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

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

and ∞� k=2 + + + =

and ∞� k=2 + + + = ǫ k ∞� k=4 ∞� k=3 ∞� k=3 ∞� k=0 k� j=2 ǫ k ǫ k ǫ k ǫ k � − 1 � 2 k� j=4 k� j=3 k� j=3 k� j=0 j� l=4 j� l=3 j� l=3 P (j−2) � ⊥ ∆X, Y ψ (k−j) � ⊥ � − 1 �� 2 � � − P (j−l) ⊥ − P (j−l) ⊥ + ∞� k=0 ǫ k P (j−l) � � (l−4) ⊥ (∆x, y Ψ1) + P (j−l) � � (l−4) ⊥ (∆x, y Ψ2) � �∂Ψ1 � � (l−3) − P (j−l) ⊥ ∂x ∂ ∂X ��∂Ψ1 � � (l−3) ∂y � �∂Ψ2 � � (l−3) − P (j−l) ⊥ ∂x ∂ ∂X ��∂Ψ2 � � (l−3) ∂y k� j=0 f (k−j) � (k−j) g ∂ ∂Y ∂ ∂Y � � (hP⊥) (j) ψ (k−j) ⊥ f (k−j) g (k−j) E (j) ψ (k−j) ⊥ . (3.15) We now collect terms at each order of ǫ. Recall there is an equation along Ψ2 analogous to (3.14). At each order, we will combine these two similar equations into one matrix equation. Order 0 The ǫ0 terms require ⎛ ⎝ h(0) 11 h (0) 12 h (0) 21 h (0) 22 ⎞ ⎠ ⎛ ⎝ f(0) g (0) ⎞ ⎠ = E (0) ⎛ ⎝ f(0) ( hP⊥) (0) ψ (0) ⊥ = E (0) ψ (0) ⊥ g (0) ⎞ ⎠, (3.16) . (3.17) The hij(x, y) vanish until second order, so this forces E (0) = 0 in (3.16), and consequently ψ (0) ⊥ after applying the reduced resolvent [( h(x,y)P⊥(x,y)) (0) ] −1 r in (3.17). Order 1 As above, the ǫ1 terms reduce to ⎛ E (1) ⎝ f(0) g (0) ⎞ ⎠ = 0, (hP⊥) (0) ψ (1) ⊥ = 0. 12 = 0

So we get E (1) = 0 and ψ (1) ⊥ = 0. Order 2 Using the known second order terms for the hij(x,y), the ǫ2 terms require ⎛ ⎞ ⎛ ⎞ where H2 = ⎛ ⎜ ⎝ − 1 2 ∆X, Y + H2 ⎝ f(0) g (0) ⎠ = E (2) (hP⊥) (0) ψ (2) ⊥ = 0, a + b 2 X2 + a − b 2 Y 2 bX Y − 1 ⎝ f(0) g (0) ⎠ , bX Y a − b a + b 2 Y 2 ∆X, Y + 2 X2 + 2 Recall we have assumed the +bxy case for the off diagonal entries. By again applying the reduced resolvent in the last equation we have ψ (2) ⊥ = 0. In chapter 4 we show that H2 is selfadjoint (on the correct domain) and has purely discrete spectrum with infinitely many eigenvalues for a > b > 0. We are only able to solve for some of them exactly. In chapter 5 we show that there is at most a two-fold degeneracy in the eigenstates of H2, but that no splitting occurs in the quasimode eigenvalues, i.e., the degeneracy remains to all orders of ǫ. We can therefore proceed as if the eigenstates of H2 were non-degenerate, since we can take any linear combination of degenerate states for f (0) and g (0) , and we know it will lead to a valid quasimode and energy E(ǫ). Fix E (2) , f (0) and g (0) corresponding to one of the states of H2. Order 3 The ǫ3 terms require ⎛ H3 ⎝ f(0) g (0) ⎞ ⎛ � � ⎠ (2) + H2 − E ⎝ f(1) g (1) ⎞ ⎠ = E (3) ⎛ (hP⊥) (0) ψ (3) ⊥ where H3 = ⎛ + = � � P (0) ⊥ ⎝ h(3) 11 h (3) 12 h (3) 21 h (3) 22 P (0) ⊥ ��∂Ψ1 � � (0) ∂x � �∂Ψ2 � � (0) ⎞ ∂x ⎠ + ⎛ ⎜ ⎝ ∂ ∂X − A (0) 21 ∂ ∂X ∂ ∂X + P (0) ⊥ + P (0) ⊥ ⎝ f(0) g (0) ⎞ ��∂Ψ1 � � (0) ∂y ��∂Ψ2 � � (0) ∂y 0 − A (0) 12 − B(0) 21 ⎞ ⎟ ⎠ . ⎠ , (3.18) ∂ ∂Y ∂ ∂X ∂ ∂Y � � f (0) g (0) , (3.19) − B(0) 12 Since H2 is self-adjoint, we can take inner products of both sides in (3.18) with obtain E (3) = � � f (0) g (0) � 13 , H3 ∂ ∂Y � f (0) g (0) � � H . 0 ∂ ∂Y ⎞ ⎟ ⎠. � f (0) g (0) � to

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