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

The functions {ei l φ r |l| L |l| K (r2 )e−r2 /2 } form a basis for L l∈Z K=0,1,2,··· 2 (IR 2 ) by theorem XIII.64 **of** [19]. Then for fixed l ∈ Z, the functions {r |l| L |l| K (r2 )e−r2 /2 }K=0,1,2,··· form a basis for the projection **of** L2 (IR 2 ) onto r-dependent multiples **of** ei l φ . We can then use the follow**in**g orthonormal basis for the perturbation expansion: ⎧⎛ ⎨ ⎩ ⎝ BN,|l|−1 r |l|−1 L |l|−1 N (r2 )e−r2 /2 0 ⎞ ⎠, ⎛ ⎝ 0 B N,|l|+1 r |l|+1 L |l|+1 N (r2 )e −r2 /2 ⎞⎫ ⎬ ⎠ ⎭ N=0,1,2,··· where the B N,|l| are constants **of** normalization. The matrix elements **of** the perturbation ˜ V **in** this basis can be obta**in**ed explicitly [9]. 5.2.1 The Non-Degenerate Perturbation Calculation Recall from (5.4), for fixed |l| �= 0, the lowest ly**in**g eigenvalue **of** H [|l|] 0 is E 0,|l| 0 degenerate). S**in**ce H |l| UZ = |l| (non- is an analytic family, we use non-dengenerate, regular perturbation theory. Us**in**g the Mathematica s**of**tware package, we easily computed the exact perturbation coefficients up to 28th order for the non-degenerate, lowest ly**in**g eigenvalue E 0,|l| , for several values **of** |l| (See Figure 3). 8 6 4 2 0.2 0.4 0.6 0.8 1.0 Figure 3: A plot **of** the perturbation series versus ˜b, **of** the non-degenerate, � lowest ly**in**g eigenvalue **of** up to order 28, for |l| = 1, 2, · · · , 8. The dashed curves are (2N + 2) 1 − ˜b, for N = 0, 1, 2, 3, H [|l|] UZ which are the l = 0 states that we have solved for exactly. Recall that we are concerned with the case where 0 < b < a, so that 0 < ˜b = b a < 1. The functions E0,|l| ( ˜b) likely do not exist as eigenvalues **of** H [|l|] UZ if ˜b ≥ 1. Seem**in**gly the radii **of** convergence **of** the series are smaller as |l| **in**creases. It appears that for |l| = 1,2, and 3 the radius **of** convergence is likely close to 1 (if not larger). For |l| > 4, the series are behav**in**g erratically for values **of** ˜ b < 1. The |l| = 4 case appears to be borderl**in**e, with radius **of** convergence possibly only slightly smaller than 1. This divergent behavior was seen even from the low order coefficients 36

for the larger values **of** |l|. The s**in**gularities are likely caused by avoided cross**in**gs between two states with the same value **of** |l|, as suggested **in** [20, 2, 11]. We highlight the cross**in**g between the |l| = 1 state and the lowest ly**in**g l = 0 state near ˜ b = 0.925. Recall that for l �= 0, the eigenvalue E J,|l| **of** H [|l|] UZ is also an eigenvalue **of** H[−|l|] UZ . Together these states correspond to a degenerate eigenvalue **of** the orig**in**al operator H2. The l = 0 states are all non-degenerate for b > 0. So, this cross**in**g implies that the ground state **of** H2 is degenerate for approximately 0 < b < 0.925a and non-degenerate for 0.925a < b < 1. 5.2.2 The Degenerate Perturbation Calculation Recall from (5.4), that only the ground state **of** H [|l|] 0 is non-degenerate if |l| �= 0. In the perturbation calculation described **in** section 5.2.1, we used regular non-degenerate perturbation theory to obta**in** the perturbation coefficients for these eigenvalues. From (5.5), we have that for fixed |l| �= 0, H [|l|] 0 also has two-fold degenerate eigenvalues **of** E N,|l| 0 = 2N + |l| for N = 1,2, · · · So we must use degenerate perturbation theory to calculate the perturbation coefficients **of** these eigenvalues. Recall from (5.5), the degenerate pair **of** eigenfunctions correspond**in**g to E N,|l| 0 given by Ψ N,|l| 0,up and ΨN,|l| 0,dwn . Employ**in**g degenerate perturbation theory **in** the usual manner, we f**in**d there is splitt**in**g that occurs at first order (we omit the details). Armed with the proper l**in**ear comb**in**ations we can then proceed as **in** the non-degenerate case. Us**in**g the Mathematica s**of**tware package, we easily computed the exact perturbation coefficients up to 12th order for the first few eigenvalues E N,|l| that are degenerate at zeroth order, for several values **of** |l| (See Figures 4, 5). 10 8 6 4 2 0.2 0.4 0.6 0.8 1.0 b� Figure 4: A plot **of** the perturbation series versus ˜b up to order 12, **of** the first 8 eigenvalues **of** H [|l|] UZ that � are degenerate at zeroth order, for |l| = 1. The dashed curves are (2N +2) 1 − ˜ � b and (2N +2) 1 + ˜b, for N = 0, 1, 2, 3, 4, which are the l = 0 states that we have solved for exactly. While the splitt**in**g is nicely illustrated, we see that all **of** the series likely have radii **of** con- vergence well below 1. The radius **of** convergence appears to decrease as |l| or N **in**crease. The divergent behavior was seen even at low orders **of** the perturbation coefficients. 37 are

- 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: Denote the eigenfunctions of H [±|
- 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