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Example text

107) Note that wn 2L2 = ω k(λnn ) . By the asymptotic behavior of w(x, λ), v(x, λ) and ω(λ) as λ → ∞, it can be shown that the Green function G(x, y; λ) can be expressed by the Mittlag-Leffler expansion ∞ G(x, y; λ) = wn (x< )vn (x> ) , ω (λn )(λ − λn ) n=0 (108) see Ahlfors (1984). Then, for every f ∈ H we have u(x, λ) = (λ − L)−1 f = ∞ Λ 1 kn wn (x) 0 wn (y)f (y)dy , ω (λn )(λ − λn ) (109) and applying T we get kn wn (x) T (λ − L)−1 f = Λ x 0 f (y)wn (y)dy + wn (x) ω (λn )(λ − λn ) 1 x f (y)vn (y)dy .

Then, recalling that g(λ) has exactly one zero ( λn = nπ) inside Cn , one obtains (18). Now, inserting the eigenvalue asymptotic estimate (18) in estimate (17) we get √ sin λn x 1 , as n → ∞. (20) y(x, λn ) = √ +O n2 λn Recalling that gn (x) = estimate y(x,λn y(x,λn ) L2 gn (x) = √ we obtain the asymptotic eigenfunction 2 sin(nπx) + O 1 n , (21) 34 A. Morassi which holds uniformly on bounded subsets of [0, 1] × L2 (0, 1) as n → ∞. Finally, by iterating the above procedure, the eigenvalue estimate (18) can be improved to obtain λn = (nπ)2 + 3 1 0 q(x)dx − 1 cos(2nπx)q(x)dx + O 0 1 n , as n → ∞.

Morassi which holds uniformly on bounded subsets of [0, 1] × L2 (0, 1) as n → ∞. Finally, by iterating the above procedure, the eigenvalue estimate (18) can be improved to obtain λn = (nπ)2 + 3 1 0 q(x)dx − 1 cos(2nπx)q(x)dx + O 0 1 n , as n → ∞. 1 Symmetric Potential and Dirichlet Boundary Conditions Let us consider the Dirichlet eigenvalue problem y (x) + λy(x) = q(x)y(x), y(0) = 0 = y(1), in (0, 1), (23) (24) where q ∈ L2 (0, 1) is a real-valued potential. Denote by {gn (x, q), λn }∞ n=1 , 1 2 g dx = 1, the eigenpairs of (23)-(24).

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A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson


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