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

By assumption, we have � � (0) � ∂Ψ1 ∂x ∈ Hel, and (hP⊥) (0)�−1 and P r (0) ⊥ are bounded operators on Hel. So �� �� � �ψ (3) ⊥ �� �� for some positive real numbers A and B and � proposition 4.7. Also, ψ (3) � �� � �� � ∂f �� ≤ A � el � (0) � � � � � � � + B � ∂X � � �ψ (3) ⊥ ∂f (0) ∂Y � � � � � , �� �� �� is exponentially decay**in**g for arbitrary γ by el ⊥ ∈ C∞ (IR 2 ) s**in**ce its (X,Y ) dependence comes strictly from derivatives **of** f (0) and g (0) �� �� . By a similar argument, we see that �� Dα ψ (3) �� �� ⊥ �� is exponentially decay**in**g with el arbitrary γ, from proposition 4.7. One can now use **in**duction on k to show the conclusion. For the **in**duction hypothesis, assume given by the perturbation formulas **of** chapter 3 satisfy the conclusions. that Ψ (k−3) and ψ (k−1) ⊥ Us**in**g equations (3.27) and (3.28) to determ**in**e Ψ (k−2) and ψ (k) ⊥ propositions and lemmas previously proved. � , the conclusion follows from the 5 The Eigenstates **of** the Lead**in**g Order Hamiltonian We adopt the follow**in**g notation throughout: 1. The operator **of** nuclear angular momentum about the z-axis is denoted by L nuc ∂ z = −i . The operator **of** total electronic angular momentum about the z-axis is ∂φ . The operator **of** total angular momentum about the z-axis is denoted by denoted by L el z L TOT z = (L nuc z ⊗ I) + (I ⊗ Lel z ). 2. We let L k n(x) be the associated Laguerre polynomials, as def**in**ed **in** [15]. The first non-vanish**in**g terms **in** our perturbation expansion are E (2) , f (0) (X, Y ), g (0) (X, Y ) aris**in**g from the eigenvalue equation ⎛ where H2 H2 = − 1 2 ∆X, Y ⊗ I2 + ⎝ f(0) ⎛ ⎜ ⎝ g (0) ⎞ ⎠ = E (2) a + b 2 X2 + bX Y a − b 2 ⎛ ⎝ f(0) Y 2 g (0) ⎞ ⎠ , bX Y a − b 2 X2 + a + b 2 Let (ρ,φ) be the usual polar coord**in**ates associated with (X,Y ). Def**in**e the unitary operators U, Z : H → H by: ⎛ ⎞ cos(φ) − s**in**(φ) U = ⎝ ⎠ s**in**(φ) cos(φ) and Z = 1 ⎛ ⎞ 1 1 √ ⎝ ⎠. 2 i −i 32 Y 2 ⎞ ⎟ ⎠ .

Let r = a1/4 ρ, ˜b = b , and a HU = 1 √ a U −1 H2 U = �� − 1 2 ∂2 1 − ∂r2 2r ∂ ∂r HUZ = 1 √ a (UZ) −1 H2 (UZ) = �� − 1 2 ∂2 1 − ∂r2 2r ∂ ∂r + 1 2 r2 + (Lnuc z ) 2 + 1 2r2 1 + 2 r2 � � ⊗ I2 + ⎛ ⎜ ⎝ ⎛ � � ⎜ ⊗ I2 + ⎜ ⎝ (L nuc z − 1)2 2r2 ˜ b 2 r2 ˜ b 2 r2 − i Lnuc r2 z (L nuc z ˜ b 2 r2 + 1)2 2r2 i Lnuc r2 z ⎞ − ˜ b 2 r2 Both HU and HUZ commute with Lnuc z ⊗ I2. So, we search for eigenfunctions **of** these operators ⎛ **of** the form ⎝ e±i|l|φ ψ1(r) e ±i|l|φ ⎞ ⎠ , |l| = 0,1,2, · · · We warn the reader that although l arises here ψ2(r) as an eigenvalue **of** L nuc z , at this po**in**t we should not associate any physical mean**in**g to l. Here we are deal**in**g with the operators HU and HUZ, which are related to H2 by the operations **of** U and Z. The physical mean**in**g **of** l will become apparent **in** theorem 6.1. We note that (U −1 H2 U)Ψ = E Ψ was the lead**in**g order equation obta**in**ed by **Renner** [20], which is unitarily equivalent to our lead**in**g order equation H2 Ψ = E Ψ. **Renner** showed that some **of** the eigenvalues can be solved for exactly, and used regular perturbation theory up to second order to approximate the other eigenvalues. These equations have been studied by several other authors, for **in**stance [2, 11]. We repeat some **of** **Renner**’s results here, but we calculate the perturbation series to much higher orders, demonstrat**in**g that many **of** the series are diverg**in**g **in**side the region **of** **in**terest. We also illustrate that there is likely a cross**in**g **in**volv**in**g the ground state eigenvalue **of** H2 near b ≈ 0.925a. The ground state appears to be degenerate for 0 < b < 0.925a and non-degenerate for 0.925a < b < a. 5.1 The Exactly Solvable l = 0 States The l = 0 states (no angular dependence) are exactly solvable. In this case HU reduces to H [l=0] U = ⎛ ⎜ ⎝ − 1 2 ∂2 1 − ∂r2 2r ∂ ∂r + 1 + ˜b r 2 2 + 1 2r2 0 − 1 2 33 ∂2 1 − ∂r2 2r 0 ⎟ ⎠ . ∂ ∂r + 1 − ˜b r 2 2 + 1 2r2 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ .

- 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: form (see Theorem XII.5 in [19])
- 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 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