×

The Maslov index in weak symplectic functional analysis. (English) Zbl 1307.53063

This paper is devoted to an interpretation of the concept of Maslov index in functional analysis. The algebraic definition of Maslov index is a canonical \(W(\mathbb{K})\)-valued mapping on \(n\)-tuples of Lagrangian subspaces of a symplectic \(\mathbb{K}\)-vector space \((V,\omega)\), where \(\mathbb{K}\) is a field with characteristic \(\not=2\) and \(W(\mathbb{K})\) is the Witt group of \(\mathbb{K}\). In the case where \(\mathbb{K}=\mathbb{R}\), one has \(W(\mathbb{K})=\mathbb{Z}\), and with \(\mathbb{K}=\mathbb{C}\), one has \(W(\mathbb{K})=\mathbb{Z}/2\mathbb{Z}\).
This function is a topological invariant, related by means of Lagrangian correspondences to the concept of cobordism between suitable cell complexes. More precisely let \(\mathrm{Lagr}(V,\omega)=\{L<V\, |\, L=L^\bot\}\) denote the set of Lagrangian subspaces of \((V,\omega)\), where \(E^\bot=\{v\in V\, |\, \omega(v,u)=0,\, \forall u\in E\}\). Let \(\mathcal{Q}_+\) denote the category whose objects are quadratic spaces, namely vector spaces with non-degenerate, symmetric bilinear forms. Set \(W(\mathbb{K})=\pi_0(\mathcal{Q}_+)\). We say that two quadratic spaces \(V_1,\, V_2\in Ob(\mathcal{Q}_+)\), are Witt-equivalent if there exists a Lagrangian correspondence between them, more precisely a morphism \(f\in \mathrm{Hom}_{\mathcal{Q}_+}(V_1,V_2):=\mathrm{Lagr}(V_1^o\oplus V_2)\), called the space of Lagrangian correspondences. Here \((V,q)^o:=(V,-q)\), with \(q\) the quadratic structure. Composition of morphisms is meant in the sense of composition of general correspondences. For example if \(f:V_1\to V_2\) is an isometry then the graph \(\Gamma_f\subset V_1^o\oplus V_2\) is Lagrangian. Then, \(W(\mathbb{K})\) is the group whose elements are Witt equivalence-classes of quadratic spaces, with addition induced by direct sum (The inverse of \((V,q)\) is \(-(V,q)=(V,q)^o\).) Given a \(n\)-tuple \(L=(L_1,\dots,L_n)\) of Lagrangian subspaces of \((V,\omega)\), we have a cochain complex \(C_L\mathop{\to} \limits^{\partial}\bigoplus_iL_i\mathop{\to}\limits^{\Sigma} V\), where \(C_L=\bigoplus(L_i\cap L_{i+1})\), \(\Sigma\) is the sum of the components, and \(\partial(a)=(a,-a)\in L_i\oplus L_{i+1}\), \(\forall a\in L_i\cap L_{i+1}\). Then we get a quadratic space \((T_L,q_L)\), with \(T_L=\ker\sum/\text{im }\partial\) and \(q_L(a,b)=\sum_{i>j}\omega(a_i,b_j)\), (Maslov form), where \(a,\, b\in T_L\) are lifted to the representative \((a_i),\, (b_i)\in \bigoplus_{i} L_i\). Then the Maslov index is \(\tau(L)=\tau(L_1,\dots,L_n)=(T_L,q_L)\in W(\mathbb{K})\). One has the following properties:
(a) Isometries: \[ T(L_1,\dots,L_n)=T(L_n,L_1,\dots,L_{n-1})=T(L_1,\dots,L_1)^o. \]
(b) Lagrangian correspondences: \[ T(L_1,\dots,L_n)\oplus T(L_1,L_k,\dots,L_{n})\to T(L_1,\dots,L_n),\;k<n. \]
By considering the cell complex \(C_L=C(L_1,\dots,L_n)\), as \(n\)-gon, with the face labelled by \(V\), edges labelled by \(L_i\) and vertices labelled by \(L_i\cap L_{i+1}\), property (b) allows us to reduce to the case of three Lagrangian subspaces. Furthermore, Lagrangian correspondences induce cobordism properties. For example \(C(L_1,L_2,L_3,L_4)\) cobords with \(C(L_1,L_2,L_3)\cup C(L_1,L_3,L_4)\).
(c) Cocycle property:
\[ \tau(L_1,L_2,L_3)-\tau(L_1,L_2,L_4)+\tau(L_1,L_3,L_4)-\tau(L_2,L_3,L_4)=0. \]
Furthermore the Maslov index allows to identify a central extension \(\mathrm{Mp}(V)\) of the group \(\mathrm{Sp}(V)\) that when \(\mathbb{K}=\mathbb{R}\) is the unique double cover of \(\mathrm{Sp}(V)\), namely the metaplectic group.
The authors carefully study the Maslov index for weak symplectic Banach spaces in the sense of P. R. Chernoff and J. E. Marsden [Properties of infinite dimensional Hamiltonian systems. York: Springer-Verlag (1974; Zbl 0301.58016)]. More precisely, they extend in this framework the definition of Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces. After a detailed introduction, the paper splits in two more sections and one appendix. 2. Weak symplectic functional analysis. (2.1 Basic symplectic functional analysis; 2.2 Fredholm pairs of Lagrangian subspaces; 2.3 Open topological problems.) 3. Maslov index in weak symplectic analysis. (3.1 Definition and properties of the Maslov index; 3.2 Comparison with real (and strong) category; 3.3 Invariance of the Maslov index under embedding.) Appendix: Spectral flow. (A.1 Gap between subspaces; A.2 Closed linear relations; A.3 Spectral flow for closed linear relations.)
The main result is a rigorous definition of the Maslov index for continuous curves of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with varying weak symplectic structures and continuously varying symplectic splittings. Part of their results are formulated and proved for relations instead of operators to admit wider application. The strategy adopted by the authors follows Floer’s suggestion to express the spectral flow of a curve of self-adjoint operators by the Maslov index of corresponding curves of Lagrangian subspaces [A. Floer, Commun. Pure Appl. Math. 41, No. 4, 393–407 (1988; Zbl 0633.58009)]. Notice that this idea stimulated interesting results relating Maslov index and spectral flow, see, e.g., [T. Yoshida, Ann. Math. (2) 134, No. 2, 277–323 (1991; Zbl 0748.57002); L. I. Nicolaescu, Duke Math. J. 80, No. 2, 485–533 (1995; Zbl 0849.58064); S. E. Cappell et al., Commun. Pure Appl. Math. 49, No. 9, 869–909 (1996; Zbl 0871.58081)].
Reviewer’s remark: It is worth to emphasize that the definition of Maslov index can be recast in the framework of the PDE’s geometry. In fact the metasymplectic structure of the Cartan distribution of \(k\)-jet-spaces \(J^k_n(W)\) over a fiber bundle \(\pi:W\to M\), \(\dim W=n+m\), \(\dim M=n\), allows us to recognize a Maslov index associated to maximal isotropic subspaces of the Cartan distribution of \(J^k_n(W)\), and by restriction on any PDE \(E_k\subset J^k_n(W)\). For details on the metasymplectic structure of the Cartan distribution and its relations with (singular) solutions of PDEs, see the following work by the reviewer: [A. Prástaro, Acta Appl. Math. 59, No. 2, 111–201 (1999; Zbl 0949.35011)].

MSC:

53D12 Lagrangian submanifolds; Maslov index
58J30 Spectral flows
46C99 Inner product spaces and their generalizations, Hilbert spaces
47A53 (Semi-) Fredholm operators; index theories
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ambrose, W.: The index theorem in Riemannian geometry. Ann. Math. 73, 49-86 (1961) · Zbl 0104.16401 · doi:10.2307/1970282
[2] Arnold V.I.: Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics), vol. 60 (1978); Title of the Russian Original Edition: Matematicheskie Metody Klassicheskoǐ Mekhaniki, Nauka, Moscow. Springer, New York (1974) · Zbl 0932.37063
[3] Atiyah M.F.: Circular symmetry and stationary-phase approximation. In: Colloquium in Honour of Laurent Schwartz, Vol. 2, pp. 43-60. Astérisque (1985); reprinted in Atiyah M.F., Collected Works, vol. 5, pp. 667-685. Oxford University Press, Oxford (2005) · Zbl 0578.58039
[4] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry, III. Math. Proc. Cambridge Philos. Soc. 79, 71-99 (1976) · Zbl 0325.58015 · doi:10.1017/S0305004100052105
[5] Bennewitz, C.: Symmetric relations on a Hilbert space. In: Everitt, W.N., Sleeman, B.D. (eds.) Conference on the Theory of Ordinary and Partial Differential Equations (Dundee, Scotland, 1972), Lecture Notes in Mathematics, vol. 280, pp. 212-218. Springer, Berlin (1972) · Zbl 0241.47022
[6] Booß-Bavnbek, B., Chen, G., Lesch, M., Zhu, C.: Perturbation of sectorial projections of elliptic pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3(1), 49-79 (2012); arXiv:1101.0067 [math.SP] · Zbl 1257.58017
[7] Booss-Bavnbek, B., Furutani, K.: The Maslov index: a functional analytical definition and the spectral flow formula. Tokyo J. Math. 21, 1-34 (1998) · Zbl 0932.37063 · doi:10.3836/tjm/1270041982
[8] Booß-Bavnbek, B., Furutani, K.: Symplectic functional analysis and spectral invariants. In: Booß-Bavnbek, B., Wojciechowski, K.P. (eds.) Geometric Aspects of Partial Differential Equations, American Mathematical Society Series Contemporary Mathematics, vol. 242, pp. 53-83. American Mathematical Society, Providence (1999) · Zbl 0942.58030
[9] Booß-Bavnbek, B., Furutani, K., Otsuki, N.: Criss-cross reduction of the Maslov index and a proof of the Yoshida-Nicolaescu theorem. Tokyo J. Math. 24, 113-128 (2001) · Zbl 1038.53072
[10] Booß-Bavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Canad. J. Math. 57/2, 225-250 (2005); arXiv: math.FA/0108014 · Zbl 1085.58018
[11] Booß-Bavnbek, B., Wojciechowski, K.P.: Elliptic Boundary Problems for Dirac Operators. Birkhäuser, Boston (1993) · Zbl 0797.58004
[12] Booß-Bavnbek, B., Zhu, C.: Weak symplectic functional analysis and general spectral flow formula (2004); arXiv:math.DG/0406139 · Zbl 1307.53063
[13] Booß-Bavnbek, B., Zhu, C.: General spectral flow formula for fixed maximal domain. Cent. Eur. J. Math. 3(3), 558-577 (2005) · Zbl 1108.58022
[14] Booß-Bavnbek B., Zhu C.: Symplectic reduction and general spectral flow formula, In preparation · Zbl 1108.58022
[15] Brown, B.M., Grubb, G., Wood, I.G.: \[M\]-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr. 282(3), 314-347 (2009) · Zbl 1167.47057 · doi:10.1002/mana.200810740
[16] Brüning J., Lesch M.: On boundary value problems for Dirac type operators. I. Regularity and self-adjointness. J. Funct. Anal. 185, 1-62 (2001); arXiv:math.FA/9905181 · Zbl 1023.58013
[17] Cappell, S.E., Lee, R., Miller, E.Y.: On the Maslov index. Comm. Pure Appl. Math. 47, 121-186 (1994) · Zbl 0805.58022 · doi:10.1002/cpa.3160470202
[18] Cappell, S.E., Lee, R., Miller, E.Y.: Selfadjoint elliptic operators and manifold decompositions, part II: Spectral flow and Maslov index. Comm. Pure Appl. Math. 49, 869-909 (1996) · Zbl 0871.58081 · doi:10.1002/(SICI)1097-0312(199609)49:9<869::AID-CPA1>3.0.CO;2-5
[19] Chernoff, P.R., Marsden, J.E.: Properties of Infinite Dimensional Hamiltonian Systems, LNM 425. Springer, Berlin (1974) · Zbl 0301.58016
[20] Cross, R.: Multivalued Linear Operators. Dekker Inc., New York (1998) · Zbl 0911.47002
[21] Dai, X., Zhang, W.: Higher spectral flow. J. Funct. Anal. 157, 432-469 (1998) · Zbl 0932.37062 · doi:10.1006/jfan.1998.3273
[22] Duistermaat, J.J.: On the Morse index in variational calculus. Adv. Math. 21, 173-195 (1976) · Zbl 0361.49026 · doi:10.1016/0001-8708(76)90074-8
[23] Floer, A.: A relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41, 393-407 (1988) · Zbl 0633.58009 · doi:10.1002/cpa.3160410402
[24] Furutani, K., Otsuki, N.: Maslov index in the infinite dimension and a splitting formula for a spectral flow. Japan. J. Math. 28(2), 215-243 (2002) · Zbl 1022.58004
[25] de Gosson, M.: The Principles of Newtonian and Quantum Mechanics: With a Forword by Basil Hiley. Imperial College/World Scientific Publishing Co., London (2001) · Zbl 1358.70001 · doi:10.1142/9781848161429
[26] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976) · Zbl 0342.47009 · doi:10.1007/978-3-642-66282-9
[27] Kirk, P., Lesch, M.; The \[\eta \]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary. Forum Math. 16, 553-629 (2004); arXiv:math.DG/0012123 · Zbl 1082.58021
[28] Leray J.: Analyse Lagrangiénne et mécanique quantique: Une structure mathématique apparentée aux développements asymptotiques et à l’indice de Maslov. In: Série Math. Pure et Appl. I.R.M.P., Strasbourg (1978); English translation 1981, MIT Press · Zbl 0748.57002
[29] Lesch, M., Malamud, M.: On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J. Differential Equations 189(2), 556-615 (2003) · Zbl 1016.37026 · doi:10.1016/S0022-0396(02)00099-2
[30] Morse M.: The Calculus of Variations in the Large, vol. 18. , A.M.S. College Publications, American Mathematical Society, New York (1934) · Zbl 0011.02802
[31] 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
[32] Nicolaescu, L.: The Maslov index, the spectral flow, and decomposition of manifolds. Duke Math. J. 80, 485-533 (1995) · Zbl 0849.58064 · doi:10.1215/S0012-7094-95-08018-1
[33] Phillips, J.: Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39, 460-467 (1996) · Zbl 0878.19001 · doi:10.4153/CMB-1996-054-4
[34] Piccione, P., Tausk, D.V.: The Maslov index and a generalized Morse index theorem for non-positive definite metrics. C. R. Acad. Sci. Paris Sér. I Math. 331, 385-389 (2000) · Zbl 0980.53095
[35] Piccione, P., Tausk, D.V.: The Morse index theorem in semi-Riemannian Geometry. Topology 41, 1123-1159 (2002); arXiv:math.DG/0011090 · Zbl 1040.53052
[36] Prokhorova M.: The spectral flow for Dirac operators on compact planar domains with local boundary conditions, pp. 33. arXiv:1108.0806v3 [math-ph] · Zbl 1281.58015
[37] Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827-844 (1993) · Zbl 0798.58018 · doi:10.1016/0040-9383(93)90052-W
[38] Schulze, B.-W.: An algebra of boundary value problems not requiring Shapiro-Lopatinskij conditions. J. Funct. Anal. 179, 374-408 (2001) · Zbl 0984.58013
[39] Waterstraat, N.: A \[K\]-theoretic proof of the Morse index theorem in semi-Riemannian geometry. Proc. Amer. Math. Soc. 140(1), 337-349 (2012) · Zbl 1241.58010 · doi:10.1090/S0002-9939-2011-10874-8
[40] Wojciechowski, K.P.: Spectral flow and the general linear conjugation problem. Simon Stevin 59, 59-91 (1985) · Zbl 0577.58029
[41] Yoshida, T.: Floer homology and splittings of manifolds. Ann. Math. 134, 277-323 (1991) · Zbl 0748.57002 · doi:10.2307/2944348
[42] Zhu, C.: Maslov-type index theory and closed characteristics on compact convex hypersurfaces in \[{\mathbb{R}}^{2n} \]. PhD Thesis (in Chinese), Nankai Institute, Tianjin (2000) · Zbl 0104.16401
[43] Zhu C.: The Morse Index Theorem for Regular Lagrangian Systems. Preprint September 2001 (math.DG/0109117) (first version); MPI-Preprint no. 55 (2003) (modified version)
[44] Zhu, C.: A generalized Morse index theorem. In: Booß-Bavnbek, B. et al. (eds.) Analysis, Geometry and Topology of Elliptic Operators, pp. 493-540. World Scientific, London (2006) · Zbl 1121.58013
[45] Zhu, C., Long, Y.: Maslov-type index theory for symplectic paths and spectral flow. Chinese Ann. Math. 20B, 413-424 (1999) · Zbl 0959.58016 · doi:10.1142/S0252959999000485
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.