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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#7, Rev. B, due Wednesday, 2024-02-28 }
\end{center}

\vspace{.3 in}

{\bf 1.} The {\bf \href{https://en.wikipedia.org/wiki/Legendre_polynomials}{Legendre Polynomials}} $P_{\ell}(x)$ appear in the solution of Laplace's equation in spherical coordinates. They can be defined as {\it the} series of orthogonal polynomials of degree $\ell=0,1,2,\ldots$ with respect to the weight $w(x)=1$ over the range $-1\leq x \leq 1$, {\it standardized} to $P_{\ell}(1)=1$.

{\bf \quad a)} Show that $\langle x^m|x^n\rangle = \frac{2}{m+n+1}$ [$m{+}n$
is even] or 0 [$m{+}n$ is odd].  Apply the \href{https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process}{Gram-Schmidt} procedure to the basis
of monomials $1, x, x^{2}, \ldots$ to generate the series of $P_{\ell}(x)$ up to order $\ell=4$.  Remmeber to standardize, not normalize, each basis function, so that $P_{\ell}(1)=1$. Calculate the {\it normalization} $h_{\ell}^{2}=\langle P_{\ell}|P_{\ell}\rangle$.  Expand $f(x)=x^{4}$ in the Legendre polynomial basis.

{\bf \quad b)} Show that these basis functions $P_{\ell}(x)$ are eigenfunctions of the linear Legendre differential operator $L=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$ by direct calculation of their eigenvalues $L[P_{\ell}(x)]=\lambda P_{\ell}(x)$.


{\bf 2.} The \href{https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory}{\bf Sturm-Liouville} 2nd order linear differential operator, which generalizes 1b) to
\begin{equation}
  L[u(x)] \equiv \frac{1}{w(x)}\left[\frac{d}{dx} p(x) \frac{d}{dx}
    - q(x)\right] u(x),
\end{equation}
is self-adjoint, $L\ad=L$, [it's a {\bf stretch}!] with respect to the inner product
\begin{equation}
  \langle u_1|u_2\rangle \equiv \int_a^b w(x) dx \; u_1^{*}(x)\, u_2(x)
\end{equation}
if we impose the boundary conditions $u(a)w(a)=u(b)w(b)=0$.  Thus $L$ has real eigenvalues $\lambda_i$ and a complete set of orthogonal eigenfunctions. %$u_i(x)=\langle x|u_i\rangle$.  In particular, $L|u_i\rangle=\lambda_i |u_i\rangle$, with $\langle u_i|u_j\rangle=\delta_{ij}$ [orthonormality], and $\sum_i|u_i\rangle\langle u_i|=I$ [completeness].
% This means that
Any function $f(x)$ has the expansion %in the basis $|u_i\rangle$ as
$|f\rangle = \sum_i |u_i\rangle f_i$ or $f(x) = \sum_i u_i(x) f_i$, with components $f_i=\langle u_i|f\rangle/ \langle u_i|u_i \rangle = \int_a^b w(x)dx\; u_i^{*}(x) f(x)/h_{i}^{2}$.

{\bf \quad a)} Show that $L$ is {\it self-adjoint} or Hermitian.  {\it Hint:} use the definition $\langle f|H\ad g\rangle \equiv \langle Hf|g\rangle$ to show that the operator $D=\frac{1}{w}\frac{d}{dx}$ is anti-Hermitian and apply it to the composition \mbox{of operators in $L$.}  [{\it bonus}: Show that any linear 2$^{\rm nd}$-order differential operator over any weight can be \mbox{put in this form.]}

{\bf \quad b)} Given eigenfunctions $L|u_i\rangle=\lambda_i |u_i\rangle$, i) show that $\lambda_i\in \mathbb{R}$ and that $\langle u_i|u_j\rangle = 0$ for all $\lambda_i\neq\lambda_j$.  ii) Show the {\it closure} relation $I=\sum_i |u_i\rangle \langle u_i|/h_{i}^{2}$ by finding the components $f_{i}$ of the linear combination $|f\rangle=\sum_{i}|u_i\rangle f_{i}$ in the basis $|u_i\rangle$ and substituting $f_{i}$ back into this expansion to obtain an equation of the form $|f\rangle=I |f\rangle$.  iii) Operate $L$ on the same expansion to show that its {\it spectral decomposition} is $L=\sum_i \lambda_i |u_i\rangle \langle u_i|/h_{i}^{2}$.  From ii), what are the eigenvalues of the identity $I$?

{\bf \quad c)} The Legendre polynomials $P_\ell(\cos\theta)$, used with the polar coordinate $\theta$ in spherically symmetric PDEs, are eigenfunctions of the operator $L=\frac{d^2}{d\theta^2}+\cot\theta\frac{d}{d\theta}$.  Show that this is a Sturm-Liouville system on the domain $0<\theta<\pi$, with $w(\theta)=\sin\theta$, $p(\theta)=\sin\theta$, and $q(\theta)=0$.  Change variables to $x=\cos\theta$ and calculate the new functions $w(x)$, $p(x)$, $q(x)$ and domain $a<x<b$ and compare with problem \#1b).  Note the sign change and reversal of integration limits!

{\bf \quad d)} [{\it bonus}: Compile a chart of $w(x),a,b,p(x),q(x),\lambda_{i},h_{i}^{2}$ for each of the following {\it orthogonal functions}: i) cylindrical harmonics $e^{im\phi}$; ii) associated Legendre functions $P_{\ell}^{|m|}(\cos\theta)$; iii) Fourier series $\sin(k_{n}x)$; iv) Bessel functions $J_{m}(k_{nm}\rho)$; v) spherical Bessel functions $j_{\ell}(k_{n\ell}r)$; vi) Hermite polynomials $H_n(x)$; vii) associated Laguerre polynomials $L^{(\alpha)}_n(x)$; viii) Airy functions $\rm{Ai}(x+x_{0})$.
(see the NIST \href{http://dlmf.nist.gov/}{Digital Library of Mathematical Functions})]

% \quad\begin{tabular}{llll@{\quad}r@{}c@{}l}
% i) & Cylindrical harmonics & $e^{im\phi}$ & on & $0<$&$\phi$&$<2\pi$.\\
% ii) & Associated Legendre functions & $P_\ell^{|m|}(x)$ & on & $-1<$&$x$&$<1$.\\
% iii) & Fourier series & $\sin(k_nx)$ & on & $0<$&$x$&$<b$.\\
% iv) & Bessel functions & $J_{m}(x{=}k_{nm}\rho)$ & on & $0<$&$x$&$<x_{nm}$.\\ %where $J_{m}(x_{km})=0$.
% v) & Spherical Bessel functions & $j_{\ell}(x{=}k_{n\ell} r)$ & on & $0<$&$x$&$<x_{n\ell}$.\\ %where $j_{\ell}(x_{n\ell})=0$.
% vi) & Hermite polynomials & $H_n(x)$ & on & $-\infty<$&$x$&$<\infty$.\\
% vii) & Associated Laguere polynomials & $L^{(\alpha)}_n(x)$ & on & $0<$&$x$&$<\infty$.\\
% viii) & Airy functions & $\rm{Ai}(x+x_{0})$ & on & $-\infty<$&$x$&$<\infty$.\\
% \end{tabular}


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