% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/12/20

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{9}

\begin{document}

\begin{homework}
[Answers should be submitted, as both a tex file and a pdf file, to both me and the teaching assistants. You may use this file as a template.]

\question{eigen}

Consider the differential operator
\begin{equation}
L_x = \frac{d^2}{dx^2}
\end{equation}
acting on functions $\phi : [0,2\pi] \to \C$.

What is the most general boundary condition consistent with $L_x$ being Hermitian?
Show that
\begin{enumerate}
\renewcommand{\theenumii}{\roman{enumii}}
\renewcommand{\labelenumii}{\theenumii.}
\item
the Dirichlet boundary conditions
\begin{equation}
\fn{\phi}{0} = \fn{\phi}{2\pi} = 0
\end{equation}
\item
the Neumann boundary conditions
\begin{equation}
\fn{\phi'}{0} = \fn{\phi'}{2\pi} = 0
\end{equation}
\item
the periodic boundary conditions
\begin{equation}
\fn{\phi}{0} = \fn{\phi}{2\pi}
\quad \textrm{and} \quad
\fn{\phi'}{0} = \fn{\phi'}{2\pi}
\end{equation}
\end{enumerate}
are all special cases of this general boundary condition.

In each case
\begin{enumerate}
\item
determine the eigenvalues and eigenspaces,
\item
state\footnote{It is not necessary to derive.} the orthogonality and completeness equations,
\item
check that $\fn{W}{\phi,\psi} = \textrm{constant}$ for $\phi$ and $\psi$ in the same eigenspace,
\item
express
\begin{equation}
\fn{f}{x} = 2 \pi x - x^2
\end{equation}
in terms of eigenvectors of $L_x$.
\end{enumerate}


\question{properties}
The real differential equation
\begin{equation} \label{de}
\frac{d^2y}{dx^2} + \fn{a}{x} \frac{dy}{dx} + \left[  \lambda \fn{b}{x} + \fn{c}{x} \right] y = 0
\end{equation}
with boundary conditions
\begin{equation} \label{bc}
\fn{y}{0} = \fn{y}{1} = 0
\end{equation}
has solutions $\fn{y_\lambda}{x}$. What properties do the $\fn{y_\lambda}{x}$ have?

\end{homework}

\end{document}

