- Text
- Eigenvalues,
- Degenerate,
- Perturbation,
- Electronic,
- Eigenvalue,
- Eigenfunctions,
- Theorem,
- Pair,
- Follows,
- Equation,
- Expansion,
- Neighborhood,
- Renner,
- Www.ima.umn.edu

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

12 10 8 6 4 2 0.2 0.4 0.6 0.8 1.0 b� Figure 5: 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| = 2. 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. We also used an elementary f**in**ite difference scheme to approximate the eigenvalues at several values **of** ˜ b, for 0 < ˜ b < 1. The results are given **in** Figure 6. The plot was generated by approximat**in**g the lowest ly**in**g 17 eigenvalues for a fixed ˜ b value, then the value **of** ˜ b was changed and the lowest 17 eigenvalues were calculated aga**in**. This was repeated at steps **of** ∆ ˜ b = .01 from 0 ≤ ˜ b < .99. Recall that the l = 0 states were exactly solvable. For comparison, the exact values **of** the lowest ly**in**g l = 0 states were plotted as dotted curves. We see that the f**in**ite difference scheme approximates these eigenvalues so well that the dotted curve are hardly dist**in**guishable from the f**in**ite difference approximation **of** these eigenvalues. Near ˜ b = 0, the 17 eigenvalues that are be**in**g approximated can be identified by their values at ˜ b = 0: 1. The curve that has value 1 at ˜ b = 0 is actually two overlapp**in**g eigenvalues **of** H2 corre- spond**in**g to the degenerate pair **of** lowest ly**in**g |l| = 1 states, one for l = 1 and l = −1. 2. There are three curves that have value 2 at ˜ b = 0. Two **of** the curves are the non-degenerate l = 0 states (one **in**creases with ˜ b and one decreases with ˜ b). The other curve is two overlapp**in**g eigenvalues correspond**in**g to the degenerate pair **of** lowest ly**in**g |l| = 2 states, one for l = 2 and l = −2. These curves together account for four eigenvalues **of** H2. 3. There are three curves that have value 3 at ˜ b = 0. Two **of** the curves are overlapp**in**g degenerate |l| = 1 states, (one degenerate pair **in**creases with ˜ b and one degenerate pair decreases with ˜ b). The other curve an overlapp**in**g degenerate pair **of** lowest ly**in**g |l| = 2 states. These curves together account for six eigenvalues **of** H2. 4. There are three curves that have value 4 at ˜ b = 0. One **of** the curves is a non-degenerate l = 0 state, one is a degenerate pair **of** |l| = 2 states, and one is a degenerate pair **of** |l| = 4 states. Together these curves account for five eigenvalues **of** H2. This plot supports the claim that a cross**in**g occurs **in**volv**in**g the ground state near ˜ b = 0.925. 38

4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6: A plot **of** lowest 17 eigenvalues **of** H2 as a function **of** ˜b, on 0 < ˜ � b < 1, as approximated by a f**in**ite difference scheme. The dotted curves are 2 1 − ˜ � b and 2 1 + ˜b, which are the lowest **of** the l = 0 states that we have solved for exactly. While the f**in**ite difference scheme is crude, we are **in**cl**in**ed to trust the qualitative features **of** the results consider**in**g the lowest **of** the exactly solvable l = 0 eigenvalues were so well approximated, even near ˜ b = 1 as seen **in** the figure. We note that as ˜ b **in**creases from zero, avoided cross**in**gs **in**volv**in**g states with the same value **of** |l| occur, as well as cross**in**gs **in**volv**in**g states with different values **of** |l|. When the uppermost curve is **in**volved with such a phenomenon it will appear to change behavior suddenly without reason, but this is only because we can only see the lowest 17 eigenvalues at each ˜ b. 6 Degeneracy **of** the Quasimode Energies The eigenfunctions **of** H2 � provide the zeroth order states for the quasimode expansion. Recall f that if (0) g (0) � is an eigenfunction **of** H2, we have derived perturbation formulas **in** chapter 3 that determ**in**e the functions f (k) (X, Y ), g (k) (X, Y ), and ψ (k) ⊥ (X, Y ) that enter **in** equation (3.11) as the asymptotic series Φǫ = Ψ1(ǫX, ǫY ) ∞� k=0 f (k) (X, Y )ǫ k + Ψ2(ǫX, ǫY ) 39 ∞� k=0 g (k) (X, Y )ǫ k + ∞� k=0 ψ (k) ⊥ (X, Y )ǫk

- 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: for the larger values of |l|. The s
- 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