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

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

We recognize that the

We recognize that the component equations are of the same form as the radial equation for angular momentum 1 states of the two dimensional Isotropic Harmonic Oscillator. From the first component equation, the eigenvalues and eigenfunctions (non-normalized) are EN+ = (2N+ � + 2) 1 + ˜b , ⎛ ⎝ ˜ r+ L1 ( ˜ N+ r+ 2 )e− ˜ r+ 2 ⎞ /2 ⎠ , 0 N+ = 0,1,2, · · · where r+ ˜ = (1+ ˜b) 1/4 r. From the second component equation, the eigenvalues and eigenfunctions (non-normalized) are EN− = (2N− � + 2) 1 − ˜b , where r− ˜ = (1 − ˜b) 1/4 r. ⎛ ⎝ 0 r− ˜ L1 ( ˜ N− r− 2 )e− ˜ r− 2 /2 ⎞ ⎠ , N− = 0,1,2, · · · Since H2 is unitarily equivalent to √ a HU, we see these states give rise to eigenvalues and eigenfunctions of H2 given by EN− = (2N− + 2) √ a − b Ψ [l=0] (ρ,φ) = N− where r− ˜ = (a − b) 1/4 ρ, and ⎛ ⎝ − ˜ r− sin(φ)L1 ( ˜ N− r− 2 )e− ˜ r− 2 /2 r− ˜ cos(φ)L1 ( ˜ N− r− 2 )e− ˜ r− 2 /2 EN+ = (2N+ + 2) √ a + b Ψ [l=0] (ρ,φ) = N+ where r+ ˜ = (a + b) 1/4 ρ. ⎛ ⎝ ˜ r+ cos(φ)L1 ( ˜ N+ r+ 2 )e− ˜ r+ 2 /2 r+ ˜ sin(φ)L1 ( ˜ N+ r+ 2 )e− ˜ r+ 2 /2 ⎞ ⎠ , N− = 0,1,2, · · · (5.1) ⎞ ⎠ , N+ = 0,1,2, · · · (5.2) 5.2 The Perturbation Calculation For the l �= 0 States In this case, HUZ reduces to H [±|l|] UZ = ⎛ ⎜ − ⎜ ⎝ 1 2 ∂2 1 − ∂r2 2r + ˜b 2 r2 � 0 1 1 0 ∂ ∂r � + 1 2 r2 + (|l| ∓ 1)2 2r 2 0 − 1 2 . 34 ∂2 1 − ∂r2 2r ∂ ∂r 0 + 1 2 r2 + (|l| ± 1)2 2r 2 ⎞ ⎟ ⎠

Denote the eigenfunctions of H [±|l|] UZ by eigenfunction of H [|l|] UZ with eigenvalue E, then ⎛ ⎝ f[±|l|] (r) g [±|l|] ⎞ ⎛ ⎠. It is clear that if ⎝ (r) f[|l|] (r) g [|l|] ⎞ ⎠ is an (r) ⎛ ⎝ f[−|l|] (r) g [−|l|] (r) ⎞ ⎠ = ⎛ ⎝ g[|l|] (r) f [|l|] (r) ⎞ ⎠ is an eigenfunction of H [−|l|] UZ with eigenvalue E. So we only need to find the eigenfunctions and eigenvalues of the H [|l|] UZ . We have not been able to solve for the eigenvalues and eigenfunctions in this case exactly. We use regular perturbation theory with perturbation parameter ˜b, letting H [|l|] UZ = H[|l|] 0 +˜b ˜ V , where H [|l|] ⎛ ⎜ − ⎜ 0 = ⎜ ⎝ 1 ∂ 2 2 1 ∂ 1 − + ∂r2 2r ∂r 2 r2 (|l| − 1)2 + 2r2 0 0 − 1 ∂ 2 2 1 ∂ 1 − + ∂r2 2r ∂r 2 r2 (|l| + 1)2 + 2r2 ⎞ ⎟ ⎠ ˜V = 1 2 r2 � 0 1 1 0 � . One can show using the relative bound found in equation (4.2), that ˜ V is relatively bounded with respect to H [|l|] 0 on H. So, we know that in terms of ˜b, H [|l|] UZ is an analytic family of type A for small ˜b [19]. Therefore, the eigenvalues and eigenfunctions will be analytic functions of ˜b in a neighborhood of ˜ b = 0. We expand the eigenvalues and eigenfunctions of H [|l|] UZ in a series in ˜ b: E N,|l| ( ˜ b) = ∞� k=0 E N,|l| ˜k N,|l| k b , Ψ ( ˜b) = ∞� k=0 Ψ N,|l| ˜k k b and solve for the coefficients E N,|l| k , Ψ N,|l| k recursively. Here N indexes the energy levels of H [|l|] UZ for fixed |l|. Again from the two-dimensional isotropic oscillator, the eigenfunctions of H [|l|] 0 are known exactly. The lowest state is non-degenerate, with eigenvalue and eigenfunction given by ⎛ ⎞ E 0,|l| 0 = |l|, Ψ0,|l| 0 = ⎝ r|l|−1 e −r2 /2 0 (5.3) ⎠ . (5.4) The rest of the states are two-fold degenerate, with eigenvalues and eigenfunctions given by ⎛ ⎞ Ψ N,|l| 0, up = Ψ N,|l| 0, dwn = ⎝ r|l|−1 L |l|−1 N (r2 )e −r2 /2 ⎛ ⎝ 0 0 r |l|+1 L |l|+1 N−1 (r2 )e −r2 /2 35 ⎠ , E N,|l| 0 ⎞ = 2N + |l|, ⎠ , N = 1,2, · · · (5.5)

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