Annales Henri Poincaré - Volume 11 by Vincent Rivasseau (Chief Editor)

By Vincent Rivasseau (Chief Editor)

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To see 2 [(V11 − E) f + V12 g] [V21 f + (V22 − E) g] (V11 − E) f + eγ V21 f + eγ x V12 g (V22 − E) g x 2 2 2 2 Let β > 0. Then, e2γ x 2 −2β |V21 f |2 dX dY ≤ V21 e x ∞ e2 So, eγ x V21 f ∈ L2 (R2 ) and by similar arguments eγ eγ x (V22 − E)g ∈ L2 (R2 ). Hence, eγ eγ x x Δf Δg x (γ+β) x f 2 2 <∞ (V11 − E)f, eγ x V12 g, 2 <∞ and Δf, Δg ∈ D(eγ x ). 4 of [12]: Let p ∈ C 1 (RN ) and ≤ 2C ∀ x ∈ RN . If RN (|f |2 + |Δf |2 )pdx < suppose for some C < ∞, ∇p(x) p(x) ∞, then ⎞1/2 ⎛ |∇f |2 pdx⎠ ⎝ ⎛ ⎞1/2 ≤C⎝ RN |f |2 p dx⎠ RN ⎡⎛ ⎢ + ⎣⎝ ⎞1/2 ⎛ |f |2 p dx⎠ RN ⎝ |Δf |2 p dx⎠ RN ⎤1/2 ⎞1/2 + C2 ⎥ |f |2 p dx⎦ .

11 (2010) Born–Oppenheimer Expansion Near a Renner–Teller 33 We adopt the following notation for simplicity: Tij (x, y) = Ψi , Δx,y Ψj el , ∂Ψj , Aij (x, y) = Ψi , ∂x el ∂Ψj Bij (x, y) = Ψi , . ∂y el We have identities involving these quantities since {Ψ1 , Ψ2 } are orthonormal and real valued. For instance we know the diagonal elements of A and B are zero and A12 = −A21 , B12 = −B21 . Now we expand all functions and operators with dependence. For ∞ example, f ( , X, Y ) = k=0 k f (k) (X, Y ). For functions and operators with exclusively (x, y) dependence, we know the form of the expansions.

It follows from Sobolev’s Lemma that f, g ∈ C ∞ on Ω. Since Ω was arbitrary f, g ∈ C ∞ (R2 ). We now show ∇f, ∇g ∈ L2 . We know Ψ ∈ D(H2 ). Let D(−Δ) and Q(−Δ) be the domain of self-adjointness and quadratic form domain of −Δ respectively. Then D(H2 ) ⊂ D(−Δ) ⊕ D(−Δ) ⊂ Q(−Δ) ⊕ Q(−Δ) f = Ψ= : ∇f, ∇g ∈ L2 (R2 ) . 39 of [26]) to prove that f, g ∈ D(eγ|X| ). The argument can be repeated for D(eγ|Y | ), and since Vol. 11 (2010) eγ x Born–Oppenheimer Expansion Near a Renner–Teller ≤ eγ eγ(|X|+|Y |) ≤ eγ e2γ max{|X|, |Y |} ≤ eγ 43 e2γ|X| + e2γ|Y | , we then have f, g ∈ D(eγ x ).

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