Singular Continuous Limiting Eigenvalue Distributions for Schrödinger Operators on a 2-Sphere

✍ Scribed by Lawrence E. Thomas; Carlos Villegas-Blas

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 736 KB
• Volume: 141
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0127

✦ Synopsis

Let H=&2+V be a Schro dinger operator acting in L 2 (S), with S the twodimensional unit sphere, 2 the spherical Laplacian, and V a continuous potential. As is well known, the eigenvalues of H in the l th cluster, i.e., those eigenvalues within a radius sup |V| of l(l+1), the l th eigenvalue of &2, have a limiting distribution; l Ä . We provide an alternative self-contained proof of this fact. We then exhibit Ho lder continuous potentials V, both axially-and nonaxially-symmetric, for which the limiting distributions are singular continuous.

1996 Academic Press, Inc.

with ds arclength measure. Hence, V is just the average of V over the geodesic #. Let S denote Schwartz space on the real line. Then the result of the above authors asserts the following:

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