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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#5, Rev. A, due Monday, 2015-10-12 }
\end{center}

\vspace{.3 in}

{\bf 1.}  In class we learned that the linear 2nd order \mbox{\bf
  Sturm-Liouville} differential operator
\begin{equation}
  L[y(x)] \equiv \frac{1}{w(x)}\left[\frac{d}{dx} p(x) \frac{d}{dx}
    - q(x)\right] y(x)
\end{equation}
is self-adjoint, $L\ad=L$, with respect to boundary conditions $y(a)=y(b)=0$
and \mbox{the inner product}
\begin{equation}
  \langle y_1|y_2\rangle \equiv \int_a^b w(x) dx \, y_1(x) y_2(x).
\end{equation}
Thus it has real eigenvalues $\lambda_i$ and a complete set of orthogonal
eigenfunctions $u_i(x)$, so that $L|u_i\rangle=\lambda_i |u_i\rangle$, where
$\langle u_i|u_j\rangle=\delta_{ij}$, and $|f\rangle=\sum_i|u_i\rangle f_i$
for any smooth function $f(x)=\langle x|f\rangle$.

{\bf \quad a)} Show that $L$ is self-adjoint or Hermitian.  {\it Hint:} use
the definition $\langle f|H\ad|g\rangle \equiv \langle Hf|g\rangle$ to show
that the derivative operator $\frac{d}{dx}$ is antiHermitian; apply it to the
composition \mbox{of operators in $L$.}

{\bf \quad b)} Repeat the proof that $L[u_i]=\lambda_i u_i$, where
$\lambda_i\in \mathbb{R}$ and $\langle u_i|u_j\rangle = \delta_{ij}$.  Operate
$L$ on the general expansion of $|f\rangle$ to show that its {\it spectral
  decomposition} is $L=\sum_i \lambda_i |u_i\rangle \langle u_i|$.  What is
the decomposition of the identity operator $1|f\rangle=|f\rangle$ in the same
orthogonal basis $|u_i\rangle$?

{\bf \quad c)} Legendre polynomials $P_n(\cos\theta)$ are defined by the
differential eigenvalue equation $L|f\rangle=\lambda|f\rangle$, where
$L[f(\theta)]=\left[\frac{d^2}{d\theta^2}+\cot\theta\frac{d}{d\theta}\right]
f(\theta)$.  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 $a<x<b$.  Note the sign change!

{\bf \quad d)} Show that $\langle x^m|x^n\rangle = \frac{2}{m+n+1}$ if $m+n$
is even and, 0 if $m+n$ is odd.  Use the Gram-Schmidt procedure on the basis
functions $1$, $x$, $x^2$, and $x^3$ to obtain the first four Legendre
polynomials $P_\ell(x)$, and find their eigenvalues $\lambda_\ell$.

{\bf \quad e)} Use your favorite reference to compile a chart of
$w,p,q,\lambda_i$ for the following functions $\phi_i(x)$:

\quad\quad i) azimuthal harmonics $e^{im\phi}$ on $0<\phi<2\pi$

\quad\quad ii) associated Legendre functions $P_l^m(x)$ on $-1<x<1$

\quad\quad iii) Fourier series $\sin(k_nx)$ on $0<x<b$

\quad\quad iv) Bessel functions $J_m(x)$ on $0<x<b$

\quad\quad v) spherical Bessel functions $j_l(x)$ on $0<x<b$.

{\bf 2.~Simultaneously diagonalize} the matrices %
$A=\pmatrix{ %
  -9 &     2 &     6 \cr %
  2 &    -9 &     6 \cr %
  6 &     6 &     7 }$ %
and $B=\pmatrix{ %
  54 &    10 &    -3 \cr %
  10 &   -45 &    30 \cr %
  -3 &    30 &    46 }$. %
Are the eigenvectors orthogonal? {\it Hint:}
\href{https://www.gnu.org/software/octave}{Octave} or
\href{http://www.wolfram.com/solutions/education/students}{Mathematica} is
your friend!

\end{document}
