%SourceDoc ./tesis.tex
\section{Commutation Relations for Cavity and Channel Operators}
\label{app:A1}

In this appendix we compute the (equal--time) commutation relations
for the cavity and channels position and momentum operators.  One
finds that the cavity operators obey the commutation relations
\begin{align}
\label{eq:comres}
 [Q_{\lambda},Q_{\lambda^\prime}] &=
 [Q_{\lambda},Q^{\dagger}_{\lambda^\prime}] = 0, \\
 [P_{\lambda},P_{\lambda^\prime}] &=
 [P_{\lambda},P^{\dagger}_{\lambda^\prime}] = 0, \\
 [Q_{\lambda},P_{\lambda^\prime}] &= \rmi\hbar \, \delta_{\lambda
   \lambda^\prime} \, , \\ 
[Q_{\lambda},P^{\dagger}_{\lambda^\prime}] &=
 \rmi\hbar \mathcal{N}^{*}_{\lambda \lambda^\prime} \, ,
\end{align}
while for the channel operators the following commutation relations
hold
\begin{align}
\label{eq:comout}
 [Q_{m}(\omega),Q_{n}(\omega^\prime) ] &=
 [Q_{m}(\omega),Q^{\dagger}_{n}(\omega^\prime)] = 0, \\
 [P_{m}(\omega),P_{n}(\omega^\prime)] &=
 [P_{m}(\omega),P_{n}^{\dagger}(\omega^\prime)] = 0, \\
 [Q_{m}(\omega),P_{n}(\omega^\prime)] &= \rmi\hbar\,\delta_{mn} \,
 \delta(\omega-\omega^\prime), \\
 [Q_{m}(\omega),P^{\dagger}_{n}(\omega^\prime)] &= \rmi\hbar
       \mathcal{N}^{*}_{n m}(\omega^\prime,\omega),
\end{align}
and in addition the cavity operators commute with all channel operators. This
shows that for each subsystem the operators $Q$ and $P$ behave like
the basic operators of position an momentum, respectively.

The equal--time commutation relations follow easily from the
commutation relations (\ref{eq:qpcomm}) for the operators of the total
system and the completeness of the modes $|{\bm
\phi}_m (\omega)\rangle$. As an example we show that $[Q_\lambda,
P_m(\omega)] = 0$. Using the definitions (\ref{eq:qopcav}),
(\ref{eq:popch}) of $Q_\lambda$ and $P_m(\omega)$, we obtain
\begin{equation}
\label{eq:qinpout}
\begin{split}
[Q_\lambda,P_m(\omega)] &= \sum_{m^{\prime} m^{\prime\prime}}
\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^{\prime}
\int\limits_{\omega_{m''}}^{\infty} \! \rmd\omega^{\prime\prime} \;
\alpha_{m^{\prime} \lambda}(\omega^{\prime})
\beta^{*}_{m^{\prime\prime} m} (\omega^{\prime\prime},\omega)
     [q_{m^{\prime}}(\omega^{\prime}),p_{m^{\prime\prime}}
       (\omega^{\prime\prime})] \\ &= i \hbar \sum_{m^\prime}
     \int\limits_{\omega_{m'}}^{\infty} \! d\omega^{\prime} \;
     \alpha_{m^\prime \lambda}(\omega^{\prime}) \beta^{*}_{m^\prime
       m}(\omega^\prime,\omega),
\end{split}
\end{equation}
where we used the commutation relation (\ref{eq:qpcomm}). According
to Eq.\ (\ref{eq:exa}), the coefficients $\alpha$ and $\beta$
can be written as
\begin{subequations}
\label{eq:defalbe}
\begin{align}
\label{eq:defal}
\alpha_{m^\prime \lambda}(\omega^\prime) & = \langle
{\bm \mu}_{\lambda} | {\bm \phi}_{m^\prime}(\omega^\prime) \rangle ,\\
\label{eq:defbe}
\beta_{m^\prime m}(\omega^\prime,\omega) & = \langle
{\bm \nu}_{m}(\omega) | {\bm \phi}_{m^\prime}(\omega^{\prime}) \rangle.
\end{align}
\end{subequations}
Substitution into the right hand side of Eq.\ (\ref{eq:qinpout}) yields
\begin{equation}
\label{eq:appa1}
\begin{split}
\sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^{\prime}
\; \alpha_{m^\prime \lambda}(\omega^{\prime}) \beta^{*}_{m^\prime
  m}(\omega^\prime,\omega) &= \sum_{m^\prime}
\int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^{\prime} \; \langle {\bm
  \mu}_{\lambda} | {\bm \phi}_{m^\prime} (\omega^{\prime}) \rangle
\langle {\bm \phi}_{m^\prime} (\omega^{\prime}) | {\bm
  \nu}_{m}(\omega) \rangle \\ 
&= \langle \vect{\mu}_{\lambda}|\vect{\nu}_{m}(\omega)\rangle,
\end{split}
\end{equation}
which vanishes due to the orthogonality between resonator and channel
modes. The calculation of all remaining commutators reduces to
Eq.~(\ref{eq:appa1}) or to one of the expressions
\begin{gather}
\label{eq:appa2}
\sum_m \! \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \alpha_{m
  \lambda} (\omega) \alpha_{m \lambda^\prime}^* (\omega) =
\delta_{\lambda \lambda^\prime},\\
\label{eq:appa3}
\sum_{n} \! \int\limits_{\omega_{n}}^{\infty} \!
\rmd\omega^{\prime\prime} \; \beta_{nm} (\omega^{\prime\prime},\omega)
\beta_{nm^\prime}^* (\omega^{\prime\prime}\omega') = \delta_{m m^\prime}
\delta(\omega -\omega^\prime),
\end{gather}
obtained in a similar fashion as Eq.~\eqref{eq:appa1}.


\section{The Hamiltonian}
\label{app:B1}

Here we show how the Hamiltonian (\ref{eq:aaham}) is derived from the
Hamiltonian (\ref{eq:hqp}) that involves operators associated with
the eigenmodes of the total system. We separately treat the two
contributions to the Hamiltonian (\ref{eq:hqp}) involving integrals
over momentum and position operators, respectively. We start with
the contribution
\begin{equation}
T = \frac{1}{2} \sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \;
p_m^\dagger(\omega) p_m(\omega).
\end{equation}
Substitution of the representation (\ref{eq:qpsplit}) for $p_m
(\omega)$ reduces the right hand side to three integrals which can
be done using Eqs.\ (\ref{eq:appa1})--(\ref{eq:appa3}). The result
has the form
\begin{equation}
\label{eq:appb2}
T = \frac{1}{2}\sum_{\lambda} P^\dagger_{\lambda} P_{\lambda} +
\frac{1}{2} \sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \;
  P^{\dagger}_m(\omega) P_m(\omega).
\end{equation}
The second contribution
\begin{equation}
\label{eq:qcont}
V = \frac{1}{2}\sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega
\; \omega^2 q^{\dagger}_m(\omega)q_m(\omega)
\end{equation}
is more difficult to compute due to the presence of the term
$\omega^2$ in the integral over frequency. Substituting the
representation (\ref{eq:qpsplit}) for $q_m(\omega)$ into Eq.\
(\ref{eq:qcont}), we obtain
\begin{multline}
V = \frac{1}{2} \sum_{\lambda \lambda^{\prime}} V^{(1)}_{\lambda
  \lambda^\prime} Q^{\dagger}_{\lambda} Q_{\lambda^\prime}
+\frac{1}{2} \sum_{m^\prime m^{\prime\prime}}
\int\limits_{\omega_{m'}}^{\infty} \!  \rmd\omega^{\prime} \!
  \int\limits_{\omega_{m''}}^{\infty} \!  \rmd\omega^{\prime\prime} \;
    V^{(2)}_{m^\prime m^{\prime\prime}}
    (\omega^\prime,\omega^{\prime\prime})
    Q_{m^\prime}^{\dagger}(\omega^\prime) Q_{m^{\prime\prime}}
    (\omega^{\prime\prime})\\
+\frac{1}{2} \sum_{m^\prime} \sum_{\lambda}
    \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime \left(
      V^{(3)}_{m^\prime \lambda}(\omega^\prime )
      Q_{\lambda}^{\dagger} Q_{m^\prime}(\omega^\prime) + \text{H.c.}
      \right) ,
\end{multline}
where $V^{(1)}$, $V^{(2)}$, and $V^{(3)}$ are integrals over the
coefficients $\alpha$ and $\beta$,
\begin{align}
\label{eq:defv1}
V^{(1)}_{\lambda \lambda^\prime} & = \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \omega^2 \alpha_{m
  \lambda}(\omega) \alpha^{*}_{m \lambda^\prime} (\omega) , \\
\label{eq:defv2}
V^{(2)}_{m^\prime m^{\prime\prime}}(\omega',\omega'') & = 
\sum_m \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \omega^2 \beta_{m^\prime m}
(\omega^\prime,\omega) \beta^{*}_{m^{\prime\prime}
m}(\omega^{\prime\prime},\omega) , \\
\label{eq:defv3}
V^{(3)}_{m^\prime \lambda} (\omega^\prime ) & = \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \;\omega^2 \alpha_{m
  \lambda}(\omega) \beta_{m^\prime m}^{*} (\omega^\prime,\omega).
\end{align}
We compute these integrals using relations that follow from the
eigenmode equation (\ref{eq:heveqsep}). Projecting this equation
onto the cavity states $\langle {\bm \mu}_\lambda|$ and the channel
states $\langle {\bm \nu}_m (\omega)|$, we obtain the set of coupled
equations
\begin{subequations}
\label{eq:abess}
\begin{gather}
\label{eq:exp1}
\left( \omega^2 - \omega_{\lambda}^2 \right) \alpha_{m \lambda}(\omega) = \sum_{n}\!
\int\limits_{\omega_{n}}^{\infty} \! \rmd\omega^\prime \; W_{\lambda
  n}(\omega^\prime) \beta_{mn} (\omega,\omega'),\\
\label{eq:exp2}
\left( \omega^2 - \omega^{\prime 2} \right) \beta_{mn}(\omega,\omega') = \sum_{\lambda}
W^{*}_{\lambda n}(\omega^{\prime})
\alpha_{m \lambda}(\omega).
\end{gather}
\end{subequations}
We used the definition \eqref{eq:Wlm} of $W_{\lambda m}(\omega)$ and
the definitions (\ref{eq:defalbe}) of $\alpha$ and $\beta$. As an
example we evaluate the integral $V^{(1)}$. We multiply both sides of
Eq.\ (\ref{eq:exp1}) by $\alpha^{*}_{\lambda^\prime m}(\omega)$ and
then sum over $m$ and integrate over $\omega$. This yields
\begin{multline}
\label{eq:defv4}
V^{(1)}_{\lambda \lambda^\prime} = \omega^2_{\lambda} \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \;
\alpha^{*}_{m \lambda^\prime}(\omega) \alpha_{m \lambda}(\omega) \\
+\sum_{m^\prime} \int\limits_{\omega_{m'}}^{\infty} \! \rmd\omega^\prime
\; W_{\lambda m^\prime}(\omega^\prime) \Biggl[ \sum_m
  \int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \alpha^{*}_{m
  \lambda^\prime}(\omega)\beta_{m m^\prime} (\omega, \omega^\prime)
  \Biggr].
\end{multline}
The term in the square brackets vanishes according to Eq.\
(\ref{eq:appa1}). The remaining term on the right hand side can be
simplified using Eq.\ (\ref{eq:defal}).  Equation (\ref{eq:defv4})
then reduces to
\begin{equation}
\begin{split}
V^{(1)}_{\lambda \lambda^\prime} &= \omega^2_{\lambda} \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \langle {\bm
  \mu}_{\lambda}|{\bm \phi}_m(\omega)\rangle \langle {\bm
  \phi}_{m}(\omega)|{\bm \mu}_{\lambda^\prime}\rangle \\
&= \omega^2_{\lambda} \, \delta_{\lambda \lambda^\prime}.
\end{split}
\end{equation}
In a similar fashion we obtain for $V^{(2)}$ and $V^{(3)}$
\begin{align}
V^{(2)}_{m^\prime m^{\prime\prime}}(\omega',\omega'') & = \omega'\,
\delta_{m' m''} \, \delta(\omega' - \omega''), \\
V^{(3)}_{m^\prime \lambda} (\omega^\prime ) & = W_{\lambda m'}(\omega').
\end{align}

Combining results the second contribution to the Hamiltonian takes the
form
\begin{equation}
\label{eq:qcont2}
V = \frac{1}{2} \sum_{\lambda} \omega^2_{\lambda}
Q^{\dagger}_{\lambda}Q_{\lambda} + \frac{1}{2} \sum_m
\int\limits_{\omega_{m}}^{\infty} \! \rmd\omega \; \omega^2
Q^{\dagger}_m(\omega) Q_m(\omega) + \sum_m \!  \sum_{\lambda}
\int\limits_{\omega_{m}}^{\infty} \!  \rmd\omega \, W_{\lambda m}(\omega)
Q^{\dagger}_{\lambda} Q_m(\omega).
\end{equation}
Where for the last term we used
\begin{equation}
\label{eq:QDQ}
\sum_m \! \sum_{\lambda} \int\limits_{\omega_{m}}^{\infty} \!
\rmd\omega \, W_{\lambda m}(\omega) Q^{\dagger}_{\lambda} Q_m(\omega) =
\sum_m \! \sum_{\lambda} \int\limits_{\omega_{m}}^{\infty} \!
\rmd\omega \, W_{\lambda m}^{*}(\omega) Q_{\lambda} Q_m^{\dagger}(\omega).
\end{equation}
This last identity is obtained using relations \eqref{eq:qprealres},
\eqref{eq:qprealout} for the position operators, and the alternative
definition of the coupling amplitude
\begin{equation}
\label{eq:Wlm2}
W_{\lambda m}(\omega) = \sum_{\lambda'}
\sum_{n}\int\limits_{\omega_n}^{\infty} \rmd \omega' \,
\mathcal{N}_{\lambda\lambda'} \mathcal{N}_{mn}^{\dagger}(\omega, \omega')
W_{\lambda' m}^{*}.
\end{equation}
The sum of the contributions (\ref{eq:appb2}) and (\ref{eq:qcont2})
yields the Hamiltonian (\ref{eq:aaham}). 

\section{Local Density of States Inside an Open Resonator}

For the electromagnetic field in the presence of a dielectric medium
the local density of states at at point $\vect{r}$ is defined in terms
of the Wiener--Khintchine theorem \cite{Mand95},
\begin{equation}
\label{eq:gldos1}
\rho(\omega,\vect{r}) = \frac{1}{\pi \hbar \omega}
\int\limits_{-\infty}^{+\infty} \! \rmd\tau \, \rme^{\rmi \omega \tau}
\epsilon(\vect{r}) \langle \vect{E}^{(+)}(\vect{r},t)
\vect{E}^{(-)}(\vect{r},t-\tau) \rangle_{\text{vac}}.
\end{equation}
Here $\epsilon(\vect{r})$ is the real and frequency independent
dielectric constant, $\langle\dots\rangle_{\text{vac}}$ denotes the
quantum average over the vacuum state, and $(\pm)$ refers to the
positive and negative component of the electric field. Substituting
the field representation in terms of the normal modes of Maxwell's
equations (see Eq.~\eqref{eq:evequniv}),
\begin{equation}
\vect{E}^{(+)}(\vect{r},t) = \sum_{m} \int \! \rmd\omega
\left(\frac{\hbar\omega}{2}\right)^{\frac{1}{2}} A_m(\omega)
  \rme^{-\rmi\omega t} \vect{f}_m(\omega,{\bm r}),
\end{equation}
and taking into account that $\langle A_{m}(\omega)
A_{m'}^{\dag}(\omega')\rangle_{\text{vac}} =
\delta_{mm'}\delta(\omega-\omega')$, equation \eqref{eq:gldos1}
reduces to
\begin{equation}
\label{eq:gldos2}
\rho(\omega,\vect{r}) = \sum_{m} \epsilon(\vect{r}) \vert
\vect{f}_{m}(\omega,{\bm r}) \vert^{2}. 
\end{equation}

In the case of open cavities one is mainly interested in the local
density of states at points $\vect{r}$ inside the resonator. It is
then useful to use the projection formalism to express the local
density of states inside the cavity; utilizing the mode expansion
\eqref{eq:fsep}, Eq.~\eqref{eq:gldos2} may be rewritten as
\begin{equation}
\label{eq:rldos1}
\rho(\omega,\vect{r}) = \sum_{m} \sum_{\lambda\lambda'}
\epsilon(\vect{r}) \alpha_{\lambda m}(\omega) \alpha_{\lambda'
  m}^{*}(\omega) \vect{u}_{\lambda}(\vect{r}) \cdot
\vect{u}_{\lambda'}^{*}(\vect{r}),
\end{equation}
for $\vect{r}$ inside the cavity. Here $\vect{u}_{\gl}(\vect{r})$ are
the mode functions of the cavity.

The change in the local density of states introduced by the
cavity are quantitatively studied using the cavity gain factor
\begin{equation}
\label{eq:cgf}
G^c(\omega) = \frac{\int_{{\bm r}\in I} \! \rmd{\bm r}
  \, \rho(\omega,{\bm r})}{\int_{{\bm r}\in I}\! \rmd{\bm r} \,
  \rho_{0}(\omega,{\bm r})} \, ,
\end{equation}
that is closely related to the dwell time of scattered radiation
inside the resonator \cite{Carv02}. Here, $\rho(\omega, {\bm r})$ is
the local density of states inside the cavity and $\rho_0 (\omega,{\bm
  r})$ the free space local density of states in the absence of the
cavity. In addition, we note that the integrated local density of
states can be expressed in terms of the expansions coefficients
(\ref{eq:expcoefa}) as
\begin{equation}
\label{eq:rldos}
\int_{{\bm r}\in I} \! \rmd{\bm r} \, \rho(\omega,{\bm r})
= \sum_{m}\sum_{\lambda} |\alpha_{m\lambda}(\omega)|^2.
\end{equation}



\section{Exact Modes of Maxwell's Equations}\
\label{app:appA}

In this appendix we summarize the normal modes of Maxwell's equations
for the three systems treated in Sec.~\ref{sec:examples}; for a more
detailed derivation we refer to references \cite{Ujih75,Ley87,Hent02}.
The modes--of--the--universe are taken to be scattering states with an
incoming wave in only one scattering channel. In the two dimensional
example, this channel is labeled by the angular momentum number
$m$. The scattering states are normalized according to
Eq.~\eqref{eq:forth}.


\subsubsection{One Dimensional Dielectric Cavity}
For the dielectric cavity of Fig.~\ref{fig:cavities}(a) the scattering
states are given by
\begin{equation}
\label{eq:f01}
f(k,x) = \frac{1}{\sqrt{2\pi}} \begin{cases} 
                \dfrac{I_{k}}{n} \sin(n k(x+l)) & (-l < x < 0), \vspace{5pt}\\
                \exp(- \rmi kx) + S_k \exp(\rmi kx) & (0 < x).
\end{cases}
\end{equation}
They satisfy the boundary condition $f(k,-l)=0$, imposed by the
completely reflecting mirror at $x=-l$, and are continuous with
continuous derivative for any value of $x>-l$. The
single-channel scattering matrix and the mode strength amplitude are
given, respectively, by
\begin{gather}
\label{eq:s01}
S_k = - \frac{n+ \rmi \tan(n k l)}{n - \rmi \tan(n k l)}, \\
\label{eq:I01}
I_k = - \frac{2\rmi n}{n\cos(nkl) - \rmi \sin(nkl)}\, .
\end{gather}

\subsubsection{Cavity with Semitransparent Mirror}
The scattering states for the one dimensional cavity with the
semitransparent mirror (Fig.~\ref{fig:cavities}(b)) have the form
\begin{equation}
\label{eq:f03}
f(k,x) = \frac{1}{\sqrt{2\pi}} 
           \begin{cases} I_{k}
  \sin(k(x+l)) & (-l < x < 0), \\ 
\exp(-\rmi kx) + S_k \exp(\rmi kx) & (0 < x),
\end{cases} 
\end{equation}
where the scattering matrix $S_k$ and the mode strength amplitude
$I_k$ are given by
\begin{gather}
\label{eq:s03}
S_k =\frac{\rmi -\eta k + \cot(k l)}{\rmi+\eta k - \cot(k l)}, \\
\label{eq:I03}
I_k =\frac{2\rmi}{(\rmi +k\eta)\sin(kl)-\cos(kl)}.
\end{gather}
Here $\eta$ specifies the mirror transparency. The modes satisfy
$f(k,-l)=0$ at the perfectly reflecting mirror and are continuous
everywhere else. At the semitransparent mirror their derivative has
a discontinuity proportional to the mode amplitude, $f'(k,0_{+}) -
f'(k,0_{-}) = -\eta k^2 f(k,0)$, where the prime denotes
differentiation with respect to the position.


\subsubsection{Dielectric Disk}
The exact eigenstates for a two dimensional dielectric disk of
radius $R$ and refractive index $n$ embedded in empty space read
\begin{equation}
\label{eq:f04}
f_m(k,r,\phi) = \sqrt{\frac{k}{8 \pi}} \rme^{- \rmi m\phi} 
           \begin{cases} I_{mk} J_{m}(nkr) & (0 < r < R), \\ 
H_{m}^{(2)}(kr) + S_{mk} H_{m}^{(1)}(kr) & (R < r),
\end{cases}
\end{equation}
where $m$ labels angular momentum. The channel with index $m$ is
open when $k$ exceeds the channel threshold $k_{m}=m/nR$. Due to
rotational symmetry angular momentum is conserved, and the
scattering matrix is diagonal in the angular momentum basis
\begin{equation}
\label{eq:s04}
S_{mk} =-\frac{H_{m}^{\prime(2)} (kR) - n [J_{m}^{\prime}(nkR) /
    J_{m}(nkR)] H_{m}^{(2)} (kR)} {H_{m}^{\prime(1)} (kR) - n
  [J_{m}^{\prime}(nkR) / J_{m}(nkR)] H_{m}^{(1)} (kR)}.
\end{equation}
The mode strength amplitude takes the form
\begin{equation}
\label{eq:I04}
I_{mk} =\frac{ 4 \rmi}{\pi k R \Bigl( J_{m}(nkR) H_{m}^{\prime(1)} (kR) - n
  J_{m}^{\prime}(nkR) H_{m}^{(1)} (kR) \Bigr)}.
\end{equation}


\section{External Green Functions}
\label{app:appB}

Here we evaluate the action of the non--Hermitian differential
operator $L_{\text{eff}}(k)$ on an arbitrary cavity state. According
to Eq.~\eqref{eq:Leff} $L_{\text{eff}}(k)$ is the sum of two
operators. The first contribution $L_{\mathcal{QQ}}$ has already
been computed in equations (\ref{eq:Lqq01}), (\ref{eq:Lqq02}),
(\ref{eq:Lqq03}) and (\ref{eq:Lqq04}). The second contribution has
the form
\begin{equation}
\label{eq:Leffb}
L_{\mathcal{QP}} \mathcal{G}_{\text{ch}}(k) L_{\mathcal{PQ}} \mu(r,\phi)
= L_{\mathcal{QP}} \sum_{m} \int\limits_{k_{m}}^{\infty} \!  \rmd k' \,
\frac{|{\bm \nu}_{m}(k')\rangle \langle {\bm \nu}_{m}(k')|}{k^2 - k'^2
  + \rmi \epsilon} L_{\mathcal{PQ}} \mu(r,\phi) ,
\end{equation}
where $\mathcal{G}_{\text{ch}}$ stands for the retarded Green function
of the isolated channel region and on the right hand side we used the
completeness of the channel modes with the help of
Eq.~\eqref{eq:evepp1}. Below we compute this contribution for the cavities
of interest.


\subsubsection{One Dimensional Channel}

For a one dimensional semi-infinite channel $x \geq 0$ with
Dirichlet boundary conditions at $x=0_+$ we find the retarded Green
function from the solutions (\ref{eq:ex01nu}) by contour
integration,
\begin{equation}
\begin{split}
  \mathcal{G}_{\text{ch}}(k,x,x') &= \frac{2}{\pi}
  \int\limits_{0}^{\infty} \!  \rmd k' \frac{\sin(k'x) \sin(k'x')}{k^2
  - k'^2 + \rmi \epsilon} \\ 
&= \frac{\rmi}{2k} \bigl(
  \rme^{\rmi k |x+x'|} - \rme^{-\rmi k |x-x'|} \big).
\end{split}
\end{equation}
Combination with the definitions (\ref{eq:Lpq01}) and (\ref{eq:Lqp01})
yields the real space representation of $L_{\text{eff}}(k)$. The relevant
derivatives have to be done with care as the limits $x \to 0$ and
$x^\prime \to 0$ must be taken independently. The result takes the
form
\begin{equation}
\label{eq:Gex01}
 L_{\mathcal{QP}}
\mathcal{G}_{\text{ch}}(k) L_{\mathcal{PQ}} \mu(x) =
-\frac{\rmi k \delta(x-0_{-})}{n^2} \mu(0_{-}).
\end{equation}
Together with equation (\ref{eq:Lqq01}) we arrive at
the result \eqref{eq:Leff01b} for $L_{\text{eff}}(k)$.

In a similar fashion we evaluate the retarded Green function for
the isolated channel problem with Neumann conditions at $x=0_{+}$.
Using the solutions (\ref{eq:ex02nu}) we obtain in this case
\begin{equation}
\label{eq:Gex02}
\begin{split}
\mathcal{G}_{\text{ch}}(k,x,x') &= \frac{2}{\pi} \int\limits_{0}^{\infty} \!
\rmd k' \frac{\cos(k'x) \cos(k'x')}{k^2 - k'^2 + \rmi \epsilon} \\
&= -\frac{\rmi}{2k} \bigl( \rme^{\rmi k |x+x'|} + \rme^{-\rmi k |x-x'|} \bigr).
\end{split}
\end{equation}
Combination with the real space representation (\ref{eq:Lpq02}) and
(\ref{eq:Lqp02}) of $L_{\mathcal{PQ}}$ and $L_{\mathcal{QP}}$,
respectively, yields 
\begin{equation}
\label{eq:Geff02}
 L_{\mathcal{QP}} \mathcal{G}_{\text{ch}}(k) L_{\mathcal{PQ}} \mu(x) =
\frac{\rmi \,\delta^\prime (x-0_{-})}{n^2 k} \frac{\rmd}{\rmd x^\prime} 
\mu(x^\prime) \biggl\vert_{x^\prime=0_{-}}.
\end{equation}
Combinations of equations (\ref{eq:Leff}), (\ref{eq:Lqq02}), and
(\ref{eq:Geff02}) yields $L_{\text{eff}}(k)$ given in equation
\eqref{eq:Leff02b}.


\subsubsection{Channel with Semitransparent Mirror}

The semitransparent mirror has width $d$ and refractive index $n$.
In the limit $d \to 0$, $n \to \infty$ with $n^2 d=\eta$ fixed, the
retarded Green function can be expressed in terms of an integral
over products of the scattering states (\ref{eq:ex03nu}),
\begin{equation}
\label{eq:Gex03}
\mathcal{G}_{\text{ch}} (k,x,x^\prime) = \frac{1}{2\pi}
\int\limits_{-\infty}^{\infty} \! \rmd k^\prime \frac{\rme^{\rmi k'(x-x')} 
+ S_{c}(k') \rme^{\rmi k'(x+x')}}
{k^2 - k'^2 + \rmi \epsilon} \, .
\end{equation}
The integral can be done by contour integration. Using the unitarity
of $S_{c}(k)$ and taking into account that $S_{c}(k)$ is analytic in
the upper half of the complex plane, Eq.~\eqref{eq:Gex03} reduces to
\begin{equation}
\label{eq:Gex03b}
\mathcal{G}_{\text{ch}} (k,x,x^\prime) = 
 -\frac{\rmi}{2k} \bigl( \rme^{\rmi k |x-x^\prime|} + S_{c}(k)
 \rme^{\rmi k |x+x^\prime|}
\bigr) \, .
\end{equation}
Combination with the definitions (\ref{eq:Lpq03}) and (\ref{eq:Lqp03})
yields
\begin{equation}
\label{eq:Geff03c}
 L_{\mathcal{QP}} \mathcal{G}_{\text{ch}}(k) L_{\mathcal{PQ}} \mu(x) =-
\frac{\delta^\prime(x-0_{-})}{k (\rmi + \eta k)} \frac{\rmd}{\rmd x^\prime}
 \mu(x^\prime) \biggl\vert_{x^\prime=0_{-}}.
\end{equation}
Equations (\ref{eq:Lqq03}), (\ref{eq:Geff03c}) along with
Eq.~\eqref{eq:Leff} yield Eq.~\eqref{eq:Leff03b}.


\subsubsection{Angular Momentum Channels}

The retarded Green function for the two--dimensional Helmholtz
equation with Neumann boundary conditions along a disk of radius
$R$, has the form
\begin{multline}
 \mathcal{G}_{\text{ch}}(k,r',\phi',r,\phi) = \\
 - \frac{\rmi}{2\pi}
\sum\limits_{-\infty}^{+\infty} \rme^{\rmi m(\phi-\phi')} \frac{H_{m}^{(1)}
  (kr_{>})}{H_{m}^{\prime (1)} (kR)} \bigl( H_{m}^{(2)}
(kr_{<})H_{m}^{\prime (1)} (kR) - H_{m}^{\prime (2)}
(kR) H_{m}^{(1)} (kr_{<}) \bigr),
\end{multline}
where $r_{<} \; (r_{>})$ stands for the smaller (larger) of $r$ and
$r'$. Substitution of the definitions (\ref{eq:Lpq04}) and
(\ref{eq:Lqp04}) yields
\begin{multline}
\label{eq:Geff04c}
 L_{\mathcal{QP}} \mathcal{G}_{\text{ch}}(k) L_{\mathcal{PQ}}
\mu(r,\phi) = - \frac{R}{2n^2\pi} \frac{\partial}{\partial r} \left(
\frac{1}{r} \delta (r-R_{-}) \right)  \\
\times \sum\limits_{m=-\infty}^{+\infty}
\int\limits_{0}^{2\pi} \! \rmd\phi' \; \rme^{\rmi m(\phi-\phi')}
\frac{H_{m}^{(1)}(kR)}{kH_{m}^{\prime (1)}(kR)}
\frac{\partial}{\partial r''}\mu(r'',\phi') \biggr\vert_{r''=R_{-}} \,
.
\end{multline}
Combination with equations (\ref{eq:Leff}) and (\ref{eq:Lqq04}) shows
that $L_{\text{eff}}(k)$ acts on an arbitrary resonator state $\mu
(r,\phi)$ like
\begin{multline}
\label{eq:leff04}
 L_{\text{eff}}(k) \mu(r,\phi) = -\frac{1}{n^2} \nabla^2 \mu(r,\phi) +
\frac{R}{n^2} \frac{\partial}{\partial r} \left( \frac{1}{r} \delta
(r-R_{-}) \right) \Biggl[ \mu(R_{-},\phi) \\ - \frac{1}{2\pi}
  \sum\limits_{m=-\infty}^{+\infty} \int\limits_{0}^{2\pi} \! d\phi'
  \; \rme^{\rmi m(\phi-\phi')} \frac{H_{m}^{(1)}(kR)}{kH_{m}^{\prime
      (1)}(kR)} \frac{\partial}{\partial r''}\mu(r'',\phi')
  \biggr\vert_{r''=R_{-}} \Biggr].
\end{multline}
Conservation of angular momentum and the requirement that the singular
terms must vanishes for the right eigenstates $\xi_{mj}(k,r,\phi)$
yield the boundary condition
\begin{equation}
\label{eq:rbcs04}
\xi_{mj}(k,R_{-},\phi) = 
  \frac{H_{m}^{(1)}(kR)}{kH_{m}^{\prime (1)}(kR)}
  \frac{\partial}{\partial r''}\xi_{mj}(k,r'',\phi) \biggr\vert_{r''=R_{-}}.
\end{equation}
There is a discrete set of solutions
\begin{equation}
\label{eq:rres04}
\xi_{mj}(k,r,\phi) = A_{mj}(k) \rme^{\rmi m\phi} J_m(n \sigma_{mj}(k) r),
\end{equation}
with normalization constants $A_{mj}(k)$. Substitution into
Eq.~\eqref{eq:rbcs04} yields the equation for the eigenvalues
\begin{equation}
k J_{m}(n \sigma_{mj}(k) R) H_{m}^{\prime (1)}(kR) - n\sigma_{mj}(k)
J_{m}^{\prime}(n \sigma_{mj}(k) R) H_{m}^{(1)}(kR) = 0.
\end{equation}
Combination with the fixed point equation $k = \sigma_{mj}(k)$
finally gives the resonance condition (\ref{eq:rescircle}).


\section{Local Density of States Inside a Dielectric Disk}
\label{app:appC}

Here we determine the density of states inside a dielectric disk with
radius $R$ and refractive index $n$. We start from equation
\eqref{eq:rldos}. Using the coefficients (\ref{eq:a04}), we obtain
\begin{equation}
\label{eq:C01}
\int\limits_{\text{disk}} \rmd\bm{r} \rho(k,\bm{r}) =
\sum_{m=-\infty}^{+\infty} \frac{k |I_{mk}|^2}{2 n^2} J_{m}^{2}(nkR)
\sum\limits_{\lambda=1}^{\infty}\frac{k_{m\lambda}^2}{(k^2-k_{m\lambda}^2)^2}
\, .
\end{equation}
We concentrate on the evaluation of the last sum on the right hand
side. It can be written as a derivative
\begin{equation}
\label{eq:C02}
\sum_{\lambda=1}^{\infty}
\frac{k_{m\lambda}^2}{(k^2-k_{m\lambda}^2)^2} = -\frac{1}{2}
\frac{\partial}{\partial \alpha} \sum_{\lambda=1}^{\infty}
\frac{1}{(k^2-\alpha^2 k_{m\lambda}^2)}
\biggr\vert_{\alpha=1} \, .
\end{equation}
The right hand side can be simplified by noticing that $n R
k_{m\lambda}= x_{m\lambda}$ is the $\lambda$-th zero of the Bessel
function $J_m(x)$. Thus the sum can be identify as the trace of the
Green function for the radial part of the Helmholtz equation in a
dielectric disk with Dirichlet conditions on the disk
perimeter. Eq.~\eqref{eq:C02} becomes
\begin{equation}
\label{eq:C03}
\sum_{\lambda=1}^{\infty}
\frac{k_{m\lambda}^2}{(k^2-k_{m\lambda}^2)^2} = -\frac{1}{2}
\frac{\partial}{\partial \alpha} \int\limits_{0}^{R} \! \rmd r \, r
\mathcal{G}_{\text{disk}}(k/\alpha,r,r)
\Bigr\vert_{\alpha=1} \, .
\end{equation}
The disk Green function can be found by standard methods
\cite{Jack75} and is given by
\begin{multline}
\label{eq:C04}
 \mathcal{G}_{\text{disk}}(k/\alpha,r',r) = \frac{\rmi \pi}{4
J_{m}\left(nkR/\alpha \right)} J_{m}\left(nkr_{<}/\alpha \right) \\
\times\left( J_{m}\left(nkR/\alpha \right) Y_{m}\left(nkr_{>}/\alpha
\right) - Y_{m}\left(nk R/\alpha\right) J_{m}\left(nkr_{>}/\alpha
\right) \right),
\end{multline}
where $Y_{m}$ is a Bessel function of the second kind and $r_{<}$
($r_{>}$) the smaller (larger) of $r$ and $r'$. The problem then
reduces to the evaluation of the integral in Eq.~\eqref{eq:C03}.
Using the relations
\begin{align}
 \int\limits_{0}^{R} \! \rmd r \,r J_{m}^{2}(kr) &= \frac{R^2}{2}
\left( J_{m}^{2}(kR) - J_{m+1}(kR) J_{m-1}(kR) \right), \\
\int\limits_{0}^{R} \! \rmd r \,r J_{m}(kr)Y_{m}(kr) &=
 \frac{R^2}{2}
\Bigl( J_{m}(kR)Y_{m}(kR) \nonumber\\
&\quad - \frac{1}{2} J_{m+1}(kR) Y_{m-1}(kR) -
\frac{1}{2} J_{m-1}(kR) Y_{m+1}(kR) \Bigr) \, , 
\end{align}
we find after some straightforward manipulations the result
\begin{equation}
\int\limits_{0}^{R} \! \rmd r \, r \mathcal{G}_{\text{disk}}(k/\alpha,r,r)
= \frac{\alpha R J_{m}^{\prime}(kR/\alpha)}{2kJ_{m}(kR/\alpha)}.
\end{equation}
Substitution into Eq.~\eqref{eq:C03} yields
\begin{equation}
\label{eq:C05}
 \sum_{\lambda=1}^{\infty}
\frac{k_{m\lambda}^2}{(k^2-k_{m\lambda}^2)^2} = \frac{R^2}{4
  J_{m}^{2}(nkR)} \left( J_{m}^{2}(nkR) - J_{m+1}(nkR) J_{m-1}(nkR)
\right) \, .
\end{equation}
After replacement into Eq.~\eqref{eq:C01}, one arrives at the
integrated density of states \eqref{eq:ldos04}.
