Revisit of Maslov Type Index for Symplectic Paths

Chungen Liu
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Chungen Liu: Guangzhou University, School of Mathematical and Information Science

Chapter Chapter 7 in Index theory in nonlinear analysis, 2019, pp 177-218 from Springer

Abstract: Abstract We recall that ( ℝ 2 n , ω ~ ) $$({\mathbb R}^{2n}, \tilde \omega )$$ is a symplectic space, and Sp ( 2 n , ω ~ ) $$\mathrm {Sp}(2n,\tilde \omega )$$ is the symplectic group of ( ℝ 2 n , ω ~ ) $$({\mathbb R}^{2n}, \tilde \omega )$$ . That is Sp ( 2 n , ω ~ ) = { M ∈ ℒ ( ℝ 2 n ) | M ∗ ω ~ = ω ~ } . $$\displaystyle \mathrm {Sp}(2n,\tilde \omega )=\{M\in {\mathcal L}({\mathbb R}^{2n})|\,M^*\tilde \omega =\tilde \omega \}. $$ We denote by P ( 2 n , ω ~ ) = { γ ∈ C ( [ 0 , 1 ] , Sp ( 2 n , ω ~ ) ) | γ ( 0 ) = I } $$\mathcal P(2n,\tilde \omega )=\{\gamma \in C([0,1], \mathrm {Sp}(2n,\tilde \omega ))|\; \gamma (0)=I\}$$ the set of continuous and piecewise smooth symplectic paths starting from I and Λ ( n , ω ~ ) $$\Lambda (n,\tilde \omega )$$ the set of Lagrangian subspaces of ( ℝ 2 n , ω ~ ) $$({\mathbb R}^{2n},\tilde \omega )$$ . We recall that P ( 2 n ) = P ( 2 n , ω ~ 0 ) $$\mathcal {P}(2n)=\mathcal {P}(2n,\tilde \omega _0)$$ . We also denote by P ~ ( 2 n , ω ~ ) = { γ | γ ∈ C ( [ a , b ] , Sp ( 2 n , ω ~ ) ) } $$\tilde {\mathcal P}(2n,\tilde \omega )=\{\gamma |\;\gamma \in C([a,b], \mathrm {Sp}(2n,\tilde \omega ))\}$$ the set of continuous and piecewise smooth symplectic paths.

Date: 2019
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DOI: 10.1007/978-981-13-7287-2_7

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