2 - sphere Schrodinger operators with odd potentials

The paper extends our earlier results on spectral theory of the 2-sphere Schrodinger operators H = A + Vwith even zonal potentials to a more difficult odd-potential case. We establish local spectral rigidity for odd zonal potentials and give the explicit solution of the inverse spectral problem. We are interested in the spectral theory of Schrodinger operators H = A + V on the 2-sphere S2 with oddzonal(axisymmetric)potentials V = V(x) = V( x), i.e. functions depending on the first coordinate of the vector x = (x; . . .) in S, . In particular, we seek a solution of the following inverse problem: determination of V by the spectrum of H, and the related problem of spectral rigidity: uniqueness of the map V + Spec(H,), modulo natural symmetries (rotations). The spectrum of the S2 Laplacian A is well known to consist of eigenvalues {i, = k(k + 1)}:=, of increasing multiplicity, ?$iL = 2k + 1 = O(k”-’) . Adding a potential Vdestroys (or lowers) the underlying symmetry of A, so the multiple eigen 2, splits into a cluster of simple (or less degenerate) eigens, = k(k f 1) + p k m Iml < k. (1) Spectral shifts { p k m } make up the kth cluster of Spec(H). To study their asymptotics Weinstein [ 141 introduced a sequence of discrete (cluster-distribution) measures It turned out that the sequence {dp,) converges to a continuous limit, j,(E.)di, whose density is directly linked to V. Namely, Po is equal to the distribution function of the so-called Radon transform P of V. The distribution function Po of represents a new type of spectral invariant, called the band invariant [14]. It yields some valuable spectral information about H (cf [3,4,8, 12,13]), but falls short of determining V, and consequently V itself. So the inverse and rigidity problems on S, , unlike many other geometric settings (cf [2,5]), remain largely open. Partial results are known for special classes of potentials: low-degree spherical harmonics [6], and even zonal potentials on S 2 , studied in the recent work of the author PI. The main idea of [8] for solving both problems was to replace asymptotics of cluster distribution measures dp, (2) by asymptotics of individual spectral shifts {p,, ,} . An earlier attempt along these lines was made in [l]. An essential feature of zonal Schrodinger operators, exploited in [8], was an auxiliary symmetry given by the angular momentum operator M = io?o, that commutes with H. So 0266-5611/90/030371 + 08 503.5