Evans Function, Parity and Nonautonomous Bifurcations

The paper’s Theorem 3.6 rigorously proves that, under (H0)–(H2) with m+=m− and E(a)E(b)≠0, the parity of the Fredholm path T(λ)=D1G(0,λ) on both (W^{1,∞},L^{∞}) and (W^{1,∞}_0,L^{∞}_0) equals sgn E(a)·sgn E(b). The proof establishes Fredholmness and the index via exponential dichotomies (Thm. 2.6), identifies endpoint invertibility from the Evans function (Prop. 3.4 and Thm. 2.4), and reduces parity to a finite-dimensional determinant equal to E up to a fixed sign, yielding the stated formula (Thm. 3.6) . The candidate’s solution captures the main architecture (dichotomy roughness, index formula, endpoint invertibility, finite-rank reduction) and arrives at the correct parity identity. However, it contains critical flaws in Step 5: it posits a bounded right inverse Sλ with L∘Sλ=IY and a bounded right-inverse of L|Vλ from Y onto a finite-dimensional space Vλ—neither can hold in general at singular points. It also asserts degLS(P(λ)T(λ))=sgn E(λ) without addressing the fixed orientation factor (−1)^{d−m+} that the paper handles carefully; although this factor cancels in the parity product, the candidate’s derivation of the endpoint degrees is not correct as written. Thus, while the final conclusion matches the paper, the provided proof contains nontrivial gaps and misstatements.