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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralNoTurnWeight
namespace OAI
/-! The full boundary and coupling error tends to zero as the remote
radius and the common scalar weight floor escape to infinity. -/
open Filter Topology
namespace DefocusingNLS
theorem spectralBoundaryError_tendsto (E kap : ℕ → ℝ)
(hE : Tendsto E atTop atTop) (hkap : Tendsto kap atTop atTop)
(A B C R : ℝ) :
Tendsto (fun n => (A+1)*(B/E n+C/R)/(kap n)^2) atTop (𝓝 0) := by
have hb : Tendsto (fun n => B/E n+C/R) atTop (𝓝 (C/R)) := by
simpa only [mul_zero,zero_add,div_eq_mul_inv,Function.comp_def] using
((tendsto_inv_atTop_zero.comp hE).const_mul B).add_const (C/R)
have hk : Tendsto (fun n => ((kap n)^2)⁻¹) atTop (𝓝 0) := by
simpa only [Function.comp_apply,inv_pow,zero_pow (by norm_num : 2 ≠ 0)] using
(tendsto_inv_atTop_zero.comp hkap).pow 2
simpa only [div_eq_mul_inv,mul_zero] using (hb.const_mul (A+1)).mul hk
end DefocusingNLS
end OAI
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