Neumann eigenfunctions of the Chebyshev differential operator ($Ly=-(1-x^2)y^{\prime \prime }+x{y}^{\prime },y_n=T_n,{\lambda }_n=n^2$)

With the weighted inner product ${\int }_{-1}^1\overline{f}gdx/\sqrt{1-x^2}$, 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 $p{y}^{\prime }\to 0$ become zero derivative conditions at $\theta =0,\pi$.

Consequently the eigenfunctions are the Chebyshev polynomials $T_n(x)=\cos (n\arccos x)$ with eigenvalues $n^2$, $n=0,1,2,\dots$. 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.