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A Born-Oppenheimer Expansion in a Neighborhood of a Renner ...

A Born-Oppenheimer Expansion in a Neighborhood of a Renner ...

uniformly for t

uniformly for t in compact subsets of IR n . From (4.8) and (4.10) we have � IR n \B S(t) |pj + itj | 2 |pk + itk | 2 | ˆ f(p + it) | 2 dp ≤ ≤ � < ∞ . IR n \B S(t) � �2 � 17 16 IR n \BS(t) � ||p || 2 + ||t || 2 � 2 | ˆ f(p + it) | 2 dp ||p || 4 | ˆ f(p + it) | 2 dp It follows that ∂xj ∂xk f ∈ D(eγ〈x〉 ) for any γ > 0. Again the same will hold for g. We now start an induction on the length |α| in D α f and D α g. Assume that D β f, D β g ∈ D(e γ〈x〉 ) for any γ > 0 and any |β| ≤ m−1. It suffices to prove that D α f ∈ D(e γ〈x〉 ) for any γ > 0 and any |α | = m. Following the notation in the proof of Proposition 4, the eigenvalue equation gives us ∆ f = V12 g + (V11 − E)f , ∆ g = V21 f + (V22 − E)g . where V11, V12 = V21, and V22 are polynomials in xj. Let |α ′ | = m − 2. Since the Vij are polynomial, our induction hypothesis gives us Dα′ ∆ f ∈ D(eγ〈x〉 ) for any γ > 0. It follows that for jk ∈ {1,2, · · · ,n} � IR n \B S(t) |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm−2 and from (4.9) we have � IR n \B S(t) + itjm−2 |2 � � � � � � n� (pj + itj) 2 � � � � � � j=1 2 | ˆ f(p + it) | 2 dp < ∞ , |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm−2 + itjm−2 |2 ||p || 4 | ˆ f(p + it) | 2 dp < ∞ . Since the jk are arbitrary, we have ∞ > n� j1,j2,···,jm−2=1 = n� � j1,j2,···,jm−2=1 IR n \B S(t) � IR n \B S(t) |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm−2 + itjm−2 |2 ||p || 4 | ˆ f(p + it) | 2 dp (p 2 j1 + t 2 j1 )(p 2 j2 + t 2 j2 ) · · · (p 2 jm−2 + t 2 jm−2 ) ||p ||4 | ˆ f(p + it) | 2 dp 28

� = IR n \BS(t) � ≥ IR n \BS(t) � = IR n \BS(t) (||p || 2 + ||t|| 2 ) m−2 ||p || 4 | ˆ f(p + it) | 2 dp ||p || 2(m−2) ||p || 4 | ˆ f(p + it) | 2 dp ||p || 2m | ˆ f(p + it) | 2 dp Then using (4.8), we have for any jk ∈ {1,2, · · · ,n} � ≤ ≤ IR n \BS(t) � < ∞ . IR n \B S(t) � �m � 17 16 IR n \BS(t) |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm + itjm | 2 | ˆ f(p + it) | 2 dp (||p || 2 + ||t|| 2 ) m | ˆ f(p + it) | 2 dp ||p || 2m | ˆ f(p + it) | 2 dp So, for arbitrary jk ∈ {1,2, · · · n}, p ↦→ pj1 pj2 · · · pjm ˆ f(p) is P-W and it follows that D α f ∈ D(e γ〈x〉 ) for any γ > 0 and any |α| = m. The same argument will work with f replaced by g and the proposition is proved. � Lemma 4.8. Let Ψ = � f g � , R(λ) = (H2 − λ) −1 for λ ∈ ρ(H2), and r(E) = (H2 − E) −1 r be the reduced resolvent at E. If f, g ∈ C ∞ and (D α ⊗ I2)Ψ ∈ D(e γ〈x〉 ⊗ I2), for all α ∈ N 2 and any γ > 0, then (D α ⊗ I2)R(λ)Ψ, (D α ⊗ I2)r(E)Ψ ∈ D(e γ〈x〉 ⊗ I2), for all α ∈ N 2 and any γ > 0. Proof: First note that for any γ1 > γ2 > 0 and j,k = 0,1,2, · · · , there exists M > 0 such that �� �� ��e γ2〈x〉 �� �� j k �� �� X Y φ�� ≤ M ||φ || + ��e γ1〈x〉 �� �� φ�� . This relative bound implies that if φ ∈ D(e γ〈x〉 ) for all γ > 0, then X j Y k φ ∈ D(e γ〈x〉 ) for all γ > 0, and arbitrary j,k = 0,1,2, · · · . By an argument similar to the one by which we obtained f, g ∈ C ∞ (IR 2 ) in the proof of Proposition 4.4, R(λ) and r(E) map functions from C ∞ (IR 2 ) ⊕ C ∞ (IR 2 ) to C ∞ (IR 2 ) ⊕ C ∞ (IR 2 ). The following identity holds as long as the terms on the right hand side are in L 2 (IR 2 ) ⊕ ̷L 2 (IR 2 ): (∂X ⊗ I2)R(λ)Φ = R(λ)(∂X ⊗ I2)Φ − R(λ)[(∂X ⊗ I2)(V )]R(λ)Φ, (4.11) 29

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