Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 69, pp. 1-33.
Title: The Legendre equation and its self-adjoint operators
Authors: Lance L. Littlejohn (Baylor Univ., Waco, TX, USA)
Anton Zettl (Northern Illinois Univ., DeKalb, IL, USA)
Abstract:
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.