, more recently, in [2] and [5]. It is not difficult to see (see [2]) that:

(1)

ind (P, Q) is the usual index of the Fredholm operator Q : im F + im Q,

and also that (2)

ind(P,Q)

= dimker(Id

-P+

Q) - dimker(ld

- Q+P).

From (1) and the continuity property of the Fredholm index of an operator, it follows that the index of a Fredholm pair is a homotopy invuriant: If {(P,, Qt)}tcl is a path of pairs of orthogonal projectors whose differences are compact, then ind (P,, Qt) is constant. From (2), it follows that the index of a Fredholm pair is a unitary invariant: If (P, Q) is a Fredholm pair, then ind (E’! Q) = ind (R-‘PR: R-l&R) if the isomorphism R in L(H) is unitary (i.e. R* = R-l). Two operators S and T in L(H) are called Calkin equivalent, written S 2 T, if their difference S-T is compact. If S 21 T and both are selfadjoint isom&phisms, it follows from the integral representation of spectral projectors that the orthogonal projectors onto the negative spectral subspaces, E-(S) and 1s a Fredholm pair. In this case the E-(T), are also Calkin equivalent and hence (E-(S), E-(T)) relative Morse index of S with respect to T; ind hl(S, T)! is defined to be ind (E-(S), E-(T)). The homotopy and unitary invariance properties of the index of Fredholm pairs of projectors lead to the same two properties of the relative Morse index of Calkin equivalent pairs of isomorphisms. PROPOSITION 1. - If the selfadjoint isomorphisms S and T ure Calkin equivalent, then for any isomorphism R in L(H), indkl(S;T) = indb,(R*SR, R*TR). To see this, deform R = RO through { Rt z U( (1 - t)mx + tld )}lct
P. M.

Fitzpatrick,

J. Pejsachowicz

and

1. Recht

PROPOSITION 2. - Let {LX}XEI be u path of compact symmetric perturbations of 3 with invertible ends. Then sf(L,I) = indR/I(La,&,).

This formula follows from the homotopy invariance both of the spectral flow and the relative Morse index and its validity in the finite-dimensional case.

4. Cogredient

parametrices

and spectral

flow

Let A be a topological space. By a cogredientparametrix relative to 3 for a family L : A -+ @k(H) is meant a family IV: A --+ GL(H) such that for each X in A, (MA)*Lxh/r, = J + Kx, where Kx is compact. PROPOSITION 3. - Every family L : A -+ @k(H) parametrized by a paracompact contractible space h possessesa cogredient parametrix. To seethis, let Ks( H) be the set of compact symmetric operators in L(H), let 4 F GL(H) x KS(H) and define 7r: 4 -+ Q$( H) by r(IM, h’) = A4*.11I4 + K for (M, K) in 4. The above Proposition assertsthat L : A + (ai (H) may be expressed as i.he composition L = n o a, where cy : A --+ G is continuous. To verify this, it suffices to prove that the map 7r : Q -+ @i(H) possessesthe homotopy lifting property. However, one can prove that r : 4 -+ @k(H) .IS a locally trivial bundle and therefore has this lifting property. Given a path L : I 4 @i(H) with invertible ends, let A4 : I + GL( H) be any cogredient parametrix for L. The spectral flow of the path L on I is defined by sf (L! I) F sf (fI4* LM, I). Propositions 1 and 2 imply that this definition is independenl.of the choice of parametrix. Spectral flow is invariant under the natural cogredient action of GL(H ) on the space of paths in (a;(H) and also invariant under homotopies keeping end-points invertible.

5. The bifurcation

theorem

THEOREM 4. - Let U be a neighborhood of I x (0) in R x H and II, : ZA+ R be a C* one-pammeter family of strongly-indejinite functionals. For each )\ in I, assume0 is a critical point of $(A, -) and let Lx be the Hessian of $(X, .) at 0. Assume the ,path {L x } ~~1 has invertible ends and non-zero spectral flow. Then every neighborhood of I x (0) contains nontrivial critical points of $(A; .).

The proof proceedsby using the cogredient and homotopy invariance properties of the spectral flow as one (i) usesProposition 3 to make a change of variables which reduces to the case that the path of Hessiansat the origin is a path of compact perturbations of J, then (ii) uses a Lyapunov-Schmidt reduction to recast the problem on a sufficiently large Gale&in subspace,and finally, (iii) applies Berger’s finite-dimensional argument (see [3]).

(‘) Scientific field: Analyse fonctionnelle/functional analysis. (‘) Nnfurion: Throughout, H is a real separable Hilbert space, GL(H) is the set of isomorphisms in L(H) and aS( H) is the set of selfadjoint operators in L(H) that are Fredholm. The space @s(H) has three components, consisting of the essentially positive (resp. negative) operators whose essential spectrum is contained in the positive (resp. negative) real axis, and the complement which we call the strongly indefinite operators and denote by Q;.(H). A functional is described by the properties of its Hessian. (3) ind (P, Q) coincides with the index of the pair of subspaces (im P, im (Id - Q)) in [8]. Note remise le 15 juillet

746

1997, acceptte le 28 juillet

1997.

Spectral

flow

and

bifurcation

of critical

points

References [I] [2] [3] [4] [5] [6] [7/ [S] [9]

Atiyah M. F.. Patodi V. K. and Singer I. M., 1975. Spectral asymmetry and Riemannian geometry. I. M&h. Proc. Cambridge Phil. Sm., 77. pp. 43-69. Avron J., Seiler R. and Simon B., 1994. The index of a pair of projections, /. Funcr. Am/.. 120, pp. X0-237. Berger M. S.. 1972. Bifurcation theory and the type numbers of M. Morse. Pmt. Nut Actrd. Sri., 69. pp.l737-1738. Booss B. and Wojciechowski K., 1993. Elliptic Boundwy Pmblmzs for Dirac Oper~zrors, Birkhluaer. Bunke U., 1994. On the spectral flow of families of Dirac operators with constant symbol, Math. Nachr.. 165, pp. 191.203. Fitzpatrick P. and Pejsachowicz J., 1990. Local bifurcation for C’ -Fredholm maps, Proc. AMS. 4( 105). pp. 995.1002. Floer A.. 1988. An instanton invariant for 3-manifolds. Cnm,nun. 1!4&. Pl~wics. 1 18. pp. 2 15-730. Kato T., 1980. Prrturbufiorz Theory j?w Linear 0perators. Grundlehren der mathematischen Wissenshaften, 132, Springer-Verlag. Mawhin J. and Willem M., 1989. Criticnl Point Thmp arid Hcmziltonicm S~QIIZ.Y. Applied Mathematical Sciences, 74. No. 483. Springer-Verlag.

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