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

⎛ ⎞ � (a + b)X bY where [(∂X ⊗ I2)(V )] = ⎝ ⎠. To see this, let R(λ)Φ = bY (a − b)X compute [∂X ⊗ I2, R(λ)]: {(∂X ⊗ I2)R(λ) − R(λ)(∂X ⊗ I2)} Φ = (∂X ⊗ I2) = = = � � � ∂X ψ1 ∂X ψ2 ∂X ψ1 ∂X ψ2 ∂X ψ1 ∂X ψ2 � � � � ψ1 ψ2 � − R(λ) − R(λ) − R(λ) − R(λ)(∂X ⊗ I2)(H2 − λ) � � � � = − R(λ)[(∂X ⊗ I2)(V )] ∂X 0 0 ∂X � � � (− 1 2 ∆ − λ)∂X ψ1 (− 1 2 ∆ − λ)∂X ψ2 (H2 − λ) � ψ1 ψ2 = − R(λ)[(∂X ⊗ I2)(V )]R(λ)Φ. Clearly (4.11) holds with X replaced by Y . � � ∂X ψ1 ∂X ψ2 � ψ1 ψ2 � (− 1 2 ∆ − λ)ψ1 (− 1 2 ∆ − λ)ψ2 + � � � + � � + � ψ1 ψ2 V11 ψ1 + V12 ψ2 V21 ψ1 + V22 ψ2 ∂X(V11)ψ1 + ∂X(V12)ψ2 ∂X(V21)ψ1 + ∂X(V22)ψ2 V11 ∂X(ψ1) + V12 ∂X(ψ2) V21 ∂X(ψ1) + V22 ∂X(ψ2) + [(∂X ⊗ I2)(V )] � ψ1 ψ2 �� � � � � and we From the hypotheses on Ψ and Corollary 4.5, we know that for all γ > 0, R(λ)(∂X ⊗ I2)Ψ ∈ D(e γ〈x〉 ⊗ I2) ⊂ L 2 (IR 2 ) ⊕ L 2 (IR 2 ). From Corollary 4.5 and the note above, we know that R(λ)[(∂X ⊗ I2)(V )]R(λ)Ψ ∈ D(e γ〈x〉 ⊗ I2) ⊂ L 2 (IR 2 ) ⊕ L 2 (IR 2 ) for all γ > 0. From this we see that (4.11) holds when applied to Ψ and therefore (∂X ⊗ I2)R(λ)Φ ∈ D(e γ〈x〉 ⊗ I2) for all γ > 0. Similarly, (∂Y ⊗ I2)R(λ)Φ ∈ D(e γ〈x〉 ⊗ I2) for all γ > 0. By apply**in**g (4.11) repeatedly, we see that (D α ⊗ I2)R(λ)Ψ is a l**in**ear comb**in**ation **of** terms **of** the form R(λ)[(D α1 ⊗ I2)(V )]R(λ)[(D α2 ⊗ I2)(V )] · · · R(λ)[(D αm−1 ⊗ I2)(V )]R(λ)(D αm ⊗ I2)Ψ, where � m j=1 |αj | = |α |. S**in**ce the [(D αj ⊗ I2)(V )] are matrices with polynomial entries, we use Corollary 4.5 and the note above to obta**in** (D α ⊗ I2)R(λ)Ψ ∈ D(e γ〈x〉 ⊗ I2) for all α ∈ N 2 and γ > 0. The conclusion **in**volv**in**g (D α ⊗ I2)r(E)Ψ follows by writ**in**g the reduced resolvent **in** the 30 � �

form (see Theorem XII.5 **in** [19]) � r(E) = 1 2πi � |λ−E|= Γ >0 R(λ) 1 dλ . λ − E Theorem 4.9. For k ≥ 2, let Ψ (k−2) � f (k−2) = g (k−2) � , and ψ (k) ⊥ be determ**in**ed by the perturbation formulas **of** chapter 3. Then, f (k−2) , g (k−2) , ψ (k) ⊥ ∈ C∞ (IR 2 ), f (k−2) , g (k−2) �� �� , ��ψ (k) �� �� ⊥ �� ∈ L el 2 (IR 2 ) and f (k−2) , g (k−2) �� �� , ��ψ (k) �� �� ⊥ �� ∈ D(e el γ〈x〉 ), for any γ > 0. In addition, Dα f (k−2) , Dα g (k−2) , �� �� ��Dα ψ (k) �� �� �� ∈ D(e el γ〈x〉 ) for all α ∈ N2 and any γ > 0. Pro**of**: ⊥ We refer to a function **in** D(e γ〈x〉 ) (or D(e γ〈x〉 ⊗ I2)) for any γ > 0, as exponentially decay**in**g with arbitrary γ. We first note that from the pro**of** **of** lemma 4.8, multiplication by polynomials **in** X and Y preserves exponential decay with arbitrary γ. S**in**ce Ψ (0) � f = (0) g (0) � is determ**in**ed at second order **in** ǫ as an eigenfunction **of** H2, we already know from propositions 4.4 and 4.7 that Ψ (0) satisfies the conclusion. The f (1) and g (1) given by equation (3.20) are determ**in**ed by H3 followed by a projection Q⊥, and reduced resolvent H2 − E (2) , act**in**g on Ψ (0) . By corollary 4.5 we know that the reduced resolvent preserves exponential decay with arbitrary γ. The projection Q⊥ was the projection **in** H onto the subspace perpendicular to the eigenspace **of** the eigenvalue E (2) **of** H2. From proposition 4.4, we know that the eigenvectors **of** H2 have exponential decay with arbitrary γ, and so it follows that Q⊥ will preserve exponential decay with arbitrary γ. S**in**ce the matrix entries **of** H3 only conta**in** polynomials and derivatives **in** X and Y , we know from lemma 4.8 that (H3 Ψ (0) ) will have exponential decay with arbitrary γ. It follows that Ψ (1) will have exponential decay with arbitrary γ. By a similar argument, Ψ (1) ∈ C ∞ ⊕ C ∞ . From the def**in**ition **of** Q⊥ along with proposition 4.7, we see that all **of** the derivatives **of** Q⊥ H3 Ψ (0) are exponentially decay**in**g with arbitrary γ. It then follows from lemma 4.8 that all **of** the derivatives **of** Ψ (1) are exponentially decay**in**g with arbitrary γ. + Recall that ψ (0) ⊥ �� �� �� �� �� �� �� �� �� �� �� �� � �ψ (3) ⊥ �� �� �� ≤ el � (hP⊥) (0)� −1 r � (hP⊥) (0)� −1 r = ψ(1) ⊥ P (0) ⊥ P (0) ⊥ = ψ(2) ⊥ � ∂Ψ1 ∂x � ∂Ψ2 ∂x = 0. From equation (3.21) we know that � �� (0) ���� � � � � el � �� (0) ���� � � � � el � � � ∂f � � (0) � � � � ∂X � + �� �� �� �� �� � � � ∂f � � (0) � � � � ∂X � + �� �� �� �� �� 31 � (hP⊥) (0)� −1 r � (hP⊥) (0)� −1 r P (0) ⊥ P (0) ⊥ � ∂Ψ1 ∂y � ∂Ψ2 ∂y � �� (0) ���� � � � � el � �� (0) ���� � � � � el � � � � � � � � � � ∂f (0) ∂Y ∂f (0) ∂Y � � � � � � � � � � .

- Page 1 and 2: A Born-Oppenheimer Expansion in a N
- Page 3 and 4: there exist quasimode energies of t
- Page 5 and 6: to both electronic states of the R-
- Page 7 and 8: 3 The Construction of the Quasimode
- Page 9 and 10: with eigenvalues ˜ E±(x, y) = ±
- Page 11 and 12: f ↔ g, Ψ1 ↔ Ψ2, h11 ↔ h22,
- Page 13 and 14: So we get E (1) = 0 and ψ (1) ⊥
- Page 15 and 16: and ⎛ Using what we know through
- Page 17 and 18: and ⎛ ⎝ f(k−2) g (k−2) ⎞
- Page 19 and 20: This easily follows from � �
- Page 21 and 22: H ± 1 = W −1 H ± 0 W = ⎛ ⎜
- Page 23 and 24: so that H2 = H0 + V . Note that for
- Page 25 and 26: Let β > 0. Then, � e 2γ〈x〉
- Page 27 and 28: � Proposition 4.7. Let Ψ = ≥ |
- Page 29: � = IR n \BS(t) � ≥ IR n \BS(
- Page 33 and 34: Let r = a1/4 ρ, ˜b = b , and a HU
- Page 35 and 36: Denote the eigenfunctions of H [±|
- Page 37 and 38: for the larger values of |l|. The s
- Page 39 and 40: 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2
- Page 41 and 42: 1. For l �= 0, each +|l| state ge
- Page 43 and 44: Φ A ǫ is also asymptotic to an ei
- Page 45 and 46: We have (H(ǫ) − Eǫ,K)Φǫ,K = F
- Page 47 and 48: Acknowledgement It is a pleasure to