Abstract
We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of self-adjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller’s theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.
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- Category:
- Articles
- Type:
- artykuły w czasopismach
- Published in:
-
Fixed Point Theory and Applications
no. 2019,
pages 1 - 20,
ISSN: 1687-1820 - Language:
- English
- Publication year:
- 2019
- Bibliographic description:
- Izydorek M., Janczewska J., Waterstraat N.: The Maslov index and the spectral flow—revisited// Fixed Point Theory and Applications -Vol. 2019, (2019), s.1-20
- DOI:
- Digital Object Identifier (open in new tab) 10.1186/s13663-019-0655-6
- Bibliography: test
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- Arnold, V.I.: On a characteristic class entering into conditions of quantization. Funkc. Anal. Prilozh. 1, 1-14 (1967) open in new tab
- Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Philos. Soc. 79, 71-99 (1976) open in new tab
- Booss-Bavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Can. J. Math. 57, 225-250 (2005) open in new tab
- Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math. 9, 171-206 (1956) open in new tab
- Cappell, S.E., Lee, R., Miller, E.Y.: On the Maslov index. Commun. Pure Appl. Math. 47, 121-186 (1994) open in new tab
- Duistermaat, J.J.: On the Morse index in variational calculus. Adv. Math. 21, 173-195 (1976) open in new tab
- Fitzpatrick, P.M., Pejsachowicz, J., Recht, L.: Spectral flow and bifurcation of critical points of strongly indefinite functionals-part I: general theory. J. Funct. Anal. 162, 52-95 (1999) open in new tab
- Guillemin, V., Sternberg, S.: Geometric Asymptotics, Mathematical Surveys, vol. 14. Am. Math. Soc., Providence (1977) open in new tab
- Hu, X., Portaluri, A.: Index theory for heteroclinic orbits of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 56, 167 (2017) open in new tab
- Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). Reprint of the 1980 edition, Classics in Mathematics open in new tab
- Lesch, M.: The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators. In: Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds. Contemp. Math., vol. 366, pp. 193-224. Am. Math. Soc., Providence (2005) open in new tab
- Musso, M., Pejsachowicz, J., Portaluri, A.: A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds. Topol. Methods Nonlinear Anal. 25, 69-99 (2005) open in new tab
- Nicolaescu, L.: On the space of Fredholm operators. An.Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 53, 209-227 (2007)
- Pejsachowicz, J., Waterstraat, N.: Bifurcation of critical points for continuous families of C 2 -functionals of Fredholm type. J. Fixed Point Theory Appl. 13, 537-560 (2013) open in new tab
- Phillips, J.: Self-adjoint Fredholm operators and spectral flow. Can. Math. Bull. 39, 460-467 (1996) open in new tab
- Piccione, P., Tausk, D.V.: A Student's Guide to Symplectic Spaces, Grassmannians and Maslov Index. IMPA Mathematical Publications, Rio de Janeiro (2008) open in new tab
- Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827-844 (1993) open in new tab
- Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27, 1-33 (1995) open in new tab
- Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303-1360 (1992) open in new tab
- Wahl, C.: A new topology on the space of unbounded selfadjoint operators, K-theory and spectral flow. In: C * -Algebras and Elliptic Theory II. Trends. Math., pp. 297-309. Birkhäuser, Basel (2008) open in new tab
- Waterstraat, N.: Spectral flow, crossing forms and homoclinics of Hamiltonian systems. Proc. Lond. Math. Soc. (3) 111, 275-304 (2015) open in new tab
- Waterstraat, N.: Fredholm operators and spectral flow. Rend. Semin. Mat. (Torino) 75, 7-51 (2017)
- Sources of funding:
- Verified by:
- Gdańsk University of Technology
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