Geodesic
The Legendre equation and its self-adjoint operators
Electronic journal of differential equations, Tome 2011 (2011)
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.
Classification : 05C38, 15A15, 05A15, 15A18
Keywords: Legendre equation, self-adjoint operators, spectrum, three-interval problem 
@article{EJDE_2011__2011__a27,
     author = {Littlejohn,  Lance L. and Zettl,  Anton},
     title = {The {Legendre} equation and its self-adjoint operators},
     journal = {Electronic journal of differential equations},
     year = {2011},
     volume = {2011},
     zbl = {1417.34206},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a27/}
}
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Littlejohn,  Lance L.; Zettl,  Anton. The Legendre equation and its self-adjoint operators. Electronic journal of differential equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a27/