QUANTIZED NEUMANN PROBLEM, SEPARABLE POTENTIALS ON S-N AND THE LAME EQUATION

The paper studies spectral theory of Schrodinger operators H = ($) over bar h(2) Delta + V on the sphere from the standpoint of integrability and separation. Our goal is to uncover the fine structure of spec H, i.e., asymptotics of eigenvalues and spectral clusters, determine their relation to the underlying geometry and classical dynamics and apply this data to the inverse spectral problem on the sphere. The prototype model is the celebrated Neumann Hamiltonian p(2) + V with quadratic potential V on S-n. We show that the quantum Neumann Hamiltonian (Schrodinger operator H) remains an integrable and find an explicit set of commuting integrals. We also exhibit large classes of separable potentials {V} based on ellipsoidal coordinates on Sn. Several approaches to spectral theory of such Hamiltonians are outlined. The semiclassical problem (small ($) over bar h) involves the EKB(M)-quantization of the classical Neumann flow along with its invariant tori, Maslov indices, etc., all made explicit via separation of variables. Another approach exploits Stackel-Robertson separation of the quantum Hamiltonian and reduction to certain ODE problems: the Hill S and the generalized Lame equations. The detailed analysis is carried out for S-2 where the ODE becomes the perturbed classical Lame equation and the Schrodinger eigenvalues are expressed through the Lame eigendata. (C) 1995 American Institute of Physics.