% Lecture notes for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/12/13

\documentclass[a4paper,12pt]{report}

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\begin{document}

\section{Laplace operator}

\subsection{Hermitian boundary conditions}

For the \keyword{Laplace operator}
\begin{equation}
\nabla^2 = \vect\nabla \cdot \vect\nabla
\end{equation}
acting on functions $\fn{\psi}{\vect{x}}$ on a domain $V$, Eq.~(3.4.9) becomes
\begin{equation}
\chi^* \nabla^2 \psi - \psi \nabla^2 \chi^* = \vect\nabla \cdot \fn{\vect{W}}{\chi,\psi}
\end{equation}
where
\begin{equation}
\fn{\vect{W}}{\chi,\psi} = \chi^* \vect\nabla \psi - \psi \vect\nabla \chi^*
\end{equation}
Therefore the Laplace operator is Hermitian if
\begin{equation} \label{lphbc}
\int_{\partial V} \vect{dS} \cdot \vect{W} = 0
\end{equation}
where $\partial V$ is the boundary of $V$.
Special cases of this boundary condition include the \keyword{Dirichlet} boundary condition
\begin{equation}
\left[ \psi \right]_{\partial V} = 0
\end{equation}
the \keyword{Neumann} boundary condition
\begin{equation}
\left[ \vect{n} \cdot \vect\nabla \psi \right]_{\partial V} = 0
\end{equation}
where $\vect{n}$ is the normal to the boundary, and the \keyword{no} or \keyword{periodic} boundary condition
\begin{equation}
\partial V = 0
\end{equation}


\subsection{Eigenfunctions in one dimension}

In one dimension,
\begin{equation}
\nabla^2 = \frac{d^2}{dx^2}
\end{equation}
If \eq{lphbc} is satisfied then $\nabla^2$ is a Hermitian operator and so its eigenspaces
\begin{equation} \label{leigen1}
\nabla^2 \fn{\Psi_\lambda}{x} = \lambda \fn{\Psi_\lambda}{x}
\end{equation}
are orthogonal and complete.

The general solution of \eq{leigen1} is
\begin{equation} \label{geigenspace1}
\fn{\Psi_\lambda}{x} = \left\{
\begin{array}{lcc}
A_0 + B_0 x
& \textrm{for} & \lambda = 0
\\
A_\lambda e^{\sqrt\lambda\, x} + B_\lambda e^{- \sqrt\lambda\, x}
& \textrm{for} & \lambda \neq 0
\end{array}
\right.
\end{equation}


\subsubsection{Finite domain}

For example, take
\begin{equation} \label{fd1}
x \in [-R,R]
\end{equation}
with the no boundary condition
\begin{equation} \label{fdbc1}
\fn{\psi}{-R} = \fn{\psi}{R}
\quad , \quad
\fn{\psi'}{-R} = \fn{\psi'}{R}
\end{equation}
so that \eq{lphbc} is satisfied.
In \eq{geigenspace1}, the boundary condition \eq{fdbc1} constrains $B_0 = 0$ and the eigenvalues to
\begin{equation}
\lambda = - \left( \frac{\pi n}{R} \right)^2
\qquad \left( n \in \Z \right)
\end{equation}
Therefore the eigenspaces for this finite domain are
\begin{equation}
\fn{\Psi_\lambda}{x} = A_\lambda \exp\left(\frac{i\pi nx}{R}\right) + B_\lambda \exp\left(-\frac{i\pi nx}{R}\right)
\end{equation}

Using
\begin{equation}
\int_{-R}^R \d{x} \left[ \exp\left(\frac{i\pi nx}{R}\right) \right]^* \exp\left(\frac{i\pi n'x}{R}\right) = 2R \delta_{nn'}
\end{equation}
we can choose the set of eigenfunctions
\footnote{A different choice of eigenfunctions within the eigenspaces could give sines and cosines instead of exponentials.}
\begin{equation}
\fn{\psi_n}{x} = \frac{1}{\sqrt{2R}\,} \exp\left(\frac{i\pi nx}{R}\right)
\end{equation}
which are orthonormal
\begin{equation}
\int_{-R}^R \d{x} \fn{\psi_n^*}{x} \fn{\psi_{n'}}{x} = \delta_{nn'}
\end{equation}
and complete
\begin{equation}
\sum_{n=-\infty}^\infty \fn{\psi_n}{x} \fn{\psi_n^*}{x'} = \fn{\delta_{2R}}{x-x'}
\end{equation}
where $\fn{\delta_{2R}}{x-x'}$ is the delta function on the domain \eq{fd1} satisfying the boundary conditions \eq{fdbc1}, and we use the notation
\begin{equation}
\fn{\delta_p}{x} = \fn{\delta}{x} \qquad \left( -\frac{p}{2} \leq x \leq \frac{p}{2} \right)
\end{equation}
and
\begin{equation}
\fn{\delta_p}{x} = \fn{\delta_p}{x+p}
\end{equation}
A function on the domain \eq{fd1} satisfying the boundary conditions \eq{fdbc1} can be expanded in terms of the eigenfunctions as
\begin{equation}
\fn{f}{x} = \sum_{n=-\infty}^\infty f_n \fn{\psi_n}{x}
\end{equation}
where
\begin{equation}
f_n = \int_{-R}^R \d{x} \fn{\psi_n^*}{x} \fn{f}{x}
\end{equation}
which is known as a \keyword{Fourier series}.


\subsubsection{Infinite domain}
\label{fourier}

Alternatively, taking
\begin{equation} \label{id1}
x \in [-\infty,\infty]
\end{equation}
then
\begin{equation}
\fn{W}{\chi,\psi} = \left( \chi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \chi^*}{\partial x} \right)
\end{equation}
is sufficiently well behaved at $x \to \pm\infty$, in the sense that its average value at infinity is zero, if $B_0 = 0$ and
\begin{equation}
\lambda = - k^2 \qquad \left( k \in \R \right)
\end{equation}
Therefore the eigenspaces for this infinite domain are
\begin{equation}
\fn{\Psi_\lambda}{x} = A_\lambda e^{ikx} + B_\lambda e^{-ikx}
\end{equation}

Using
\begin{equation}
\int_{-\infty}^\infty \d{x} \left( e^{ikx} \right)^* e^{ik'x} = 2\pi \fn{\delta}{k - k'}
\end{equation}
we can choose the set of eigenfunctions
\begin{equation}
\fn{\psi}{k;x} = \frac{1}{\sqrt{2\pi}\,} e^{ikx}
\end{equation}
which are orthonormal
\begin{equation}
\int_{-\infty}^\infty \d{x} \fn{\psi^*}{k;x} \fn{\psi}{k';x} = \fn{\delta}{k-k'}
\end{equation}
and complete
\begin{equation}
\int_{-\infty}^\infty \d{k} \fn{\psi}{k;x} \fn{\psi^*}{k;x'} = \fn{\delta}{x-x'}
\end{equation}
A function on the domain \eq{id1} which is sufficiently well behaved at $x \to \pm\infty$ can be expanded in terms of the eigenfunctions as
\begin{equation}
\fn{f}{x} = \int_{-\infty}^\infty \d{k} \fn{\tilde{f}}{k} \fn{\psi}{k;x}
\end{equation}
where
\begin{equation}
\fn{\tilde{f}}{k} = \int_{-\infty}^\infty \d{x} \fn{\psi^*}{k;x} \fn{f}{x}
\end{equation}
which is known as a \keyword{Fourier transform}.


\subsection{Eigenfunctions in two dimensions}

In two dimensional polar coordinates,
\begin{equation}
\nabla^2 = \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial\theta^2}
\end{equation}
We identify the operator
\begin{equation}
L_\theta = \frac{\partial^2}{\partial\theta^2}
\end{equation}
that commutes with $\nabla^2$
\begin{equation}
\left[ \nabla^2 , L_\theta \right] = 0
\end{equation}
Eq.~(3.4.9) gives
\begin{equation}
\fn{W_\theta}{\chi,\psi} = \chi^* \frac{\partial \psi}{\partial\theta} - \psi \frac{\partial \chi^*}{\partial\theta}
\end{equation}
and $L_\theta$ is Hermitian if
\begin{equation} \label{lthbc2}
\left[ \fn{W_\theta}{\chi,\psi} \right]_{\partial C_\theta} = 0
\end{equation}
where $\partial C_\theta$ is the boundary of the constant $r$ contours.
If \eqs{lphbc}{lthbc2} are satisfied then $\nabla^2$ and $L_\theta$ are commuting Hermitian operators and so their eigenspaces
\begin{eqnarray} \label{leigen2}
\nabla^2 \fn{\Psi_{\lambda\mu}}{r,\theta} & = & \lambda \fn{\Psi_{\lambda\mu}}{r,\theta}
\\ \label{lteigen2}
L_\theta \fn{\Psi_{\lambda\mu}}{r,\theta} & = & \mu \fn{\Psi_{\lambda\mu}}{r,\theta}
\end{eqnarray}
are orthogonal and complete.

To solve for the eigenspaces, we note that \eq{leigen2} reduces to the ordinary differential equation
\begin{equation} \label{lreigen2}
\left( \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + \frac{\mu}{r^2} \right) \fn{\Psi_{\lambda\mu}}{r,\theta} = \lambda \fn{\Psi_{\lambda\mu}}{r,\theta}
\end{equation}
Writing $x = \sqrt{-\lambda}\, r$ and $\mu = - n^2$, this takes the form of Bessel's equation
\begin{equation}
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left( x^2 - n^2 \right) y = 0
\end{equation}
whose solutions are the \href{http://en.wikipedia.org/wiki/Bessel_function}{\keyword{Bessel functions}}
\footnote{The Bessel functions have many properties which can be looked up as needed.}
of the first $\fn{J_n}{x}$ and second $\fn{Y_n}{x}$ kind. 
Therefore the general solution of \eqs{leigen2}{lteigen2} is
\begin{eqnarray}
\fn{\Psi_{\lambda\mu}}{r,\theta} & = &
A_{\lambda\mu} \fn{J_n}{\sqrt{-\lambda}\, r} e^{in\theta}
+ B_{\lambda\mu} \fn{J_n}{\sqrt{-\lambda}\, r} e^{-in\theta}
\nonumber \\ \label{geigenspace2} && {}
+ C_{\lambda\mu} \fn{Y_n}{\sqrt{-\lambda}\, r} e^{in\theta}
+ D_{\lambda\mu} \fn{Y_n}{\sqrt{-\lambda}\, r} e^{-in\theta}
\end{eqnarray}
where $n = \sqrt{-\mu}$.


\subsubsection{Finite domain}

For example, taking
\begin{equation} \label{fd2}
r \in [0,R]
\end{equation}
with the boundary condition
\begin{equation} \label{fdbc2}
\fn{\psi}{R} = 0
\end{equation}
so that \eqs{lphbc}{lthbc2} are satisfied.
In \eq{geigenspace2}, periodicity in $\theta$ fixes
\begin{equation}
n \in \Z
\end{equation}
regularity at $r=0$ fixes $C_{\lambda\mu} = D_{\lambda\mu} = 0$, and the boundary condition \eq{fdbc2} constrains $\lambda$ to satisfy
\begin{equation}
\fn{J_n}{\sqrt{-\lambda}\, R} = 0
\end{equation}
and so
\begin{equation}
\lambda = - \frac{\zeta_{an}^2}{R^2}
\end{equation}
where $\zeta_{an}$ is the $a$th positive zero of $J_n(x)$.
Therefore the eigenspaces for this finite domain are
\begin{equation}
\fn{\Psi_{\lambda\mu}}{r,\theta} = \fn{J_n}{\frac{\zeta_{an}r}{R}} \left( A_{\lambda\mu} e^{in\theta} + B_{\lambda\mu} e^{-in\theta} \right)
\end{equation}

Using
\begin{equation}
\int_0^1 \d{x} x \left[ \fn{J_n}{\zeta_{an} x} \right]^2 = \frac{1}{2} \left[ \fn{J_{n+1}}{\zeta_{an}} \right]^2
\end{equation}
we can choose the set of eigenvectors
\begin{equation}
\fn{\psi_{an}}{r,\theta} = \frac{1}{\sqrt{\pi}\, R \fn{J_{n+1}}{\zeta_{an}}} \fn{J_n}{\frac{\zeta_{an} r}{R}} e^{in\theta}
\end{equation}
which are orthonormal
\begin{equation}
\int_0^R \d{r} r \int_0^{2\pi} \d\theta \fn{\psi_{an}^*}{r,\theta} \fn{\psi_{a'n'}}{r,\theta} = \delta_{aa'} \delta_{nn'}
\end{equation}
and complete
\begin{equation}
\sum_{a=1}^\infty \sum_{n=-\infty}^\infty \fn{\psi_{an}}{r,\theta} \fn{\psi_{an}^*}{r',\theta'} = \frac{1}{r} \fn{\delta}{r-r'} \fn{\delta_{2\pi}}{\theta-\theta'}
\end{equation}
A function on the domain \eq{fd2} satisfying the boundary conditions \eq{fdbc2} can be expanded in terms of the eigenfunctions as
\begin{equation}
\fn{f}{r,\theta} = \sum_{n=-\infty}^\infty \sum_{a=1}^\infty f_{an} \fn{\psi_{an}}{r,\theta}
\end{equation}
where
\begin{equation}
f_{an} = \int_0^R \d{r} r \int_0^{2\pi} \d\theta \fn{\psi_{an}^*}{r,\theta} \fn{f}{r,\theta}
\end{equation}
which is related to \keyword{Fourier-Bessel series}.


\subsubsection{Infinite domain}

Alternatively, taking
\begin{equation} \label{id2}
r \in [0,\infty)
\end{equation}
and noting that
\begin{equation}
\fn{J_n}{x} \to \sqrt{ \frac{2}{\pi x} }\, \cos \left( x - \frac{n\pi}{2} - \frac{\pi}{4} \right)
\qquad \textrm{as $x \to \infty$}
\end{equation}
then
\begin{equation}
\fn{W_r}{\chi,\psi} = r \left( \chi^* \frac{\partial \psi}{\partial r} - \psi \frac{\partial \chi^*}{\partial r} \right)
\end{equation}
is sufficiently well behaved at $r \to \infty$, in the same sense as in Section~\ref{fourier}, if
\begin{equation}
\lambda = - \zeta^2 \qquad \left( \zeta \in \R \right)
\end{equation}
Therefore the eigenspaces for this infinite domain are
\begin{equation}
\fn{\Psi_{\lambda\mu}}{r,\theta} = \fn{J_n}{\zeta r} \left( A_{\lambda\mu} e^{in\theta} + B_{\lambda\mu} e^{-in\theta} \right)
\end{equation}

Using
\begin{equation}
\int_0^\infty \d{r} r \fn{J_n}{\zeta r} \fn{J_n}{\zeta' r} = \frac{1}{\zeta} \fn{\delta}{\zeta - \zeta'}
\end{equation}
we can choose the set of eigenvectors
\begin{equation}
\fn{\psi_n}{\zeta;r,\theta} = \frac{1}{\sqrt{2\pi}\,} \fn{J_n}{\zeta r} e^{in\theta}
\end{equation}
which are orthonormal
\begin{equation}
\int_0^\infty \d{r} r \int_0^{2\pi} \d\theta \fn{\psi_n^*}{\zeta;r,\theta} \fn{\psi_{n'}}{\zeta';r,\theta} = \frac{1}{\zeta} \fn{\delta}{\zeta-\zeta'} \delta_{nn'}
\end{equation}
and complete
\begin{equation}
\int_0^\infty \d{\zeta} \zeta \sum_{n=-\infty}^\infty \fn{\psi_n}{\zeta;r,\theta} \fn{\psi_n^*}{\zeta;r',\theta'} = \frac{1}{r} \fn{\delta}{r-r'} \fn{\delta_{2\pi}}{\theta-\theta'}
\end{equation}
A function on the domain \eq{id2} which is sufficiently well behaved at $r \to \infty$ can be expanded in terms of the eigenfunctions as
\begin{equation}
\fn{f}{r,\theta} = \int_0^\infty \d{\zeta} \zeta \sum_{n=-\infty}^\infty \fn{f_n}{\zeta} \fn{\psi_n}{\zeta;r,\theta}
\end{equation}
where
\begin{equation}
\fn{f_n}{\zeta} = \int_0^\infty \d{r} r \int_0^{2\pi} \d\theta \fn{\psi^*_n}{\zeta;r,\theta} \fn{f}{r,\theta}
\end{equation}
which is related to the \keyword{Hankel transform}.


\subsection{Eigenfunctions in three dimensions}

In three dimensional spherical polar coordinates,
\begin{equation} \label{laplacertp}
\nabla^2 =\frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial\theta} + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2}{\partial\phi^2}
\end{equation}
We identify the operators
\begin{equation} \label{Lttp}
L_\theta = \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial\theta} + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial\phi^2}
\end{equation}
and
\begin{equation} \label{Lpp}
L_\phi = \frac{\partial^2}{\partial\phi^2}
\end{equation}
that commute with $\nabla^2$ and each other
\begin{equation}
\left[ \nabla^2 , L_\theta \right] = \left[ \nabla^2 , L_\phi \right] = \left[ L_\theta , L_\phi \right] = 0
\end{equation}
Eq.~(3.4.9) becomes
\begin{equation}
\chi^* L_\theta \psi - \psi L_\theta \chi^* = \frac{1}{\sin\theta} \vect\nabla_\theta \cdot \fn{\vect{W}_\theta}{\chi,\psi}
\end{equation}
with
\begin{equation}
\vect\nabla_\theta = \left( \frac{\partial}{\partial\theta} , \frac{1}{\sin\theta} \frac{\partial}{\partial\phi} \right)
\end{equation}
and
\begin{equation}
\fn{\vect{W}_\theta}{\chi,\psi} = \sin\theta \left( \chi^* \vect\nabla_\theta \psi - \psi \vect\nabla_\theta \chi^* \right)
\end{equation}
and
\begin{equation}
\chi^* L_\phi \psi - \psi L_\phi \chi^* = \frac{\partial}{\partial\phi} \fn{W_\phi}{\chi,\psi}
\end{equation}
with
\begin{equation}
\fn{W_\phi}{\chi,\psi} = \chi^* \frac{\partial \psi}{\partial\phi} - \psi \frac{\partial \chi^*}{\partial\phi}
\end{equation}
Therefore $L_\theta$ is Hermitian if
\begin{equation} \label{lthbc3}
\int_{\partial S_\theta} \d{l} \vect{n} \cdot \vect{W}_\theta = 0
\end{equation}
where $\partial S_\theta$ is the boundary of the constant $r$ surfaces, and $L_\phi$ is Hermitian if
\begin{equation} \label{lphbc3}
\left[ \fn{W_\phi}{\chi,\psi} \right]_{\partial C_\phi} = 0
\end{equation}
where $\partial C_\phi$ is the boundary of the constant $r,\theta$ lines.
If \eqss{lphbc}{lthbc3}{lphbc3} are satisfied then $\nabla^2$, $L_\theta$ and $L_\phi$ are commuting Hermitian operators and so their eigenspaces
\begin{eqnarray} \label{leigen3}
\nabla^2 \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} & = & \lambda \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi}
\\ \label{lteigen3}
L_\theta \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} & = & \mu \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi}
\\ \label{lpeigen3}
L_\phi \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} & = & \nu \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi}
\end{eqnarray}
are orthogonal and complete.

To solve for the eigenspaces, we note that \eq{leigen3} reduces to the ordinary differential equation
\begin{equation} \label{lreigen}
\left( \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} + \frac{\mu}{r^2} \right) \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} = \lambda \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi}
\end{equation}
Writing $x = \sqrt{-\lambda}\, r$ and $\mu = -l(l+1)$, this takes the form of the spherical Bessel equation
\begin{equation}
x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + \left[ x^2 - l(l+1) \right] y = 0
\end{equation}
whose solutions are the \href{http://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions:_jn.2C_yn}{\keyword{spherical Bessel functions}} of the first $\fn{j_l}{x}$ and second $\fn{y_l}{x}$ kind.
Also, \eq{lteigen3} reduces to the ordinary differential equation
\begin{equation}
\left( \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial\theta} + \frac{\nu}{\sin^2 \theta} \right) \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} = \mu \fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi}
\end{equation}
Writing $x = \cos\theta$, $\mu = -l(l+1)$ and $\nu = -m^2$, this takes the form of the general Legendre equation
\begin{equation}
\left( 1 - x^2 \right) \frac{d^2 y}{dx^2} - 2 x \frac{dy}{dx} + \left[ l(l+1) - \frac{m^2}{1-x^2} \right] y = 0
\end{equation}
whose solutions are the \href{http://en.wikipedia.org/wiki/Associated_Legendre_polynomials}{\keyword{associated Legendre polynomials}} of the first $\fn{P_l^m}{x}$ and second $\fn{Q_l^m}{x}$ kind.
Therefore the general solution of \eqss{leigen3}{lteigen3}{lpeigen3} is
\begin{eqnarray}
\fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} & = &
A_{\lambda\mu\nu} \fn{j_l}{\sqrt{-\lambda}\, r} \fn{P_l^m}{\cos\theta} e^{im\phi}
+ B_{\lambda\mu\nu} \fn{j_l}{\sqrt{-\lambda}\, r} \fn{P_l^m}{\cos\theta} e^{-im\phi}
\nonumber \\ && {}
+ C_{\lambda\mu\nu} \fn{j_l}{\sqrt{-\lambda}\, r} \fn{Q_l^m}{\cos\theta} e^{im\phi}
+ D_{\lambda\mu\nu} \fn{j_l}{\sqrt{-\lambda}\, r} \fn{Q_l^m}{\cos\theta} e^{-im\phi}
\nonumber \\ && {}
+ E_{\lambda\mu\nu} \fn{y_l}{\sqrt{-\lambda}\, r} \fn{P_l^m}{\cos\theta} e^{im\phi}
+ F_{\lambda\mu\nu} \fn{y_l}{\sqrt{-\lambda}\, r} \fn{P_l^m}{\cos\theta} e^{-im\phi}
\nonumber \\ && {}
+ G_{\lambda\mu\nu} \fn{y_l}{\sqrt{-\lambda}\, r} \fn{Q_l^m}{\cos\theta} e^{im\phi}
+ H_{\lambda\mu\nu} \fn{y_l}{\sqrt{-\lambda}\, r} \fn{Q_l^m}{\cos\theta} e^{-im\phi}
\nonumber \\ \label{geigenspace3}
\end{eqnarray}
where $l = (\sqrt{1-4\mu}\,-1) / 2$ and $m = \sqrt{-\nu^2}$.


\subsubsection{Finite domain}

For example, taking
\begin{equation} \label{fd3}
r \in [0,R]
\end{equation}
with the boundary condition
\begin{equation} \label{fdbc}
\fn{\psi}{R} = 0
\end{equation}
so that \eqss{lphbc}{lthbc3}{lphbc3} are satisfied.
In \eq{geigenspace3}, periodicity in $\phi$ fixes
\begin{equation}
m \in \Z
\end{equation}
regularity at $\theta = 0,\pi$ fixes $C_{\lambda\mu\nu} = D_{\lambda\mu\nu} = G_{\lambda\mu\nu} = H_{\lambda\mu\nu} = 0$, and
\begin{equation}
l \in \Z \quad \textrm{and} \quad |m| \leq l
\end{equation}
regularity at $r=0$ fixes $E_{\lambda\mu\nu} = F_{\lambda\mu\nu} = G_{\lambda\mu\nu} = H_{\lambda\mu\nu} = 0$, and the boundary condition \eq{fdbc} constrains $\lambda$ to satisfy
\begin{equation}
\fn{j_l}{\sqrt{-\lambda}\, R} = 0
\end{equation}
and so
\begin{equation}
\lambda = - \frac{\zeta_{al}^2}{R^2}
\end{equation}
where $\zeta_{al}$ is the $a$th positive zero of $j_l(x)$.
Therefore the eigenspaces for this finite domain are
\begin{equation}
\fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} = \fn{j_l}{\frac{\zeta_{al}r}{R}} \fn{P_l^m}{\cos\theta} \left( A_{\lambda\mu\nu} e^{im\phi} + B_{\lambda\mu\nu} e^{-im\phi} \right)
\end{equation}

Using
\begin{equation}
\int_0^1 \d{x} x^2 \left[ \fn{j_l}{\zeta_{al} x} \right]^2 = \frac{1}{2} \left[ \fn{j_{l+1}}{\zeta_{al}} \right]^2
\end{equation}
and
\begin{equation}
\int_{-1}^1 \d{x} \left[ \fn{P_l^m}{x} \right]^2 = \frac{2 (l+m)!}{(2l+1) (l-m)!}
\end{equation}
we can choose the set of eigenvectors
\begin{equation}
\fn{\psi_{alm}}{r,\theta,\phi} = \sqrt{\frac{2}{R^3}}\, \frac{1}{\fn{j_{l+1}}{\zeta_{al}}} \fn{j_l}{\frac{\zeta_{al} r}{R}} \fn{Y_l^m}{\theta,\phi}
\end{equation}
where the $\fn{Y_l^m}{\theta,\phi}$ are the \href{http://en.wikipedia.org/wiki/Spherical_harmonics}{\keyword{spherical harmonics}}
\begin{equation}
\fn{Y_l^m}{\theta,\phi} = \sqrt{\frac{(2l+1) (l-m)!}{4\pi (l+m)!}}\, \fn{P_l^m}{\cos\theta} e^{im\phi}
\end{equation}
These eigenvectors are orthonormal
\begin{equation}
\int_0^R \d{r} r^2 \int_0^\pi \d\theta \sin\theta \int_0^{2\pi} \d\phi \fn{\psi_{alm}^*}{r,\theta,\phi} \fn{\psi_{a'l'm'}}{r,\theta,\phi} = \delta_{aa'} \delta_{ll'} \delta_{mm'}
\end{equation}
and complete
\begin{equation}
\sum_{a=1}^\infty \sum_{l=0}^\infty \sum_{m=-l}^l \fn{\psi_{alm}}{r,\theta,\phi} \fn{\psi_{alm}^*}{r',\theta',\phi'} = \frac{1}{r^2 \sin\theta} \fn{\delta}{r-r'} \fn{\delta}{\theta-\theta'} \fn{\delta}{\phi-\phi'}
\end{equation}


\subsubsection{Infinite domain}

Alternatively, taking
\begin{equation} \label{fd2}
r \in [0,\infty)
\end{equation}
and noting that
\begin{equation}
\fn{j_l}{x} \to \frac{1}{x} \sin \left( x - \frac{l\pi}{2} \right)
\qquad \textrm{as $x \to \infty$}
\end{equation}
then
\begin{equation}
\fn{W_r}{\chi,\psi} = r^2 \left( \chi^* \frac{\partial \psi}{\partial r} - \psi \frac{\partial \chi^*}{\partial r} \right)
\end{equation}
is sufficiently well behaved at $r \to \infty$, in the same sense as in Section~\ref{fourier}, if
\begin{equation}
\lambda = - \zeta^2 \qquad \left( \zeta \in \R \right)
\end{equation}
Therefore the eigenspaces for this infinite domain are
\begin{equation}
\fn{\Psi_{\lambda\mu\nu}}{r,\theta,\phi} = \fn{j_l}{\zeta r} \fn{P_l^m}{\cos\theta} \left( A_{\lambda\mu\nu} e^{im\phi} + B_{\lambda\mu\nu} e^{-im\phi} \right)
\end{equation}

Using
\begin{equation}
\int_0^\infty \d{x} x^2 \fn{j_l}{\zeta x} \fn{j_l}{\zeta' x} = \frac{\pi}{2 \zeta^2} \fn{\delta}{\zeta - \zeta'}
\end{equation}
we can choose the set of eigenvectors
\begin{equation}
\fn{\psi_{lm}}{\zeta;r,\theta} = \sqrt{\frac{2}{\pi}}\, \fn{j_l}{\zeta r} \fn{Y_l^m}{\theta,\phi}
\end{equation}
which are orthonormal
\begin{equation}
\int_0^\infty \d{r} r^2 \int_0^\pi \d\theta \sin\theta \int_0^{2\pi} \d\phi \fn{\psi_{lm}^*}{\zeta;r,\theta,\phi} \fn{\psi_{l'm'}}{\zeta';r,\theta,\phi} = \frac{1}{\zeta^2} \fn{\delta}{\zeta-\zeta'} \delta_{ll'} \delta_{mm'}
\end{equation}
and complete
\begin{equation}
\int_0^\infty \d{\zeta} \zeta^2 \sum_{l=0}^\infty \sum_{m=-l}^l \fn{\psi_n}{\zeta;r,\theta,\phi} \fn{\psi_n^*}{\zeta;r',\theta',\phi'} = \frac{1}{r^2 \sin\theta} \fn{\delta}{r-r'} \fn{\delta}{\theta-\theta'} \fn{\delta}{\phi-\phi'}
\end{equation}

\end{document}


