Electron. J. Diff. Equ., Vol. 2011 (2011), No. 69, pp. 1-33.

The Legendre equation and its self-adjoint operators

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 $L^2(-1,1)$. We characterize all self-adjoint Legendre operators in $L^2(-1,1)$ as well as those in $L^2(-\infty,-1)$ and in $L^2(1,\infty)$ and discuss their spectral properties. Then, using the "three-interval theory", we find all self-adjoint Legendre operators in $L^2(-\infty,\infty)$. 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
email: lance_littlejohn@baylor.edu
Anton Zettl
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL 60115-2888, USA
email: zettl@math.niu.edu

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