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

used above, − ∆X X 2 − X 2 ∆X and − ∆Y Y 2 − Y 2 ∆Y each are positive operators m**in**us twice the identity. Thus for all φ ∈ DHO, � �� φ, − ∆X,Y + X 2 + Y 2�2 2 − ∆X,Y + ∆X,Y and hence, ≤ � φ, � − ∆X,Y + X 2 + Y 2� � 2 φ � X 2 + Y 2 � + � X 2 + Y 2� ∆X,Y + 4 〈 φ, φ 〉 � � � X 2 + Y 2� φ � � 2 ≤ � � � − ∆X,Y + X 2 + Y 2� φ � � 2 + 4 �φ� 2 . � � φ It follows that (4.3) holds for all ψ ∈ DHO ⊕DHO . This proves (4.2) and the theorem follows. � Unless otherwise stated, it is assumed that by H2 we are referr**in**g to this operator with doma**in** D(H2) = DH0 ⊕ DHO. We now show that H2 has purely discrete spectrum. Theorem 4.2. If a > b > 0, H2 has purely discrete spectrum, with countably many eigenvalues {µj(H2)} ∞ j=1 satisfy**in**g N √ a − b ≤ µ N(N−1)+1(H2) ≤ µ N(N−1)+2(H2) ≤ ... ≤ µ N(N+1)(H2) ≤ N √ a + b, for N = 1,2,3,... Pro**of**: Let (ρ,φ) be the usual polar coord**in**ates associated with (X,Y ). Def**in**e the unitary operators U, W : H → H by (def**in**ed as multiplication operators on H): Def**in**e and ⎛ ⎞ cos(φ) − s**in**(φ) U = ⎝ ⎠ and W = s**in**(φ) cos(φ) 1 ⎛ √ 2 H0 = U −1 H2 U = ⎛ ⎜ − ⎜ ⎝ 1 2 ∆ρ,φ + 1 a + b + ρ 2ρ2 2 2 − 1 ρ2 ∂ ∂φ H ± 0 = ⎛ ⎜ − ⎜ ⎝ 1 2 ∆ρ,φ + 1 a ± b + ρ 2ρ2 2 2 − 1 ρ2 ∂ ∂φ and note that H − 0 ≤ H0 ≤ H + 0 . Now we def**in**e 20 ⎝ eiφ e −iφ ie iφ −ie −iφ 1 ρ 2 ∂ ∂φ ⎞ ⎠ . ⎞ − 1 2 ∆ρ,φ + 1 a − b + ρ 2ρ2 2 2 ⎟ ⎠ , 1 ρ 2 ∂ ∂φ ⎞ − 1 2 ∆ρ,φ + 1 a ± b + ρ 2ρ2 2 2 ⎟ ⎠ ,

H ± 1 = W −1 H ± 0 W = ⎛ ⎜ ⎝ − 1 2 ∆ρ,φ + a ± b 2 ρ 2 0 − 1 2 ∆ρ,φ + In the context **of** the m**in**/max pr**in**ciple [19], for all n ∈ N, 0 a ± b 2 µn(H − 1 ) = µn(H − 0 ) ≤ µn(H0) = µn(H2) = µn(H0) ≤ µn(H + 0 ) = µn(H + 1 ). The operators H ± 1 have purely discrete spectrum, with 2N-fold degenerate eigenvalues **of** N√ a ± b for N = 1,2,... So, H2 must have purely discrete spectrum with eigenvalues µ1(H2) ≤ µ2(H2) ≤ ... satisfy**in**g the required bound. � To prove the quasimode can be expanded to any order **in** ǫ, we must show the terms aris**in**g at arbitrary order **in** the equations **of** chapter 3 are **in** H. This follows from the propositions and lemmas we now prove. A similar analysis was needed **in** [8] and the pro**of**s presented here are analogous to those found **in** [8]. For our purposes it must be shown that the details can be extended to this situation on H. Lemma 4.3. Let T(α) be def**in**ed on a dense doma**in** **of** a separable Hilbert space H, and suppose that T(α) is an analytic family **in** the sense **of** Kato for all α ∈ C, and self-adjo**in**t for all α ∈ IR. If, for all α ∈ IR, T(α) has purely discrete spectrum with eigenvalues accumulat**in**g at ∞, then T(α) has purely discrete spectrum for all α ∈ C. Pro**of**: First we note that if a self-adjo**in**t operator has purely discrete spectrum with eigenvalues accumulat**in**g at ∞, then it has compact resolvent by Theorem XIII.64 **of** [19]. We also note that for any closed operator A, (A − µ) −1 is compact for some µ ∈ ρ(A) if and only if (A − µ) −1 is compact for all µ ∈ ρ(A). This follows from the first resolvent formula. We first show that if T(α) has compact resolvent for all α ∈ IR, then T(α) has compact resolvent for all α ∈ C. We then show that if a closed operator def**in**ed on a separable Hilbert space has compact resolvent, then it must have purely discrete spectrum. S**in**ce T(α) is an entire analytic family, the resolvent Rα(λ) = (T(α) − λ) −1 is analytic **in** both α and λ **in**side the set R = {(α, λ) : α ∈ C, λ ∈ ρ(T(α)) }. From Theorem XII.7 **of** [19], R is open **in** both α and λ. Let B(H) and C(H) denote the bounded operators and compact operators on the Hilbert space H respectively. It follows from the Hahn-Banach Theorem [17], that for any B ∈ B(H) \ C(H), there exists lB ∈ B(H) ∗ such that lB(B) �= 0, and lB = 0 on C(H). 21 ρ 2 ⎞ ⎟ ⎠ .

- 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: This easily follows from � �
- 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 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