= Neumann eigenfunctions of the Chebyshev differential operator
{c}
{title2=$Ly=-(1-x^2)y''+xy',\quad y_n=T_n,\quad\lambda_n=n^2$}
With the <weighted inner product> $\int_{-1}^1\overline f g\,dx/\sqrt{1-x^2}$, the <Chebyshev differential equation> has the <Sturm-Liouville form> with coefficients $p=\sqrt{1-x^2}$, $w=(1-x^2)^{-1/2}$ and $q=0$. The substitution $x=\cos\theta$ converts its operator to $-d^2/d\theta^2$ on $(0,\pi)$ and its inner product to the ordinary one in $d\theta$. The natural endpoint conditions $py'\to0$ become zero derivative conditions at $\theta=0,\pi$.
Consequently the <eigenfunctions> are the <Chebyshev polynomials> $T_n(x)=\cos(n\arccos x)$ with <eigenvalues> $n^2$, $n=0,1,2,\ldots$. Their squared norms are $\pi$ for $n=0$ and $\pi/2$ otherwise. In particular $T_1=x$ and $T_3=4x^3-3x$ are orthogonal. The endpoints of the original differential expression are singular; specifying these transformed <Neumann boundary conditions> defines its self-adjoint realization. The <Dirichlet eigenfunctions of the Chebyshev differential operator> use a different endpoint condition and have sine rather than cosine profiles.
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