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Spectral flow, Brouwer degree and Hill’s determinant formula. (English) Zbl 1444.58007

The paper presents a definition of a new topological invariant, defined by means of a suspension of the complexified family of Morse-Sturm boundary value problems. This invariant is defined in terms of the Brouwer degree of an associated determinant map and generalizes the invariant defined in [M. Musso et al., Topol. Methods Nonlinear Anal. 25, No. 1, 69–99 (2005; Zbl 1101.58012)]. The authors prove a new spectral flow formula and illustrate the relation between this spectral flow formula and the Hill’s determinant formula. Moreover, they exploit this invariant for detecting instability of periodic orbits of a Hamiltonian system. The final appendix contains an overview of basic definitions and properties on the Maslov index and the spectral flow.

MSC:

58J30 Spectral flows
47H11 Degree theory for nonlinear operators
55M25 Degree, winding number
58J20 Index theory and related fixed-point theorems on manifolds
58J32 Boundary value problems on manifolds

Citations:

Zbl 1101.58012
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References:

[1] Abbondandolo, Alberto; Portaluri, Alessandro; Schwarz, Matthias, The homology of path spaces and Floer homology with conormal boundary conditions, J. Fixed Point Theory Appl., 4, 2, 263-293 (2008) · Zbl 1171.53344
[2] Atiyah, Michael F., Anomalies and index theory, Lect. Notes Phys., 208, 313-322 (1984)
[3] Barutello, Vivina; Jadanza, Riccardo D.; Portaluri, Alessandro, Linear instability of relative equilibria for n-body problems in the plane, J. Differ. Equ., 257, 6, 1773-1813 (2014) · Zbl 1291.70028
[4] Barutello, Vivina; Jadanza, Riccardo D.; Portaluri, Alessandro, Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory, Arch. Ration. Mech. Anal., 219, 387-444 (2016) · Zbl 1395.70016
[5] Bavnbek, Booss-Bernhelm; Lesch, Matthias; Phillips, John, Unbounded Fredholm operators and spectral flow, Can. J. Math., 57, 2, 225-250 (2005) · Zbl 1085.58018
[6] Cappell, Sylvain E.; Lee, Ronnie; Miller, Edward Y., On the Maslov index, Commun. Pure Appl. Math., 47, 2, 121-186 (1994) · Zbl 0805.58022
[7] Duistermaat, J. J., On the Morse index in variational calculus, Adv. Math., 21, 2, 173-195 (1976) · Zbl 0361.49026
[8] Gesztesy, F.; Simon, B.; Teschl, G., Zeros of the Wronskian and renormalized oscillation theory, Am. J. Math., 118, 3, 571-594 (1996) · Zbl 0858.47027
[9] Gohberg, I.; Goldberg, S.; Kaashoek, M., Classes of Linear Operators, vol. 1 (1990), Birkauser Verlag: Birkauser Verlag Basel, Boston, Berlin · Zbl 0745.47002
[10] Hu, Xijun; Portaluri, Alessandro, Index theory for heteroclinic orbits of Hamiltonian systems, Calc. Var. Partial Differ. Equ., 56, 6, Article 167 pp. (2017) · Zbl 1390.53089
[11] Hu, Xijun; Portaluri, Alessandro, Bifurcation of heteroclinic orbits via an index theory, Math. Z., 292, 1-2, 705-723 (2019) · Zbl 1474.37053
[12] Hu, Xijun; Portaluri, Alessandro; Yang, Ran, Instability of semi-Riemannian closed geodesics, Nonlinearity, 32, 11, 4281-4316 (2019) · Zbl 1427.58005
[13] Hu, Xijun; Sun, Shanzhong, Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Commun. Math. Phys., 290, 737-777 (2009) · Zbl 1231.37031
[14] Hu, Xijun; Sun, Shanzhong, Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223, 1, 98-119 (2010) · Zbl 1354.70026
[15] Hu, Xijun; Wang, Penghui, Eigenvalue problem of Sturm-Liouville systems with separated boundary conditions, Math. Z., 283, 1-2, 339-348 (2016) · Zbl 1352.34037
[16] Kato, Tosio, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, vol. 132 (1980), Springer-Verlag · Zbl 0435.47001
[17] Kavle, Henry; Offin, Daniel; Portaluri, Alessandro, Keplerian orbits through the Conley-Zehnder index, preprint available at · Zbl 1498.70011
[18] Klingenberg, W., Closed Geodesics on Riemannian Manifolds, CBMS Regional Conference Series in Mathematics, vol. 53 (1983) · Zbl 0539.53003
[19] Long, Yiming, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, vol. 207 (2002), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1012.37012
[20] Lesch, Matthias; Tolksdorff, Jürgen, On the determinant of one dimensional elliptic boundary value problems, Commun. Math. Phys., 193, 3, 643-660 (1998) · Zbl 0920.47046
[21] Long, Yiming; Zhu, Chaofeng, Maslov-type index theory for symplectic paths and spectral flow. II, Chin. Ann. Math., Ser. B, 21, 1, 89-108 (2000) · Zbl 0959.58017
[22] Magnus, Jan R.; Neudecker, Heinz, Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley Series in Probability and Statistics (1999), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester · Zbl 0912.15003
[23] Musso, M.; Pejsachowicz, J.; Portaluri, A., A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds, Topol. Methods Nonlinear Anal., 25, 1, 69-99 (2005) · Zbl 1101.58012
[24] Musso, Monica; Pejsachowicz, Jacobo; Portaluri, Alessandro, Morse index and bifurcation of p-geodesics on semi Riemannian manifolds, ESAIM Control Optim. Calc. Var., 13, 3, 598-621 (2007) · Zbl 1127.58005
[25] Portaluri, Alessandro, Indefinite Sturm theory, Funkc. Anal. Prilozh.. Funkc. Anal. Prilozh., Funct. Anal. Appl., 43, 4, 316-319 (2009), translation in · Zbl 1188.34042
[26] Portaluri, Alessandro, A K-theoretical invariant and bifurcation for a parameterized family of functionals, J. Math. Anal. Appl., 377, 2, 762-770 (2011) · Zbl 1215.58008
[27] Piccione, Paolo; Portaluri, Alessandro; Tausk, Daniel V., Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics, Ann. Glob. Anal. Geom., 25, 2, 121-149 (2004) · Zbl 1050.58015
[28] Portaluri, Alessandro; Waterstraat, Nils, On bifurcation for semilinear elliptic Dirichlet problems and the Morse-Smale index theorem, J. Math. Anal. Appl., 408, 2, 572-575 (2013) · Zbl 1310.35121
[29] Portaluri, Alessandro; Waterstraat, Nils, On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls, J. Math. Anal. Appl., 415, 1, 240-246 (2014) · Zbl 1318.35017
[30] Portaluri, Alessandro; Waterstraat, Nils, Bifurcation results for critical points of families of functionals, Differ. Integral Equ., 27, 3-4, 369-386 (2014) · Zbl 1324.47108
[31] Portaluri, Alessandro; Wu, Li; Ran, Yang, Linear instability for periodic orbits of non-autonomous Lagrangian systems, preprint available at · Zbl 1456.58013
[32] Robbin, Joel; Salamon, Dietmar, The Maslov index for paths, Topology, 32, 4, 827-844 (1993) · Zbl 0798.58018
[33] Robbin, Joel; Salamon, Dietmar, The spectral flow and the Maslov index, Bull. Lond. Math. Soc., 27, 1, 1-33 (1995) · Zbl 0859.58025
[34] Smale, S., On the Morse index theorem, J. Math. Mech., 14, 1049-1055 (1965) · Zbl 0166.36102
[35] Smale, S., Corrigendum: “On the Morse index theorem”, J. Math. Mech., 16, 1069-1070 (1967)
[36] Zhu, Chaofeng; Long, Yiming, Maslov-type index theory for symplectic paths and spectral flow. I, Chin. Ann. Math., Ser. B, 20, 4, 413-424 (1999) · Zbl 0959.58016
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