The L-Index Theory

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

Chapter Chapter 5 in Index theory in nonlinear analysis, 2019, pp 95-159 from Springer

Abstract: Abstract In this subsection for L ∈ Λ(n) we will define an index pair for any symplectic path γ ∈ P τ ( 2 n ) $$\gamma \in \mathbb {P}_{\tau }(2n)$$ with L-boundary condition(L-index for short). Comparing with the Maslov-type index theory for periodic boundary condition which is suitable to be used in the study of periodic solution of a Hamiltonian systems, the L-index theory is suitable to be used in the study of Hamiltonian systems with L-boundary condition. Firstly, we consider a special case. Suppose L = L 0 = { 0 } ⊕ ℝ n ⊂ ℝ 2 n $$L=L_0=\{0\}\oplus {\mathbb R}^n\subset \mathbb R^{2n}$$ which is a Lagrangian subspace of the linear symplectic space ( ℝ 2 n , ω 0 ) $$(\mathbb R^{2n}, \omega _0)$$ .

Date: 2019
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DOI: 10.1007/978-981-13-7287-2_5

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