Lance L. Littlejohn, Anton Zettl
The Legendre equation has interior singularities at -1 and +1. The celebrated classical Legendre polynomials are the eigenfunctions of a particular self-adjoint operator in . We characterize all self-adjoint Legendre operators in as well as those in and in and discuss their spectral properties. Then, using the "three-interval theory", we find all self-adjoint Legendre operators in . These include operators which are not direct sums of operators from the three separate intervals and thus are determined by interactions through the singularities at -1 and +1.
Submitted April 17, 2011. Published May 25, 2011.
Math Subject Classifications: 05C38, 15A15, 05A15, 15A18.
Key Words: Legendre equation; self-adjoint operators; spectrum; three-interval problem.
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| Lance L. Littlejohn |
Department of Mathematics, Baylor University
One Bear Place # 97328, Waco, TX 76798-7328, USA
| Anton Zettl |
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL 60115-2888, USA
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