File size: 3,427 Bytes
fac6777 | 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 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowMatchingColumns
namespace OAI
/-! # A nonsingular normalization of the slow tail
The only vanishing Pochhammer factor in the relevant half-plane is `q`.
Coefficients of `H(q+1,m+1)` give the tail divided by this factor without
division. The resulting recurrence remains meaningful at `q = 0`.
-/
open Filter Topology
namespace DefocusingNLS
noncomputable def normalizedSlowB (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ) : ℂ :=
slowLaguerreCoefficient (q + 1) (m + 1) s (n - 1)
noncomputable def normalizedSlowC (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ) : ℂ :=
slowLaguerreB (q + 1) (m + 1) s n
noncomputable def normalizedSlowCoefficient (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ) : ℂ :=
slowLaguerreCoefficient (q + 1) (m + 1) s (n - 1) -
slowLaguerreCoefficient (q + 1) (m + 1) s n
theorem normalizedSlowB_succ (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ) :
normalizedSlowB q m s (n + 1) =
normalizedSlowB q m s n - normalizedSlowCoefficient q m s n := by
simp only [normalizedSlowB, normalizedSlowCoefficient, Nat.add_sub_cancel]
ring
theorem normalizedSlowC_succ (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ) :
normalizedSlowC q m s (n + 1) = normalizedSlowC q m s n - normalizedSlowB q m s (n + 1) := by
simp only [normalizedSlowC, normalizedSlowB, Nat.add_sub_cancel, slowLaguerreB_succ]
/-- The normalized recurrence holds at the removable parameter `q = 0`. -/
theorem normalizedSlow_recurrence_succ (q : ℂ) (M : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
(q + (n + 1 : ℕ)) * normalizedSlowCoefficient q (M + 1) s (n + 1) =
-(M : ℂ) * normalizedSlowB q (M + 1) s (n + 2) -
s * normalizedSlowC q (M + 1) s (n + 1) := by
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have h0 := slowLaguerre_recurrence (q + 1) (M + 1) s n hq' hsre hsim
have h1 := slowLaguerre_recurrence (q + 1) (M + 1) s (n + 1) hq' hsre hsim
rw [slowLaguerreB_succ, slowLaguerreC_succ] at h1
simp only [normalizedSlowCoefficient, normalizedSlowB, normalizedSlowC,
Nat.add_sub_cancel, show n + 2 - 1 = n + 1 by omega]
push_cast at h0 h1 ⊢
linear_combination h0 - h1
theorem normalizedSlow_recurrence (q : ℂ) (M : ℕ) (s : ℂ) (n : ℕ)
(hn : 1 ≤ n) (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
(q + n) * normalizedSlowCoefficient q (M + 1) s n =
-(M : ℂ) * normalizedSlowB q (M + 1) s (n + 1) -
s * normalizedSlowC q (M + 1) s n := by
obtain ⟨j, rfl⟩ := Nat.exists_eq_add_of_le' hn
simpa only [add_comm, add_left_comm, add_assoc] using
normalizedSlow_recurrence_succ q M s j hq hsre hsim
theorem normalizedSlowB_tendsto_zero (q : ℂ) (m : ℕ) (s : ℂ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
Tendsto (normalizedSlowB q m s) atTop (𝓝 0) := by
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
exact (shiftedSlowDerivative_laguerre_tendsto_zero 0 (q + 1) (m + 1) s hq' hsre hsim).comp
(tendsto_sub_atTop_nat 1)
theorem normalizedSlowC_tendsto_zero (q : ℂ) (m : ℕ) (s : ℂ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
Tendsto (normalizedSlowC q m s) atTop (𝓝 0) := by
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
exact slowLaguerreB_tendsto_zero (q + 1) (m + 1) s hq' hsre hsim
end DefocusingNLS
end OAI
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