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

L TOT z with eigenvalue l TOT z . The Φ0 function is given by Φ0 = Ψ1(0,0)f (0) + Ψ2(0,0)g (0) = Ψ1(0,0) = α + β � α F (0) + β F (0) � � + Ψ2(0,0) α G (0) + β G (0) � � e i(|l|−1)φ f [|l|] (Ψ1(0,0) + iΨ2(0,0)) + e i(|l|+1)φ g [|l|] � (Ψ1(0,0) − iΨ2(0,0)) � e −i(|l|−1)φ f [|l|] (Ψ1(0,0) − iΨ2(0,0)) + e −i(|l|+1)φ g [|l|] � (Ψ1(0,0) + iΨ2(0,0)) . We now plug this **in**to the equation L TOT z Φ0 − l TOT z Φ0 = 0, and for |l| ≥ 2, we project along e i(|l|−1)φ f [|l|] , e i(|l|+1)φ g [|l|] , e −i(|l|−1)φ f [|l|] , and e −i(|l|+1)φ g [|l|] , and obta**in** the follow**in**g four equa- tions: L el z (Ψ1(0,0) + iΨ2(0,0)) = (l TOT z − |l| + 1) (Ψ1(0,0) + iΨ2(0,0)) (6.3) L el z (Ψ1(0,0) − iΨ2(0,0)) = (l TOT z − |l| − 1) (Ψ1(0,0) − iΨ2(0,0)) (6.4) L el z (Ψ1(0,0) − iΨ2(0,0)) = (l TOT z + |l| − 1) (Ψ1(0,0) − iΨ2(0,0)) (6.5) L el z (Ψ1(0,0) + iΨ2(0,0)) = (l TOT z + |l| + 1) (Ψ1(0,0) + iΨ2(0,0)) (6.6) Equations (6.3) and (6.4) hold as long as α �= 0 and equations (6.5) and (6.6) hold as long as β �= 0. By comb**in****in**g (6.3) and (6.6) we obta**in** l TOT z − |l| + 1 = l TOT z + |l| + 1 which contradicts our assumption that |l| �= 0. So, either α = 0 or β = 0. Assume that β = 0 and take α = 1, so that equations (6.3) and (6.4) still hold. S**in**ce LTOT z −lTOT z Φ0 = l TOT z Φ0, we know that L TOT z Φ0 = Φ0. Us**in**g this equation and project**in**g along e −i(|l|−1)φ f [|l|] and e −i(|l|+1)φ g [|l|] , we obta**in** equations similar to (6.5) and (6.6), but with l TOT z replaced by −l TOT z : L el z (Ψ1(0,0) − iΨ2(0,0)) = (−l TOT z + |l| − 1) (Ψ1(0,0) − iΨ2(0,0)) (6.7) L el z (Ψ1(0,0) + iΨ2(0,0)) = (−l TOT z + |l| + 1) (Ψ1(0,0) + iΨ2(0,0)) . (6.8) By comb**in****in**g (6.3) and (6.8) we obta**in** l TOT z = |l| and these equations now reduce to L el z (Ψ1(0,0) − iΨ2(0,0)) = − (Ψ1(0,0) − iΨ2(0,0)) (6.9) L el z (Ψ1(0,0) + iΨ2(0,0)) = Ψ1(0,0) + iΨ2(0,0). (6.10) By repeat**in**g the argument with α = 0, β = 1, we would **in**stead f**in**d l TOT z = −|l|. From this analysis we see that α = 1, β = 0 and α = 0, β = 1 are correct l**in**ear comb**in**ations that will generate two orthogonal quasimodes Φ A ǫ and Φ B ǫ respectively. These quasimodes satisfy L TOT z Φ A ǫ = |l|ΦA ǫ H(ǫ). We note that Φ B 0 and LTOT z ΦB ǫ = −|l|ΦBǫ = ΦA0 and that they are asymptotic to eigenfunctions **of** . S**in**ce H(ǫ) commutes with complex conjugation, we have that 42

Φ A ǫ is also asymptotic to an eigenfunction with the same eigenvalue as Φ A ǫ . S**in**ce quasimodes are determ**in**ed by their zeroth order eigenfunctions through the perturbation formulas **of** chapter 3, this imples that Φ B ǫ = ΦA ǫ s**in**ce ΦB 0 = ΦA 0 . So, the ΦA ǫ and ΦB ǫ correspond to a degenerate pair and we see that no splitt**in**g occurs **in** the perturbation expansion. As a result, any l**in**ear generated by the comb**in**ations comb**in**ation would be a correct one. The ΦA ǫ and ΦBǫ = ΦAǫ α = 1, β = 0 and α = 0, β = 1 respectively, are the quasimodes that satisfy LTOT z ΦA ǫ = |l|ΦA and L ǫ TOT z ΦB ǫ = −|l|ΦB ǫ . If |l| = 1, we take projections **of** L TOT z Φ0 − l TOT z Φ0 = 0 along f [1] , e 2iφ g [1] , and e −2iφ g [|1|] , and obta**in** three equations. By proceed**in**g **in** a similar manner to the analysis **in** the |l| ≥ 2 case above, we would obta**in** l TOT z = 1 if α = 1, β = 0 and l TOT z = −1 if α = 0, β = 1. In either case, we would obta**in** (6.9) and (6.10) and the desired results follow as **in** the |l| ≥ 2 case above. Case 2: |l| = 0 If |l| = 0, we have non-degenerate eigenfunctions **of** H2 **of** the form **in** equations (5.1) or (5.2). In any case, we see that Φ0 is real. Then from L TOT z it is clear that l TOT z Φ0 = l TOT z = 0 **in** this case. By plugg**in**g **in**to L TOT z Φ0 and L TOT z Φ0 = −l TOT z Φ0, Φ0 = 0 and tak**in**g projections along e iφ and e −iφ **in** a manner similar to the |l| �= 0 case, we obta**in** the same relations for the electronic basis vectors at zeroth order given by equations (6.9) and (6.10). � Corollary 6.2. For all ˜ b **in** some **in**terval (0,δ), if ǫ is sufficiently small, the ground state **of** H(ǫ) (correspond**in**g to the R-T pair **of** states we are consider**in**g) is degenerate. Pro**of**: From our perturbation analysis, we have that for all ˜ b **in** some **in**terval (0,δ), the ground state **of** H2 is degenerate, aris**in**g from the l = ±1 states **of** HUZ. The previous theorem tells us that these generate a degenerate pair **of** quasimodes Φ A ǫ and Φ B ǫ that satisfy L TOT z Φ A ǫ = Φ A ǫ , Φ B ǫ = ΦA ǫ and LTOT z Φ B ǫ = − ΦB ǫ . If the quasimode energy lies below the essential spectrum, then this will correspond to the lowest ly**in**g eigenvalue **of** H(ǫ) correspond**in**g to the R-T pair. � Our perturbation calculations suggest that there is a cross**in**g **in**volv**in**g this eigenvalue with the lowest ly**in**g l = 0 eigenvalue, somewhere near ˜ b = 0.925. The ground state seem**in**gly corresponds to these l = ±1 states for 0 < ˜ b < 0.925 and corresponds to the non-degenerate, lowest ly**in**g l = 0 state for 0.925 < ˜ b < 1. We now prove that the ground state **of** H2 cannot arise from any other |l| states. Proposition 6.3. Let 0 < ˜ b < 1. Then the ground state **of** HUZ is either an |l| = 1 state or an l = 0 state. Pro**of**: As previously mentioned, s**in**ce [HUZ,L nuc z ] = 0 we assume Ψ = 43 ⎛ ⎝ e±i|l|φ ψ1(r) e ±i|l|φ ψ2(r) ⎞ ⎠ and then

- 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 and 30: � = IR n \BS(t) � ≥ IR n \BS(
- Page 31 and 32: form (see Theorem XII.5 in [19])
- 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: 1. For l �= 0, each +|l| state ge
- Page 45 and 46: We have (H(ǫ) − Eǫ,K)Φǫ,K = F
- Page 47 and 48: Acknowledgement It is a pleasure to