The left-definite Legendre type boundary problem
Tóm tắt
The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM
k
[−] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation
$$M_k [y] = \lambda y$$
has orthogonal polynomial eigenfunctions, called the Legendre type polynomials, associated with the eigenvalues
$$\lambda _n = n(n + 1)(n^2 + n + 4\alpha - 2) + k.$$
In this paper, we show that the spaceC
2[−1, 1] is dense inH, from which it follows that the spectrum of the self-adjoint left-definite operatorS
k
[·] associated withM
k
[·] is a purely point spectrum and consists only of the eigenvaluesλ
n
. Comparisons betweenS
k
[·] and the associated right-definite operatorT
k
[·] are made. This work extends earlier work of Everitt, Krall, Littlejohn, and Williams.
Tài liệu tham khảo
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