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

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

In the appendix we argue

In the appendix we argue that all of the odd order E (k) are zero. Let Q⊥ be the projection in H onto the subspace perpendicular to the eigenspace of the eigenvalue E (2) of H2. Adopting “intermediate normalization” we may choose the non-zero order wave functions perpendicular to the eigenspace of E (2) (note that this will produce a non-normalized quasimode), so that � f (k) g (k) � � f = Q⊥ (k) g (k) � , for k ≥ 1. Then from (3.18) we get ⎛ ⎞ From (3.19) we have ψ (3) ⊥ = ⎝ f(1) g (1) � ( hP⊥) (0)� −1 + � P (0) ⊥ r ⎠ = − � � P (0) ⊥ ��∂Ψ2 � � (0) ∂x � H2 − E (2)� −1 ��∂Ψ1 � � (0) ∂x ∂ ∂X + P (0) ⊥ Order 4 The ǫ4 terms require ⎛ � H2 − E (2)� ⎝ f(2) g (2) ⎞ ⎛ � ⎠ + H3 − E (3)� ⎝ f(1) g (1) where ( h P⊥) (0) ψ (4) ⊥ = − (hP⊥) (1) ψ (3) ⊥ + 1 � 2 + + 4� j=3 4� j=3 P (0) � � (0) ⊥ (∆x, y Ψ1) j� l=3 j� l=3 H4 = � � P (j−l) ⊥ P (j−l) ⊥ � − 1 � 2 ⎛ ⎝ ⎛ ⎜ ⎝ ⎞ r Q⊥ H3 ∂ ∂X � + P (0) ⊥ ��∂Ψ2 � � (0) ⎠ + ∂y f (0) + P (0) � � (0) ⊥ (∆x,y Ψ2) � �∂Ψ1 � � (l−3) ∂x � �∂Ψ2 � � (l−3) ∂x (0) T 11 T (0) 12 T (0) 21 T (0) 22 − A (1) 21 ∂ ∂X ⎞ ⎠ + ∂ ∂X ∂ ∂X ⎛ f (0) g (0) � ��∂Ψ1 � � (0) ∂y ∂ ∂Y � ⎛ � H4 − E (4)� ⎝ f(0) g (0) + P (j−l) ⊥ + P (j−l) ⊥ ⎝ h(4) 11 h (4) 12 h (4) 21 h (4) 22 0 − A (1) 12 − B(1) 21 14 ∂ ∂Y � (0) g . (3.20) g (0) � �∂Ψ1 � � (l−3) ∂y � �∂Ψ2 � � (l−3) ⎞ ∂y ⎠ + ∂ ∂X 0 − B(1) 12 ⎞ � ∂ ∂Y � f (0) . (3.21) ⎠ = 0, (3.22) ∂ ∂Y ∂ ∂Y ∂ ∂Y � � ⎞ ⎟ ⎠. f (4−j) g (4−j) , (3.23)

and ⎛ Using what we know through order 3, we can solve (3.22) and (3.23). From (3.22) we obtain: E (4) = � ⎛ ⎝ f(0) g (0) ⎞ ⎛ � ⎠ , H3 − E (3)� ⎝ f(1) g (1) ⎞ � ⎠ � + ⎛ ⎝ f(0) g (0) ⎞ ⎠ , H4 ⎛ ⎝ f(0) g (0) ⎞ � ⎠ ⎝ f(2) g (2) ⎞ ⎠ = − From (3.23) we get ψ (4) ⊥ = + 1 � 2 + + 4� j=3 4� j=3 � H2 − E (2)� −1 � ( h P⊥) (0)� −1 r � P (0) � � (0) ⊥ (∆x, y Ψ1) j� l=3 j� l=3 � � P (j−l) ⊥ P (j−l) ⊥ r Q⊥ H ⎡ ⎣ � ⎛ H3 − E (3)� ⎝ f(1) g (1) − (hP⊥) (1) ψ (3) ⊥ f (0) + P (0) � � (0) ⊥ (∆x, y Ψ2) ��∂Ψ1 � � (l−3) ∂x ��∂Ψ2 � � (l−3) ∂x ∂ ∂X ∂ ∂X + P (j−l) ⊥ + P (j−l) ⊥ ⎞ ⎠ + H4 � (0) g ⎛ ��∂Ψ1 � � (l−3) ∂y ��∂Ψ2 � � (l−3) ∂y ⎝ f(0) g (0) ∂ ∂Y ∂ ∂Y ⎞ ⎤ ⎠ ⎦ . � � f (4−j) H ⎤ g (4−j) ⎦ . Order k ≥ 5 We now show that we can proceed in this manner to any order of ǫ desired. In chapter 4 we will show that all of the quantities involved exist in the relevant Hilbert space. If k ≥ 5, the ǫk terms require ⎛ � H2 − E (2)� ⎝ f(k−2) g (k−2) ⎞ ⎠ + + ⎛ � Hk − E (k)� ⎝ f(0) g (0) ⎞ ⎠ + ⎛ � H3 − E (3)� ⎝ f(k−3) g (k−3) �k−3 j=2 � − 1 � 2 ⎛ ⎝ ⎞ ⎠ + �k−1 j=4 〈 Ψ(j−2) 1 , ∆X, Y ψ (k−j) ⊥ 〉el 〈 Ψ (j−2) 2 , ∆X, Y ψ (k−j) ⊥ 〉el 15 ⎛ � Hj − E (j)� ⎝ f(k−j) g (k−j) ⎞ ⎞ ⎠ ⎠ = 0, (3.24)

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