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.