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

A **Born**-**Oppenheimer** **Expansion** **in** a **Neighborhood** **of** a **Renner**-Teller Intersection Mark S. Herman ∗ Institute for Mathematics and its Applications, University **of** M**in**nesota M**in**neapolis, M**in**nesota 55455-0134, U.S.A. October 13, 2008 Abstract We perform a rigorous mathematical analysis **of** the bend**in**g modes **of** a l**in**ear triatomic molecule that exhibits the **Renner**-Teller effect. Assum**in**g the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions **in** powers **of** ǫ, where ǫ 4 is the ratio **of** an electron mass to the mass **of** a nucleus. To prove the validity **of** the expan- sion, we must prove various properties **of** the lead**in**g order equations and their solutions. The lead**in**g order eigenvalue problem is analyzed **in** terms **of** a parameter ˜ b, which is equivalent to the parameter orig**in**ally used by **Renner**. For 0 < ˜ b < 1, we prove self-adjo**in**tness **of** the lead**in**g order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and f**in**ite difference calculations suggest that the ground bend**in**g vibrational state is **in**volved **in** a level cross**in**g near ˜ b = 0.925. We also discuss the degeneracy **of** the eigenvalues. Because **of** the cross**in**g, the ground state is degenerate for 0 < ˜ b < 0.925 and non-degenerate for 0.925 < ˜ b < 1. ∗ This research was supported **in** part by National Science Foundation Grant DMS–0600944 while at Virg**in**ia Poly- technic Institute and State University, and also by the Institute for Mathematics and its Applications at the University **of** M**in**nesota, with funds provided by the National Science Foundation. 1

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