Crossing Condition

If λ is a simple eigenvalue of a matrix A(α) which depends on parameter α, then on applying Jacobi’s formula for derivative of determinant of a matrix,

we get dλ dα = uT dA dαv uT_v

where uT _{is row eigenvector and v is column eigenvector of matrix A corre-}

sponding to λ. Applying this to the matrix E(κ) + D(κ)e−λr _{we get}

dλ dκ = uT₍dE dκ + e −λr dD dκ)v uT_{(I + re}−λr_D)v

evaluated at λ = iωc, κ = κc. Here uT is the first row of Ψ(0) and v is the

first column of Φ(0). The real part of dλ dκ is δc.

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