Lance L. Littlejohn, Anton Zettl
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
.
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 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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