\section{Introduction}
\label{sec:Intro}

The exchange interactions control the magnetic order and properties of a vast number of materials
\cite{White2006Dec}
and lead to many fascinating phenomena, such as various types of the Kondo effect
\cite{Kondo,NozieresBlandin,Pustilnik_Glazman}.
Double quantum dots (DQDs), and in general multi-impurity systems, constitute
a convenient and controllable playground,
where nearly as much different exchange mechanisms compete with each other to
shape the ground state of the system.
\emph{Local exchange} between the spin of a quantum dot (QD)
and the spin of conduction band electrons gives rise to the
Kondo effect \cite{Kondo,Hewson_book}.
\emph{Direct exchange} arriving with an additional side-coupled QD may destroy it or lead to the
two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia,Granger,ZitkoBonca,ZitkoPRB2010,Ferreira}.
In a geometry where the two QDs contact the same lead, conduction band electrons
mediate the \emph{RKKY exchange} \cite{RK,K,Y}. The RKKY interaction competes
with the Kondo effect and leads to the quantum phase transition of a still debated nature
\cite{Doniach,Jones,Affleck,Bork,Neel,KondoRKKYexp,Hans,Hans2,Fabian}.
Moreover, in DQDs coupled in series also \emph{superexchange} can alter the Kondo physics significantly
\cite{Zitko_2QDEx,Sela}.

Recently, hybrid quantum devices, in which the interplay between various magnetic correlations
with superconductivity (SC) plays an important role, have become an important direction of research
\cite{hybridQDs,SCspintronics}. In particular, chains of magnetic atoms on SC surface have proven
to contain self-organized Majorana quasi-particles and exotic spin textures
\cite{Braunecker,Klinovaja,Vazifeh,Yazdani},
while hybrid DQD structures have been used to split the Cooper pairs coherently into two entangled
electrons propagating to separated normal leads \cite{CPS1,CPS2,CPS4,CPS5,CPS9}.
The latter is possible due to non-local (\emph{crossed}) Andreev reflections (CARs),
in which each electron of a Cooper pair tunnels into different QD, and
subsequently to attached lead. Such processes give rise to an exchange mechanism \cite{Yao},
that we henceforth refer to as \emph{the CAR exchange}, which can greatly modify
the low-temperature transport behavior of correlated hybrid nanostructures.

The CAR exchange may be seen as RKKY-like interaction between
two nearby impurities on SC surface \cite{Yao}.
The effect can be understood as a consequence
of spin-dependent hybridization of the Yu-Shiba-Rusinov (YSR)
states \cite{Yu,Shiba,Rusinov} in SC contact,
caused both by the overlap of their wave functions
and their coupling to Cooper-pair condensate.
This process is the most effective when the YSR states
are close to the middle of the SC gap, {\it e.g.} in the YSR-screened phase \cite{YSRscreening}.
The mechanism presented here is essentially the same,
yet in the considered regime can be understood
perturbatively without referring to YSR states,
as a consequence of the non-local pairing induced by SC electrode.
In particular, the presence of YSR bound states close to the Fermi level
is not necessary for significant consequences for the Kondo physics,
as long as some inter-dot pairing is present.

The proximity of SC induces pairing in QDs \cite{RozhkovArovas,Buitelaar}
and tends to suppress the Kondo effect if the superconducting energy gap $2\Delta$
becomes larger than the relevant Kondo temperature $T_K$
\cite{Buitelaar2002Dec,adatomsSC,Kondo_vs_SC1,Kondo_vs_SC2,Zitko_Kondo-Andreev,Zitko_S-QD-N,IW_Sau,YSRscreening}.
Moreover, the strength of SC pairing can greatly affect the Kondo physics in the sub-gap transport regime:
For QDs attached to SC and normal contacts, it can enhance the Kondo effect
\cite{DomanskiIW,KWIW,part1}, while
for DQD-based Cooper pair splitters, it tends to suppress both the $\mathrm{SU}(2)$ and $\mathrm{SU}(4)$ Kondo effects \cite{IW_Kacper}.
Our main result is that the non-local pairing induced by superconducting
proximity effect, which gives rise to CAR exchange, can be the sole cause of the Kondo screening.
Moreover, relatively small values of coupling to SC, $\GS{}\ll U$, are sufficient for the effect to occur.
This is in contrast to the DQD system considered in Ref.~\cite{part1},
where only one of the quantum dots is proximized, such that
CAR exchange cannot arise,
and the Kondo physics becomes qualitatively
affected only for $\GS{}\sim U/2$.%

\begin{figure}[bt]
\centering
\includegraphics[width=1\linewidth]{Fig1.png}
\caption{
		 (a) Schematic of the considered system. Left/right (L/R) lead
		 is coupled to the first quantum dot (QD1), while superconductor
		 is attached to both QD1 and QD2.
		 (b)-(d) illustrate an example of direct spin exchange:
		 spin-up electron from the initial state (b) hops to the other QD (c) and spin-down electron
		 hops back (d). Note, that the final state is in fact the same singlet state,
		 only with opposite sign.
		 (e)-(g) show an example of process contributing to crossed Andreev reflection (CAR) exchange.
		 A Cooper pair from SC approaches DQD (e) and two singlets of the same charge
		 are formed (f), before the Cooper pair is re-emitted (g).
		 (h)-(j) present an example of RKKY process: an electron scattered off
		 one QD (h) mediates the spin exchange towards the other (i), before it is finally scattered
		 off there, too (j).
		 }
\label{fig:system}
\end{figure}

In this paper we discuss the CAR-induced Kondo screening in a setup comprising T-shaped DQD
with normal and superconducting contacts, see \fig{system}(a).
We note that despite quite generic character of CAR exchange,
and its presence in systems containing at least two localized electrons
coupled close to each other to the same SC bath,
to best of our knowledge CAR-induced screening
has hardly been identified in previous studies
\cite{CPS1,CPS2,CPS4,CPS5,CPS9,IW_Kacper,IW_Sau,Zitko_Josephson,Zitko_S2QD,Martinek2017}.
In the system proposed here [\fig{system}(a)], its presence is evident.
Moreover, CAR exchange magnitude can be directly related to the relevant energy scales, such as the Kondo
temperature, which provides a fingerprint for quantitative experimental verification of our predictions.

The paper is organized as follows. In \Sec{model} we describe the considered system
and present the model we use to study it. In \Sec{scales} the relevant energy scales are estimated
to make the discussion of main results concerning CAR-induced Kondo effect in \Sec{main} more clear.
Finally, the influence of effects neglected in \Sec{main} are presented in the following sections,
including CAR exchange interplay with RKKY interaction (\Sec{RKKY}), particle-hole asymmetry (\Sec{asym}),
couplings asymmetry (\Sec{x}) and reduced efficiency of CAR coupling (\Sec{coef}). In summary,
the effects discussed in \Sec{main} remain qualitatively valid in all these cases.
The paper is concluded in \Sec{conclusions}.

\section{Model}
\label{sec:model}

The schematic of the considered system is depicted in \fig{system}(a).
It contains two QDs attached to a common SC lead.
Only one of them (QD1) is directly attached to the left (L) and right (R) normal leads,
while the other dot (QD2) remains coupled only through QD1.
The SC is modeled by the BCS Hamiltonian,
$H_{\mathrm{S}}=\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}}a_{\mathbf{k}\sigma}^{\dag}a_{\mathbf{k}\sigma}-\Delta\sum_{\mathbf{k}}(a^\dag_{\mathbf{k}\uparrow}a_{-\mathbf{k}\downarrow}^{\dag}+a_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow})$,
with energy dispersion $\xi_{\mathbf{k}}$, energy gap $2\Delta>0$ and $a_{\mathbf{k}\sigma}$ annihilation operator
of electron possessing spin $\sigma$ and momentum $\mathbf{k}$. The coupling between
SC and QDs is described by the hopping Hamiltonian
$H_{\mathrm{TS}}=\sum_{i\mathbf{k}\sigma}v_{\mathrm{S}i}(d^\dagger_{i\sigma}a^{}_{\mathbf{k}\sigma}+h.c.)$,
with $d^\dagger_{i\sigma}$ creating a spin-$\sigma$ electron at QD$i$. The matrix element
$v_{\mathrm{S}i}$ and the normalized density of states of SC in normal state, $\rho_{\rm S}$,
contribute to the coupling of QD$i$ to SC electrode as $\GS{i} = \pi \rho_{\rm S} |v_{{\rm S}i}|^2$.
We focus on the sub-gap regime, therefore, we integrate out SC degrees of freedom lying outside the energy gap \cite{RozhkovArovas}.
This gives rise to the following effective Hamiltonian,
$H_{\mathrm{eff}}=H_{\mathrm{SDQD}}+H_{\rm L}+H_{\rm R}+H_{\rm T}$,
where
\begin{eqnarray}
H_{\rm SDQD} 	& = &
				\sum_{i\sigma} \varepsilon_{i} n_{i\sigma}
				+\sum_{i} U n_{i\uparrow} n_{i\downarrow}
				+U' (n_1-1)(n_2-1)
				\nonumber\\
				&+&\sum_\sigma t(d^\dagger_{1\sigma}d^{}_{2\sigma} + h.c.)
				+J \vec{S}_1\vec{S}_2
				\nonumber\\
				&+&\sum_{i} \!\!\left[ \Gamma_{{\rm S}i} (d^\dagger_{i\uparrow} d^\dagger_{i\downarrow} \!+\! h.c.)
				+\Gamma_{\rm SX} (d^\dagger_{i\uparrow} d^\dagger_{\bar{i}\downarrow} \!+\! h.c.) \right]
	\label{H_DQD}
\end{eqnarray}
is the Hamiltonian of the SC-proximized DQD
\cite{IW_Kacper,Walldorf2018Feb}, with QD$i$ energy level $\varepsilon_i$,
inter-site (intra-site) Coulomb interactions $U'$ ($U$),
inter-dot hopping $t$, and CAR coupling $\GS{\rm X}$.
$n_{i\sigma}=d^\dagger_{i\sigma}d^{}_{i\sigma}$ denotes the electron number operator
at QD$i$, $n_i=n_\uparrow+n_\downarrow$, and $\bar{i}\equiv 3-i$.
Our model is strictly valid in the regime where $\Delta$ is the largest
energy scale. Nevertheless, all discussed phenomena are
present in a full model for energies smaller than SC gap.
Moreover, by eliminating other consequences of the presence of SC lead,
our model pinpoints the fact that the non-local pairing is
sufficient for the occurrence of the CAR exchange.
The presence of out-gap states shall result mainly in additional broadening of DQD energy levels,
changing the relevant Kondo temperatures.
We note that the procedure of integrating out out-gap states neglects the
RKKY interaction mediated by SC lead and other possible indirect exchange mechanisms%
 \footnote{
 Note, that by RKKY interaction we mean only such an effective exchange,
 which arises due to multiple scattering of a single electron or hole, see \fig{system}(h)-(j).
 Other mechanisms leading to the total indirect exchange are considered separately.
 In particular, in the large gap limit, exchange described in Ref.~\cite{Yao} is in fact reduced to
 the CAR exchange, and additional antiferromagnetic contribution would arise for finite gap.
 }.
To compensate for this,
we explicitly include the Heisenberg term $ J \vec{S}_1\vec{S}_2$ in
$H_{\rm SDQD}$, with $\vec{S}_i$ denoting the spin operator of QD$i$
and a Heisenberg coupling $J$ substituting the genuine RKKY exchange.

The normal leads are treated as reservoirs of noninteracting electrons,
$H_{r}=\sum_{\mathbf{k}\sigma}\varepsilon_{r\mathbf{k}}c^\dagger_{r\mathbf{k}\sigma}c^{}_{r\mathbf{k}\sigma}$,
where $c^{}_{r\mathbf{k}\sigma}$ annihilates an electron of spin
$\sigma$ and momentum $\mathbf{k}$ in lead $r$ ($r={\rm L,R}$) with the corresponding energy $\varepsilon_{r\mathbf{k}\sigma}$.
The tunneling Hamiltonian reads,
$H_{\rm T} = \sum_{r\mathbf{k}\sigma} v_{r} (d^\dagger_{1\sigma}c^{}_{r\mathbf{k}\sigma} + h.c.)$,
giving rise to coupling between lead $r$ and QD$i$ of strength $\Gamma_r = \pi \rho_r |v_r|^2$,
with $\rho_r$ the normalized density of states of lead $r$ and $v_r$ the
local hopping matrix element, assumed momentum-independent.
We consider a wide-band limit, assuming constant $\Gamma_r=\Gamma/2$
within the cutoff $\pm D = \pm 2U$ around the Fermi level.

For thorough analysis of the CAR exchange mechanism and its consequences
for transport, we determine the linear conductance between the two normal leads from
\begin{equation}
G = \frac{2e^2}{h} \pi \Gamma \int \left[ -\frac{\partial f_T}{\partial\omega} \right] \mathcal{A}(\omega) {\rm d} \omega ,
\label{G}
\end{equation}
where $f_T$ is the Fermi function at temperature $T$,
while $\mathcal{A}(\omega)$ denotes the normalized local spectral density
of QD1 \cite{fn1}.
Henceforth, unless we state otherwise, we assume a maximal CAR coupling,
$\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$ \cite{IW_Kacper,Walldorf2018Feb},
$\GS{1}=\GS{2}=\GS{}$ and consider DQD tuned to the particle-hole symmetry point,
$\varepsilon_1=\varepsilon_2=-U/2$. However, these assumptions are not crucial for the results presented
here, as discussed in Secs.~\ref{sec:asym}-\ref{sec:coef}.

\section{Estimation of relevant energy scales}
\label{sec:scales}

Since we analyze a relatively complex system, let us build up the understanding of its behavior starting
from the case of a QD between two normal-metallic leads, which can be obtained in our
model by setting $t=\GS{}=J=U'=0$. Then, the conductance as a function of temperature, $G(T)$, grows
below the Kondo temperature $T_K$ and reaches maximum for $T\to 0$, $G(T\!=\!0)=G_{\rm max}$.
At particle-hole symmetry point, the unitary transmission is achieved, $G_{\rm max}= G_0 = 2e^2/h$;
see short-dashed line in \fig{G-T}(a).
An experimentally relevant definition of $T_K$ is that at $T=T_K$
$G(T)=G_{\rm max}/2$. $T_K$ is exponentially small in
the local exchange $J_0 = 8\Gamma / (\pi \rho U)$, and is approximated by
$T_K \approx D \exp[-1/(\rho J_0)]$ \cite{Hewson_book}.

The presence of a second side-coupled QD, $t,U'>0$, significantly enriches the physics of the system
by introducing direct exchange between QDs, see \fig{system}(b-d).
In general, effective inter-dot exchange can be defined as energy difference between
the triplet and singlet states of isolated DQD,
$J^{\mathrm{eff}} = E_{S=1} - E_{\rm GS}$. Unless $U$ becomes very large, superexchange can be neglected
\cite{Zitko_2QDEx} and $J^{\mathrm{eff}}$ is determined by \emph{direct exchange}, $J^{\mathrm{eff}}\approx 4t^2/(U-U')>0$.
When the hopping $t$ is tuned small \cite{CPS1}, one can expect $J^{\mathrm{eff}}\lesssim T_K$, which
implies the two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia}.
Then, for $T \ll T_K$, the local spectral density of QD1 serves as a band of width $\sim T_K$ for QD2.
The spin of an electron occupying QD2
experiences the Kondo screening below the associated Kondo temperature
\begin{equation}
T^* = a T_K \exp(- b T_K / J_{\rm eff})
\label{Tstar}
\end{equation}
with $a$ and $b$ constants of order of unity \cite{Pustilnik_Glazman,Cornaglia}.
This is reflected in conductance, which drops to $0$ with lowering $T$, maintaining characteristic
Fermi-liquid
$G\sim T^2$ dependence \cite{Cornaglia}; see the curves indicated with squares
in \fig{G-T}(a). Similarly to $T_K$, experimentally relevant definition of $T^*$ is that
$G(T\!=\!T^*) = G_{\rm max}/2$. Even at the particle-hole
symmetry point $G_{\rm max} < G_0$, because the single-QD strong-coupling fixed point
is unstable in the presence of QD2 and $G(T)$ does not achieve $G_0$ exactly,
before it starts to decrease.

The proximity of SC gives rise to two further exchange mechanisms that
determine the system's behavior. First of all, the (conventional)
\emph{RKKY interaction} appears, $J \sim \GS{}^2$ \cite{RK,K,Y}.
Moreover, the \emph{CAR exchange} emerges as a consequence of finite $\GS{}$ \cite{Yao}.
It can be understood on the basis
of perturbation theory as follows. DQD in the inter-dot singlet state may absorb
and re-emit a Cooper pair approaching from SC; see \fig{system}(e)-(g). As a second-order
process, it reduces the energy of the singlet, which is the ground state of isolated DQD.
A similar process is not possible in the triplet state due to spin conservation.
Therefore, the singlet-triplet energy splitting $J^{\mathrm{eff}}$ is increased (or generated for $t=J=0$).
More precisely, the leading ($2$nd-order in $t$ and $\GS{}$) terms
in the total exchange are
\begin{equation}
J^{\mathrm{eff}} 	\approx 	J + \frac{4t^2}{U-U'+\frac{3}{4}J} + \frac{4\GS{}^2}{U+U'+\frac{3}{4}J}.
\label{Jeff}
\end{equation}
Using this estimation, one can predict $T^*$ for finite $\GS{}$, $t$ and $J$ with \eq{Tstar}.
Apparently, from three contributions corresponding to:
(i) RKKY interaction, (ii) direct exchange and (iii) CAR exchange, only the first may bear a negative (ferromagnetic) sign.
The two other contributions always have an anti-ferromagnetic nature.
More accurate expression for $J^{\mathrm{eff}}$ is derived in Appendix~\ref{sec:downfolding}
[see \eq{A_J}] by the Hamiltonian down-folding procedure. The relevant terms differ
by factors important only for large $\GS{}/U$.
Finally, it seems worth stressing that normal leads are not necessary for CAR exchange to occur.
At least one of them is inevitable for the Kondo screening though, and two symmetrically coupled
normal leads allow for measurement of the normal conductance.

It is also noteworthy that inter-dot Coulomb interactions
decrease the energy of intermediate states contributing to direct exchange
[\fig{system}(c)], while increasing the energy of intermediate
states causing the CAR exchange [\fig{system}(f)].
This results in different dependence of corresponding terms in \eq{Jeff} on $U'$.
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), it has a significant effect
on the actual values of $T^*$.

\begin{figure}
\includegraphics[width=1\linewidth]{Fig2.pdf}
\caption{(a) Linear conductance $G$ as function of $T$ calculated for
		 $\varepsilon_1=\varepsilon_2=-U/2$, $\Gamma=U/5$, $U'=U/10$ and different situations,
		 as indicated. The quantity $\xi\equiv\sqrt{\GS{}^2+t^2}$ is fixed
		 for different curves drawn with the same dashing style.
		 Note the logarithmic scale on both axes.
		 %
		 (b) Points show $T^*/T_K$ calculated by NRG from curves in subfigure (a).
		 Lines present the fit to \eq{Tstar} with $J^{\mathrm{eff}}$ obtained from \eq{Jeff}.
		 %
		 (c) The same as (b), only for $U'=0$.
		 %
		 (d) and (e) show the residual conductance $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of
		 $\GS{}$ for $t=0$ (denoted "CAR") and $t=\GS{}$ (denoted "Both").
		 Dotted line is a guide for eyes. $U'=U/10$ in (b) and (d) and $U'=0$ in (c) and (e).
		}
\label{fig:G-T}
\end{figure}

\section{CAR exchange and Kondo effect}
\label{sec:main}

To verify \eqs{Tstar}-(\ref{Jeff}) we calculate $G$ using
accurate full density matrix numerical renormalization group (NRG) technique \cite{WilsonNRG,Weichselbaum,FlexibleDMNRG,fn2}.
We compare $U'=0$ case with experimentally relevant value $U'=U/10$ \cite{Keller2013Dec}.
While for two close adatoms on SC surface RKKY interactions may lead to prominent consequences
\cite{Klinovaja}, the conventional ({\it i.e.} non-CAR) contribution should
vanish rapidly when the inter-impurity distance $r$ exceeds a few lattice constants \cite{RKKYrange,SC_RKKY}.
Meanwhile, the CAR exchange may remain significant for $r$ of the order
of coherence length of the SC contact \cite{Yao}. Therefore, we first neglect the conventional RKKY coupling and analyze its consequences in Sec.~\ref{sec:RKKY}.

The main results are presented in \fig{G-T}(a), showing the temperature dependence of $G$
for different circumstances.
For reference, results for $\GS{}=0$ are shown, exhibiting
the two-stage Kondo effect caused by \emph{direct} exchange mechanism.
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), an excellent agreement of $T^*$ found from NRG calculations and \eq{Tstar}
is obtained with $a=0.42$ and $b=1.51$, the same for both $U'=0$ and $U'=U/10$. Note,
however, that $J^{\mathrm{eff}}$ is different in these cases, cf. \eq{Jeff},
and $U'$ leads to increase of $T^*$.

Furthermore, for $t=0$ and $\GS{}>0$ the two-stage Kondo effect caused solely by the \emph{CAR
exchange} is present; see \fig{G-T}(a).
Experimentally, this situation
corresponds to a distance between the two QDs smaller than the superconducting coherence length,
but large enough for the exponentially suppressed direct hopping to be negligible.
While intuitively one could expect pairing to compete with any kind of magnetic ordering,
the Kondo screening induced by CAR exchange is a beautiful example of a superconductivity
in fact leading to magnetic order, namely the formation of the Kondo singlet.
This CAR-exchange-mediated Kondo screening is our main finding.
For such screening, \eq{Tstar} is still fulfilled with very similar
parameters, $a=0.37$ ($a=0.35$) and $b=1.51$ ($b=1.50$) for $U'=0$ ($U'=U/10$),
correspondingly; see \figs{G-T}(b-c).
Moreover, as follows from \eq{Jeff}, $U'$ reduces CAR exchange, and therefore diminishes $T^*$.
For the same values of $J^{\mathrm{eff}}$, the dependence of $G(T)$ for $t=0$ and $\GS{}>0$ is hardly different
from the one for $\GS{}=0$ and $t>0$ for $T\geq T^*$ (results not shown).
However, $G(T)$ saturates at residual value $G_{\mathrm{min}}$ as $T\to 0$ only for finite
$\GS{}$, which at particle-hole symmetry makes $G_{\mathrm{min}}$
the hallmark of SC proximity and the corresponding CAR exchange processes.
From numerical results, one can estimate it as
\begin{equation}
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{}^2}{U^2}
	\qquad {\scriptstyle (\GS{1}=\GS{2}=\GS{})} ,
\label{Gmin}
\end{equation}
with $c\approx 2.25$, barely depending on $U'$ and getting smaller for $t>0$.
This is illustrated in \figs{G-T}(d-e), where the dotted line corresponds to \eq{Gmin} with $c=2.25$.

Lastly, in \fig{G-T}(a) we also present the curves obtained for $t=\GS{}$ chosen such,
that the quantity $\xi=\sqrt{t^2+\GS{}^2}$ remains the same
in all the cases.
This is to illustrate what happens when \emph{both} (direct and CAR) exchange interactions are
present. \fig{G-T}(c) clearly shows that $T^*$ remains practically unaltered for $U'=0$.
The comparison with \fig{G-T}(b) proves that in this case it practically does not depend
on $U'$. The enhancement of direct exchange is compensated by the decrease of the CAR one.
On the contrary, $G_{\mathrm{min}}$ decreases for larger $t$ below the estimation given by Eq.~(\ref{Gmin}),
as can be seen in \figs{G-T}(d-e).

While analyzing the results concerning $G_{\mathrm{min}}(\GS{})$ plotted in \figs{G-T}(d-e)
one needs to keep in mind that $G_{\mathrm{min}}$ is obtained at deeply cryogenic conditions. To illustrate
this better, $G(\GS{})$ obtained for $t=0$ and $T=10^{-6}U$ is plotted with solid line
in \fig{3}. Clearly, for weak $\GS{}$ the system exhibits rather conventional (single-stage)
Kondo effect with $G=G_{\mathrm{max}}\approx 2e^2/h$, while QD2 is effectively decoupled ($G_{\mathrm{max}}<2e^2/h$
in the proximity of SC lead \cite{KWIW}). Only for larger values of $\GS{}$
the CAR exchange is strong enough, such that $T^*>T$ and the dependence $G(\GS{})$ continuously
approaches the $T=0$ limit estimated by \eq{Gmin} and presented in \figs{G-T}(d-e).

\section{CAR-RKKY competition}
\label{sec:RKKY}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig3.pdf}
\caption{Linear conductance $G$ vs. $\GS{}$ calculated
		 for $t=0$, $\Gamma=U/5$, $U'=U/10$, finite $T=10^{-6}U$
		 and different values of RKKY coupling $J$, as indicated.
		 Inset shows QD1 spectral function $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0.1U$ curve, indicated with corresponding symbols.
		}
\label{fig:3}
\end{figure}

Let us now discuss the effects introduced by the conventional RKKY interaction.
We choose $t=0$ for the sake of simplicity and
analyze a wide range of $\GS{}$, starting from the case of anti-ferromagnetic
RKKY interaction ($J>0$). Large $J>0$ leads to the formation of a molecular singlet in the
nanostructure. This suppresses the conductance, unless $\GS{}$ becomes of the order of $U/2$,
when the excited states of DQD are all close to the ground state. This is illustrated
by double-dotted line in \fig{3}.
Smaller value of $J>0$ causes less dramatic consequences, namely it just increases $J^{\mathrm{eff}}$ according
to \eq{Jeff}, leading to enhancement of $T^*$, cf. \eq{Tstar}. This is presented with
dot-dashed line in \fig{3}.

The situation changes qualitatively for ferromagnetic RKKY coupling, $J<0$.
Then, RKKY exchange and CAR exchange have opposite signs and compete with each other.
Depending on their magnitudes and temperature, one
of the following scenarios may happen.

For $J^{\mathrm{eff}} > 0$, {\it i.e.} large enough $\GS{}$, and $T<T^*$, the system is in the
singlet state due to the two-stage Kondo screening of DQD spins. $G(T\!=\!0)$ is reduced
to $G_{\mathrm{min}}$, which tends to increase for large negative $J$; see dashed lines in \fig{3}.
In the inset to \fig{3}, the spectral density of QD1 representative for this regime is plotted
as curve indicated by triangle. It corresponds to a point on the $J=-0.1U$ curve in the main
plot, also indicated by triangle. The dip in $\mathcal{A}(\omega)$ has width of order of $T^*$.

For finite $T$, there is always a range of sufficiently small $|J^{\mathrm{eff}}|$, where QD2 becomes effectively
decoupled, and, provided $T<T_K$, $G$ reaches $G_{\mathrm{max}}$ due to conventional Kondo effect
at QD1. This is the case for sufficiently small $\GS{}$ for $J=0$ or $J=-0.01U$, and in the narrow
range of $\GS{}$ around the point indicated by a circle in \fig{3} for $J=-0.1U$ (for $J=0.01U$,
the considered $T$ is close to $T^*$ and $G$ does not reach $G_{\rm max}$). The conventional Kondo effect manifests itself with
a characteristic peak in $\mathcal{A}(\omega)$, as illustrated in the inset in \fig{3} with line denoted by circle.

Finally, large enough $J^{\mathrm{eff}} < 0$ and low $T$, give rise to an effective ferromagnetic coupling of DQDs
spins into triplet state. Consequently, the underscreened Kondo effect occurs
\cite{Mattis,NozieresBlandin} for weak $\GS{}$ and, {\it e.g.}, $J=-0.1U$;
see the point indicated by square in \fig{3}.
This leads to $G=G_{\mathrm{max}}$ and a peak in $\mathcal{A}(\omega)$, whose shape is significantly different from the
Kondo peak, cf. the curve denoted by square in the inset in \fig{3}.

\section{Effects of detuning from the particle-hole symmetry point}
\label{sec:asym}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig4.pdf}
\caption{
         (a) Linear conductance between the normal leads $G$ as a function of temperature $T$
         for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, and additional curves for finite
         detuning from particle-hole symmetry point, $\delta_1=-\delta_2$,
         and two values of $\xi=\sqrt{t^2+\GS{}^2}$, as indicated in the figure.
         (b) $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of QD1 detuning $\delta_1$ for different
         exchange mechanisms, $\xi=U/10$ and $\delta_2=\pm\delta_1$ (as indicated).
		}
\label{fig:asym}
\end{figure}

At PHS $G_{\mathrm{min}}=G(T \!=\! 0)=0$ in the absence of superconducting lead, making $G_{\mathrm{min}} > 0$ a hallmark
of SC-induced two-stage Kondo effect. However, outside of PHS point $G_{\mathrm{min}} > 0$ even in the case of
the two-stage Kondo effect caused by the direct exchange.
Exact PHS conditions are hardly possible in real systems, and the fine-tuning of the QD energy
levels to PHS point is limited to some finite accuracy.
Therefore, there may appear a question, if the results obtained at PHS are of any importance for the
realistic setups. As we show below --- they are,
in a reasonable range of detunings $\delta_i=\varepsilon_i +U/2$.

In \fig{asym}(a) we present the $G(T)$ dependence in and outside the PHS, corresponding to
parameters of \fig{G-T}(a).
Clearly, for considered small values of $\delta_1=\delta_2=\delta$,
$G_{\mathrm{min}}<10^{-3}e^2/h$ for direct exchange only, while $G_{\mathrm{min}}$ in the presence of a superconductor is
significantly increased and close to the PHS value. Furthermore, for $|\delta_1| \sim |\delta_2|
\sim \delta$, the residual conductance caused by the lack of PHS, $G_{\mathrm{min}} \approx e^2/h \cdot (\delta/U)^2$,
which is a rapidly decreasing function in the vicinity of PHS point, as illustrated in \fig{asym}(b)
with lines denoted by a square. Evidently, in the regime $|\delta_i| < 0.01U$ the residual conductance
caused by SC is orders of magnitude larger, leading to the plateau in $G_{\mathrm{min}}(\delta_1)$ dependence,
visible in \fig{asym}(b).
Taking into account that the realistic values of $U$ in the semiconductor quantum dots are rather
large, this condition seems to be realizable by fine-tuning of QD gate voltages.

Lastly, let us point out that while in the presence of only one exchange mechanism, \emph{CAR} or
\emph{direct}, $G_{\mathrm{min}}(\delta_1)$ dependencies depicted in \fig{asym}(b) are symmetrical with respect
to sign change of $\delta_1$, for \emph{both} exchange mechanisms the dependence is non-symmetric.

\section{Effects of asymmetry of couplings to superconductor}
\label{sec:x}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig5.pdf}
\caption{
		 (a) Linear conductance between the normal leads, $G$, as a function of temperature, $T$,
		 for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, for different values
		 of asymmetry coefficient $x$ [see \eq{xGS}], in the presence of \emph{CAR} exchange only.
		 %
		 (b) The second-stage Kondo temperature $T^*$ normalized by $T_K$ as a function of $x$,
		 calculated with the aid of NRG (points) and a fit to \eq{Tstar} (lines)
		 with $J^{\mathrm{eff}}$ from \eq{Jeff}.
		 %
		 (c) The zero-temperature conductance $G_{\mathrm{min}}$ as a function of QD1 coupling to SC lead, $\GS{1}$,
		 compiled from data obtained at different circumstances (as indicated in the legend)
		 for different $x$. Dotted line corresponds to \eq{Gmin2} with $c=2.25$.
		}
\label{fig:x}
\end{figure}

Similarly to PHS, the ideal symmetry in the coupling between respective QDs and SC lead is hardly possible
in experimental reality. As shown below, it does not introduce any qualitatively new features.
On the other hand, it decreases the second stage Kondo temperature, which is already small, therefore,
quantitative estimation of this decrease may be important for potential experimental approaches.
To analyze the effects of $\GS{1}\neq\GS{2}$, we introduce the asymmetry parameter $x$ and extend
the definition of $\GS{}$,
\beq
x = \frac{\GS{1}-\GS{2}}{\GS{1}+\GS{2}}, \quad \GS{} = \frac{\GS{1}+\GS{2}}{2}.
\label{xGS}
 \end{equation}
Note, that even for a fixed $\GS{}$, the actual CAR coupling $\GS{\rm X}=\GS{}\sqrt{1-x^2}$ decreases
with increasing $|x|$, which is a main mechanism leading to a decrease of $T^*$ outside the $x=0$ point
visible in \figs{x}(a) and (b). To illustrate this, the curves corresponding to \emph{both} exchange
mechanisms were calculated using $x$-dependent $t=\GS{\rm X}$ instead of $t=\xi/\sqrt{2}$.
Therefore, $\xi$ was generalized for $x\neq 0$ by setting $\xi=\sqrt{t^2(1-x^2)^{-1}+\GS{}^2}$.
Clearly, in \fig{x}(b) the curves for different exchange mechanisms are very similar and differ mainly
by a constant factor, resulting from different influence of $U'$; see \Sec{scales}.
The magnitude of $T^*$ changes is quite large, exceeding an order of magnitude for $x=\pm 0.5$
and $\xi=U/20$. Moreover, $T^* \to 0$ for $x\to\pm 1$. Consequently, for strongly asymmetric
devices one cannot hope to observe the second stage of Kondo screening.

A careful observer can note that the $T^*(x)$ dependency is not symmetrical; note for example different
$T^*$ for $x=\pm 0.5$ in \fig{x}(a). This is caused by the dependence of the first stage Kondo temperature
$T_K$ on $\GS{1}$ \cite{part1,DomanskiIW},
\beq
\widetilde{T}_K(\GS{1}) = T_K \cdot \exp\!\left( \frac{\pi}{2} \frac{\GS{1}^2}{\Gamma U}\right).
 \end{equation}
Here, $T_K$ is, as earlier, defined in the absence of SC, while $\widetilde{T}_K$ is a function
of $\GS{1}$, such that $G(\widetilde{T}_K) = G_{\rm max}(\GS{1})/2$ in the absence of QD2.
As $\widetilde{T}_K$ grows for increasing $\GS{1}$ (or $x$), $T^*$ decreases according to \eq{Tstar}.
Its $\GS{}$ dependence can be accounted for by small changes in the coefficients $a$ and $b$ in \eq{Tstar},
as long as $x$ is kept constant.

To close the discussion of $T^*(x)$ dependence let us point out, that in \eq{A_J}
there appears a correction to \eq{Jeff} for $x\neq 0$. However, it is very small due to additional
factor $\GS{}^2/U^2$ in the leading order. Its influence on curves plotted in \fig{x}(b) is hardly visible.

In turn, let us examine the $x$ dependence of the $T=0$ conductance $G_{\mathrm{min}}$. As can be seen
in \fig{x}(a), it monotonically increases with $x$, as it crosses $x=0$ point. In fact, \eq{Gmin}
can be generalized to
\beq
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{1}^2}{U^2} ,
\label{Gmin2}
 \end{equation}
with $c\approx 2.25$ (indicated by a dotted line in \fig{x}(c)). Note that $G_{\mathrm{min}}$ is proportional to
$\GS{1}^2=(x+1)^2 \GS{}^2$, instead of simply $\GS{}$, cf. \eq{Gmin}. The values of $G_{\mathrm{min}}$ obtained
from all analyzed $G(T)$ dependencies for different $x$ have been compiled in \fig{x}(c).
It is evident, that \eq{Gmin2} is approximately fulfilled for all the considered cases.

Finally, it seems noteworthy that the normal-lead coupling asymmetry,
$\Gamma_{\rm L}\neq \Gamma_{\rm R}$, is irrelevant for the results except for a constant factor
diminishing the conductance $G$ \cite{KWIWJB-asym}.

\section{The role of CAR efficiency}
\label{sec:coef}

\begin{figure}[tb]
\includegraphics[width=0.98\linewidth]{Fig6.pdf}
\caption{Linear conductance between the normal leads
		 $G$ as a function of coupling to SC lead, $\GS{}$, for indicated values of RKKY exchange $J$
		 and the efficiency of CAR processes reduced by factor (a) $\mathcal{C}=0.9$ and (b) $\mathcal{C}=0.5$.
		 Other parameters as in \fig{3}.
		 Insets: QD1 local spectral density $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0.1U$ curve, indicated with corresponding symbols.
		}
\label{fig:C}
\end{figure}

Up to this point we assumed $\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$, which is valid when the two
quantum dots are much closer to each other than the coherence length in the superconductor.
This does not have to be the case in real setups, yet relaxing this assumption does not
introduce qualitative changes. Nevertheless, the model cannot be extended to inter-dot
distances much larger than the coherence length, where $\GS{\rm X}\to 0$.

To quantitatively analyze the consequences of less effective Andreev coupling we define the
CAR efficiency as $\mathcal{C} \equiv \GS{\rm X} / \sqrt{\GS{1}\GS{2}}$ and analyze $\mathcal{C} < 1$
in the wide range of $\GS{1}=\GS{2}=\GS{}$ and other parameters corresponding to \fig{3}.
The results are presented in \fig{C}.

Clearly, decreasing $\mathcal{C}$ from $\mathcal{C}=1$ causes diminishing of $\GS{\rm X}$, and consequently of CAR
exchange. For a change as small as $\mathcal{C}=0.9$, the consequences reduce to some shift of the
conventional Kondo regime, compare \fig{C}(a) with \fig{3}. Stronger suppression of CAR may,
however, increase the SC coupling necessary to observe the second stage of Kondo screening caused
by CAR outside the experimentally achievable range, see \fig{C}(b). Moreover, the reduced $T^*$
leads to narrowing of the related local spectral density dip, while the
increased critical $\GS{}$ necessary for the observation of the second stage of screening leads to the
shallowing of the dip. This is visible especially in the inset in \fig{C}(b).

\section{Conclusions}
\label{sec:conclusions}

The CAR exchange mechanism is present in any system comprising at least
two QDs or magnetic impurities coupled to the same superconducting contact
in a way allowing for crossed Andreev reflections.
In the considered setup, comprised of two quantum dots in a T-shaped geometry
with respect to normal leads and proximized by superconductor,
it leads to the two-stage Kondo
screening even in the absence of other exchange mechanisms.
This CAR induced exchange screening is characterized by a residual
low-temperature conductance at particle-hole symmetric case.
We have also shown that the competition between CAR exchange and RKKY
interaction may result in completely different Kondo screening scenarios.

The presented results bring further insight into the low-temperature
behavior of hybrid coupled quantum dot systems, which hopefully could be verified
with the present-day experimental techniques.
Moreover, non-local pairing is present also in bulk systems such as non-$s$-wave superconductors.
The question if an analogue of discussed CAR exchange may play a role there
seems intriguing in the context of tendencies of many strongly correlated materials
to possess superconducting and anti-ferromagnetic phases.

\begin{acknowledgments}
This work was supported by the National Science Centre in Poland through project no.
2015/19/N/ST3/01030.
We thank J. Barna\'{s} and T. Maier for valuable discussions.
\end{acknowledgments}


\section{Introduction}
Setting up health information systems to monitor the evolving
burden of noncommunicable diseases (NCDs) and their risk factors,
is one of the claims of the Moscow Declaration, which was approved
by the First Global Ministerial Conference on Healthy Lifestyles
and NCD control \cite{Mos11}. In that respect, secular trends of
the incidence of NCDs are especially important as they indicate
changes of the risk profile of the population under consideration.
Common ways to detect secular trends in the incidence are either
performing a series of follow-up studies or running a register.
Both approaches may be very costly and lead to a variety of
practical problems. In contrast, a series of prevalence studies
sometimes is much easier to accomplish. Based on illness-death
model (IDM), this article describes a method for detecting trends
in the incidence of NCDs without a series of follow-up studies and
without a register.

\bigskip

The next section introduces the IDM and derives the theoretical
background for the method. In the third section, the theory is
applied to data from the National Diabetes Register in Denmark.
The register observed an increasing diabetes incidence in
1995--2004. We show that the trend is detectable using the IDM and
a series of prevalence data. The last section contains a summary.

\section{Illness-Death Model}
In modelling chronic (i.e., irreversible) diseases, often the
three-state model (compartment model) in Figure
\ref{fig:CompModel} is used. The numbers of persons in the states
\emph{Normal} and \emph{Disease} are denoted by $S$ and $C$. The
transition intensities (synonymously: rates) between the states
are: the incidence rate $i$ and the mortality rates $m_0$ and
$m_1$ of the healthy or the diseased, respectively. These rates
generally depend on the calendar time $t$, the age $a$ and in the
case of the mortality $m_1$ also on the duration of the disease
$d$.

\begin{figure}[ht]
  \centering
  \includegraphics[keepaspectratio,width=0.85\textwidth]{ThreeStates.eps}
\caption{Chronic disease model with three states and the
corresponding transition rates. People in the state \emph{Normal}
are healthy with respect to the disease under consideration. At
onset of the disease, they change to state \emph{Disease}.}
\label{fig:CompModel}
\end{figure}

Although the inclusion of the disease duration $d$ is also
possible, hereinafter it is assumed that $m_1$ does not depend on
$d.$ This article analytically describes the relationship between
the prevalence of a chronic disease, the incidence and mortality
rates. This problem has existed at least since 1934 \cite{Mue34},
but so far has only been solved in special cases. The article
\cite{Hen10} presents a brief review and further references.

\bigskip

As in \cite{Bri12}, we look for the numbers $S(t, a)$ and $C(t,
a)$ of healthy and diseased persons in terms of differential
equations, which can be derived from the disease model in Figure
\ref{fig:CompModel}. For the healthy persons we get the following
initial value problem:
\begin{align}
(\partial_t + \partial_a) \, S & = - \left [ m_0 +
i \right ] \, S \label{e:PDE_S_ta} \\
S(t - a, 0) & = S_0(t - a). \nonumber
\end{align}

Here $S_0(t - a) = S(t - a, 0)$ is the number of (healthy)
newborns\footnote{This paper only considers diseases acquired
after birth.} at calendar time $t-a.$ The notation $\partial_x$
denotes the partial derivative with respect to $x, ~x \in \{t,
a\}$. The solution $S(t, a)$ is
\begin{equation}\label{e:S}
S(t, a)= S_0(t - a) \, \exp \left ( - \int_0^a m_0(t-a+\tau, \tau)
+ i(t-a+\tau, \tau) \, \mathrm{d}\tau \right ),
\end{equation}
which may be checked easily.

\bigskip

The number $C$ of diseased persons are described similarly:
\begin{align}
(\partial_t + \partial_a) \, C & = -m_1 \, C + i \, S \label{e:PDE_C_ta} \\
C(t, 0) & = 0. \nonumber
\end{align}

The solution is
\begin{equation}\label{e:C}
C(t, a) = \int_0^a i( t -\delta, a - \delta) \, S(t - \delta, a -
\delta) \exp \left ( - \int_0^\delta m_1(t - \delta + \tau, a -
\delta + \tau) \mathrm{d} \tau \right ) \mathrm{d}\delta.
\end{equation}

Equation \eqref{e:C} allows the following interpretation: Starting
from $(t, a)$ at $\delta$ time units before, i.e., at $(t -
\delta, a - \delta)$, exactly $i(t - \delta, a - \delta) \, S (t -
\delta, a - \delta)$ persons newly enter state \emph{Disease}.
Until $(t, a)$ the proportion
$$\exp \left (- \int_0 ^ \delta m_1 (t - \delta
+ \tau, a - \delta  + \tau) \mathrm{d} \tau \right)$$ of those has
survived. Integration over all possible $0 \le \delta \le a$
yields the number of diseased persons at time $(t, a).$ After
applying the quotient rule to the age-specific prevalence
$$p(t, a) = \frac{C (t, a)}{S (t, a) + C (t, a)}$$ and using
\eqref{e:PDE_S_ta} and \eqref{e:PDE_C_ta} it follows

\begin{align}
(\partial_t + \partial_a) \, p & = \left ( 1-p \right ) \, \left (
i  - p \, \left (m_1 - m_0 \right ) \right ) \label{e:PDE}\\
p(t, 0) & = 0. \nonumber
\end{align}

\begin{rem}\label{rem:nachI}
For $t, a \ge 0$ with $p(t, a) \neq 1$ it holds
\begin{equation}\label{e:inc}
    i(t, a) = \frac{(\partial_t + \partial_a) \, p (t,
    a)}{1-p(t,a)} + p(t, a) \, \left (m_1(t, a)
- m_0(t, a) \right ).
\end{equation}
\end{rem}

Furthermore, the solution of \eqref{e:PDE} can be calculated
directly via \eqref{e:S} and \eqref{e:C}:
\begin{equation}
p(t, a) = \frac{ \int \limits_0^a i(t - \delta, a - \delta) \,
\exp \left ( -\int\limits_0^\delta \Psi(t - \delta + \tau, a -
\delta + \tau) \mathrm{d} \tau \right )
\mathrm{d}\delta}{1+\int\limits_0^a i(t-\delta, a - \delta) \,
\exp \left ( -\int\limits_0^\delta \Psi(t - \delta + \tau, a -
\delta + \tau) \mathrm{d}\tau \right ) \mathrm{d}\delta},
\label{e:p}
\end{equation}
with $\Psi := m_1 - m_0 - i.$

\bigskip

This follows from
\begin{align*}
C(t, a)
&= \int_0^a i( t -\delta, a - \delta) \, S_0(t-a) \\ & \qquad \cdot \exp
\left ( - \int_0^\delta m_1(t - \delta + \tau, a - \delta + \tau)
\mathrm{d} \tau - \int_0^{a-\delta}(m_0 + i)(t - a + \tau, \tau)
\mathrm{d} \tau
\right ) \mathrm{d}\delta \displaybreak[0]\\
&= \int_0^a i( t - \delta, a - \delta) \, S_0(t-a) \\ & \qquad \cdot \exp
\left ( - \int_0^\delta m_1(t - \delta + \tau, a - \delta + \tau)
\mathrm{d} \tau - \int_0^{a}(m_0 + i)(t - a + \tau, \tau)
\mathrm{d} \tau \right. \\ & \qquad \qquad \qquad + \left.  \int_{a - \delta}^{a} (m_0 + i)(t - a + \tau, \tau)
\mathrm{d} \tau
\right ) \mathrm{d}\delta \displaybreak[0]\\
&= S_0(t-a) \, \exp
\left ( - \int_0^a (m_0 + i)(t-a+\tau, \tau) \mathrm{d}\tau
\right ) \\
& \qquad \cdot \int \limits_0^a i(t - \delta, a - \delta) \, \exp
\left ( - \int \limits_0^\delta \Psi(t - \delta + \tau, a - \delta
+ \tau) \mathrm{d} \tau \right ) \mathrm{d}\delta.
\end{align*}

The first part of the last expression equals $S(t, a)$ and
Equation \eqref{e:p} follows.

\bigskip

The usefulness of equation \eqref{e:p} is obvious: Given the
incidence $i(t, a)$ and mortalities $m_0(t, a), ~m_1(t, a)$, the
prevalence $p(t, a)$ can be calculated for all $t, a \ge 0$.

\bigskip

So we can state
\begin{rem}
The prevalence $p(t, a)$ is independent from the number $S_0$ of
newborns.
\end{rem}

\begin{rem}
For $t, a \ge 0$ it holds: $0 \le p(t, a) \le 1.$
\end{rem}

\begin{rem}\label{rem:increasing}
If for some $(t_1, a_1)$ the integral $$\Upsilon(t_1, a_1) :=
\int\limits_0^{a_1} i(t_1-\delta, a_1 - \delta) \, \exp \left (
-\int\limits_0^\delta \Psi(t_1 - \delta + \tau, a_1 - \delta +
\tau) \mathrm{d}\tau \right ) \mathrm{d}\delta$$ is lower than for
$(t_2, a_2): ~\Upsilon(t_1, a_1) < \Upsilon(t_2, a_2)$, then it
holds $p(t_1, a_1) < p(t_2, a_2).$ This follows from observing
that $x \mapsto \nicefrac{x}{1+x}, ~x\ge 0$ is strictly
increasing.
\end{rem}

\bigskip

At the end of the section we introduce the \emph{relative
mortality} $R(t, a).$ For $(t, a) \ge 0$ with $m_0(t, a) > 0,$
define
\begin{equation*}
        R(t, a) = \frac{m_1(t, a)}{m_0(t, a)}.
\end{equation*}

Now we have all the definitions and results for the next section.

\section{Diabetes in Denmark}
In the article \cite{Car08} the age-specific prevalence of
diabetes for men (and women) in Denmark in the period 1995-2007 is
presented in great detail. The results are based on a complete
survey of the Danish population ($n > 5$ million). Classifying a
person as diabetic is done by combining different health
registers, which yields a sensitivity of more than 85\% \cite[p.
2188]{Car08}. In this paper we confine ourselves to the male
population in Denmark. The age-specific incidence rate $i$ for
2004 is given for all age groups, but for the other years in the
period 1995-2007 just relatively to 2004, averaged across all age
groups; likewise, with the mortality $m_0$ of the non-diabetic
population. Mortality $m_1$ significantly depends on the disease
duration \cite[Fig. 4]{Car08}. To the apply the model of the
previous section model, the duration dependence has to be
suppressed. This is done by an initialization step: The relative
mortality $R^\star (a)$ is calculated such that the observed
incidence and the associated increase in prevalence from 1995 to
1996 are in agreement. Therefor, the Equation \eqref{e:PDE} is
solved for $m_1$. Then, $ R^\star$ is calculated by $R^\star =
\nicefrac{m_1}{m_0}.$ For the period 1996 - 2004, this relative
mortality is kept fixed and Equation \eqref{e:PDE} is solved for
$i$ as in Remark \ref{rem:nachI} with $m_1 = R^\star \, m_0.$ By
doing so, the relative mortality $R^\star$ for the period
1996-2004 is assumed to be independent from calendar time. Thus,
we have a two-step approach:
\begin{enumerate}
    \item Initialization: Calculate $R^\star$ by fitting the observed incidence rate in 1995
    and the increase in age-specific prevalence from 1995 to 1996.
    \item Application: Derivation of the incidence rates in 1996-2004
    via Remark \ref{rem:nachI} mit $m_1(t, a) = R^\star(a) \, m_0(t, a).$
\end{enumerate}

\begin{rem}
After initialization, we just use the mortality $m_0$ of the
non-diabetic population and the prevalences $p(t, a),$ $t=1996,
\dots, 2004,$ for deriving $i(t, a),$ $t=1996, \dots, 2004.$
\end{rem}

Figure \ref{fig:Inc2001} shows the results of applying the model
to 2001 (circles). For comparison the observed incidence is shown
(solid line). Obviously, the data are in good agreement.

\begin{figure}[ht]
  \centering
  \includegraphics[width=.8\textwidth,keepaspectratio]{Incidence2001.eps}\\
  \caption{Age-spefic incidence rate in 2001: observed (solid
  line) and reconstructed by the model (circles).}\label{fig:Inc2001}
\end{figure}

\bigskip

Now, the increase in the incidence can be examined. For some of
the age groups, the incidence rate over calendar time is shown in
Figure \ref{fig:Trends}. In addition, the regression lines and the
corresponding correlation coefficients $r$ are given. In all age
groups there is a significant upward trend. The higher the age,
the better the fit of the linear regression model. Table \ref{tab}
shows the numerical values for all age groups.

\begin{figure}[ht]
  \centering
  \includegraphics[width=.9\textwidth,keepaspectratio]{Trends.eps}\\
  \caption{Incidence rates over calendar time for some age groups. }\label{fig:Trends}
\end{figure}

\begin{table}[ht]
  \centering
\begin{tabular}{c|c|c}
Age-    & Annual   & Correlation- \\
group   & increase (\%) & coefficient\\ \hline
40 -- 44 & 9.2 & 0.76 \\
45 -- 49 & 9.6 & 0.86 \\
50 -- 54 & 9.3 & 0.90 \\
55 -- 59 & 8.8 & 0.93 \\
60 -- 64 & 8.3 & 0.96 \\
65 -- 69 & 8.0 & 0.98 \\
70 -- 74 & 7.8 & 0.99 \\
75 -- 79 & 7.9 & 0.99 \\
80 -- 84 & 8.2 & 0.99 \\
\end{tabular}
\caption{Parameters of the secular trends in the incidence rates
in the age groups.}\label{tab}
\end{table}

The annual increase rates in the age groups (second column in
Table \ref{tab}) are all greater than the corresponding value
5.3\% reported in \cite[p. 2190]{Car08}. However, the reported
increase of 5.3\% refers to all persons (both sexes, all age
groups). Hence, a direct comparison with the values of Table
\ref{tab} is impossible.

\section{Summary}
In this work, a novel method for deriving trends in incidence from
a sequence of prevalence studies is presented. With a view to the
tremendous effort required by collecting incidence data, the novel
method provides a simpler alternative. A typical application is
the conversion of a sequence of telephone surveys for the
collection of age-specific prevalence of a chronic disease into a
sequence of incidence data.

As a first application, the method was used with data from the
Danish National Diabetes Register. The directly observed secular
trend in the incidence is visible by the new method as well.

\bigskip

In the application to the Danish Diabetes Register, the relative
mortality $R^\star$ in 1995 has been extrapolated for the period
1996-2004. While this may be possible for a period of eight years
is a word of caution is in order:
\begin{rem}
The calendar time trend in mortality $m_0$ of the non-diabetic
population is usually much better known than the trend in
mortality $m_1$ of the patients. The reason is that $m_0$ is
surveyed on a demographic scale, while $m_1$ is investigated
sporadically in epidemiological studies only. In epidemiology, one
might try to link the time trend in $m_1$ to the time trend in
$m_0$. The idea might be to measure a relative mortality $R$ at
some time $t' < t$ and extrapolate from $t'$ to $t$ and set $m_1
(t, a) = R (t', a) \, m_0 (t, a).$ Indeed, time dependence of
$m_1$ is enforced, but this approach may lead to a possibly
unexpected increase in the prevalence. If the incidence $i$ is
independent of $t$ and the family of functions $ t \mapsto m_{0,
a} (t): = m_0(t, a) $ is decreasing for all $a \ge 0$, then, by
Remark \ref{rem:increasing}, the function $t \mapsto p_{a}(t) : =
p (t, a)$ is monotonically increasing for all $ a \ge 0.$ This
means: although the incidence $i$ is remains unchanged by the
calendar time, by increasing the life expectancy of the healthy
(decreasing $m_{0, a}$), the prevalence increases. Extrapolating
the relative mortality $R$ from $t'$ to $t$ therefore must be
viewed critically.
\end{rem}

Beside the presentation of the theoretical background, this work
is little more than a feasibility study. There are two sources of
limitations:
\begin{enumerate}
    \item Data: Due to the incomplete detection of diabetes cases
    and the shortened report of incidence trends (pooled for all
    persons), a direct comparison between the observed and derived
    trends in the incidence is impossible.
    \item Model: Although it is evident that $m_1$ depends on the
    duration $d$ of diabetes, this dependency is neglected. In
    additon, the relative mortality $R^\star$ has been calculated
    for 1995, but has been extrapolated to 1996-2004.
\end{enumerate}
Both inaccuracies interact, which makes a rigorous evaluation
difficult. Thus, a systematic evaluation of the method based on a
comprehensive simulation study is necessary.


\subsection*{\sf I. Introduction}
The physical interpretation of \pti\ operators---and hence their
relevance for the description of physical systems---conti\-nues to
be debated \cite{bender04,mostafazadeh04,weigert04}. There is, however, no doubt about the cathartic role of \pty : it has become more
evident what it means to let go hermiticity in exchange for a
weaker property such as \pte . The success and ease to describe
quantum mechanical systems in terms of hermitean operators is based
on two of their generic properties, namely the existence of
{\em real} eigenvalues and their {\em diagonalizability}, i.e. the completeness of their orthonormal eigenstates. These properties do not persist if a quantum system was described by a \ptc\ Hamiltonian: its eigenvalues could be complex, and its eigenfunctions would, in general, neither be pairwise orthogonal nor form a complete set. Given a \pti\ operator, it thus appears desirable to decide whether it is diagonalizable or not.

The purpose of this contribution is to provide an algorithm
answering the question of whether a given \pti\ Hamiltonian
operator in a finite-dimensional space {\em does} or {\em does
not} possess a complete set of eigenstates. It is convenient to
represent such an operator as a \ptc\ matrix ${\sf M}$, say. A
procedure will be outlined which, after a finite number of steps,
will announce whether the matrix ${\sf M}$ at hand is
diagonalizable or not. In principle, the algorithm can be carried
out by hand for matrices of any dimension, and no approximations
are necessary.

Often, the question of diagonalizability will arise in a more general
setting where one considers not just a single matrix but a {\em
family}\ of \ptc\ matrices ${\sf M} (\varepsilon), \varepsilon \in
\mathbb R$. The parameter $\varepsilon$ measures the strength
$\varepsilon \in \mathbb R$ of a ``perturbation'' which destroys hermiticty while repecting \pte . As the parameter varies, all of the cases described previously may occur: typically, two real eigenvalues merge into a single real one at a critical value of $\varepsilon$, subsequently splitting into a pair of two complex conjugate eigenvalues, or {\em vice versa}. These dramatic modifications are accompanied by changes in the nature of the eigenstates of the \pti\ operator, possibly no longer spanning the space on which ${\sf M}(\varepsilon)$ acts.

This behaviour can be understood in terms of so-called {\em
exceptional points} \cite{kato84} which are known to occur
when a matrix is subjected to the perturbation depending analytically on
a parameter such as $\varepsilon$. At such a point, the
corresponding matrix is not diagonalizable, and its
spectrum may undergo a qualitative change. For a {\em hermitean} operator
subjected to a parameter-dependent {\em hermitean} perturbation, exceptional points cannot occur.

If one applies the algorithm testing for diagonalizability to a
parameter-dependent matrix ${\sf M} (\varepsilon)$, it will output
a polynomial in $\varepsilon$ instead of a number. Its zeros
correspond to those values of the perturbation parameter where the
matrix family ${\sf M} (\varepsilon)$ has exceptional points.
The matrices corresponding to these values of the perturbation
are not diagonalizable, and the spectra of matrices for nearby values of the parameter differ qualitatively.

The following section summarizes the properties of \pti\ systems in
terms of ($2 \times 2$) matrices. Then, the link between
diagonalizability and the so-called minimal polynomial is
reviewed. In Section 3, the algorithmic test is presented
which consists of constructing the minimal polynomial of the matrix followed by a search for degenerate roots by means of the Euclidean algorithm. Various methods are known to effectively
calculate the minimal polynomial of a matrix, outlined in Section
4. Simple examples are studied in Section 5, leading to some
general conclusions about the structure of \ptc\ Hamiltonian
operators in finite-dimensional spaces. Section 6 summarizes the
results and discusses the challenge to extend them to state spaces
of infinite dimension.
 %
 \subsection*{\sf II. \pti\ systems}
 %
A matrix {\sf H} is \pti\ \cite{bender+98},
 %
 \begin{equation}\label{ptinv}
 %
 [{\sf H} , {\sf P} {\sf T} ] = 0 \, ,
 %
\end{equation}
 %
if it commutes with the product of parity ${\sf P}$ and the anti-unitary operation of time reversal ${\sf T}$, represented here by complex conjugation, ${\sf T}^\dagger {\sf T} {\sf H} = {\sf H}^*$.
Eq. (\ref{ptinv}) implies that the characteristic polynomial of any \ptc\ operator {\sf H} has real coefficients only. Consequently, its roots are either real or come in complex-conjugate pairs. One way to show this is to construct a basis in which the Hamiltonian
has real matrix elements only \cite{bender+02}.

Let us briefly review the properties of \ptc\ systems by
considering the most general \pti\ matrix of dimension $2$,
 %
 \begin{equation}\label{2by2}
 %
 {\sf H} = \left(
          \begin{array}{cc}
           a   & b \\
           b^* & a^*
          \end{array} \right) \, , \quad a,b \in  \mathbb C \, ,
 %
\end{equation}
 %
with parity given by the Pauli matrix $\sigma_x$ in the standard
representation. For real numbers $a$ and $b$, the
matrix ${\sf H}$ is not only \pti\ but also hermitean. Thus, its eigenvalues are {\em real}, and its orthonormal eigenstates span ${\mathbb C}^2$. For $a^* \neq a$ and $b=0$, ${\sf H}$ has a pair of complex conjugate eigenvalues and two orthonormal eigenstates. Matrices of the form
 %
 \begin{equation}\label{interesting}
 %
  {\sf H} = \left(
          \begin{array}{cc}
           i   & b \\
           b & -i
          \end{array} \right) \, ,
          \quad b \in [-1,1] \, ,
 %
\end{equation}
 %
are particularly interesting. For $|b| < 1$, one finds a pair of
two complex conjugate eigenvalues,
 %
 \begin{equation}\label{ccpair}
 %
 E_\pm = \pm \sqrt{b^2-1}  \in i \mathbb R \, ,
 %
\end{equation}
 %
associated with two non-orthogonal eigenstates,
 %
 \begin{equation}\label{nononeigenstates}
 %
 \frac{1}{\sqrt{2} b }\left(
 \begin{array}{c}
 b \\
 i - \sqrt{b^2-1}
 \end{array} \right)
 \, , \qquad
  \frac{1}{\sqrt{2} b }\left(
 \begin{array}{c}
 b \\
 i + \sqrt{b^2-1}
 \end{array} \right)
 \, .
 %
\end{equation}
 %
When $b = \pm 1 $ in (\ref{interesting}), ${\sf H}$ has a two-fold
degenerate eigenvalue, $E_0=0$, and there is only {\em one}
eigenstate, namely,
 %
 \begin{equation}\label{singleeigenstate}
 %
 \frac{1}{\sqrt{2}}\left(
 \begin{array}{c}
 \mp 1 \\
 i
 \end{array} \right)
 \, .
 %
\end{equation}
 %
This situation, impossible for a hermitean matrix, is usually
described by saying that the {\em algebraic} multiplicity of the
eigenvalue $E_0$ is two while its {\em geometric} multiplicity
equals one: the characteristic polynomial of $\sf M$ has a double root
associated with a single eigenvector only. In this case, the matrix ${\sf H}$ is not diagonalizable: a similarity transformation sending it
to a {\em diagonal} matrix cannot exist since its eigenstates {\em would} span the space ${\mathbb C}^2$.
\subsection*{\sf III. Diagonalizability and the minimal polynomial of a matrix}
Each square matrix $\sf M$ of dimension $N$ satisfies the identity
\begin{equation}
p_{\sf M}( {\sf M}) = 0 \, ,
\label{CayleyHamM}
\end{equation}
where $p_{\sf M}(\lambda)$ is the {\em characteristic polynomial} of ${\sf M}$,
\begin{equation}\label{charpolM}
p_{\sf M}( \lambda )
         = \det \left( \lambda {\sf E} - {\sf M} \right) \, ,
\ee
with ${\sf E}$ being the unit matrix of dimension $N$. In other
words, the characteristic polynomial of ${\sf M}$ {\em
annihilates} the matrix ${\sf M}$. The polynomial $p_{\sf
M}(\lambda)$ has degree $N$ and it is a {\em monic} polynomial,
that is, the coefficient multiplying the highest power of
$\lambda$ is equal to $1$. Obviously, many other monic
polynomials of {\em higher} degree also annihilate ${\sf
M}$: simply take $p^2_{\sf M}(\lambda), p^3_{\sf M}(\lambda),
\dots$ It is less obvious, however, whether one can find
polynomials of degree {\em less} than $N$ which annihilate ${\sf
M}$. This, in fact, depends on the properties of the matrix ${\sf
M}$.

Define \cite{lancaster+85} the {\em minimal
polynomial} of the matrix {\sf M} as the monic polynomial
$m_{\sf M}(\lambda)$ of {\em least} degree which
annihilates {\sf M}:
\begin{equation}\label{minpolM}
m_{\sf M}( \sf M ) = 0 \, .
\ee
The minimal polynomial $m_{\mbox{\small $\sf M$}}( \lambda)$ is unique \cite{lancaster+85}, and its degree $N_0$ is less than or equal to the degree of the characteristic polynomial, $N_0 \leq N$. The minimal polynomial divides the characteristic polynomial without a remainder, $m_{\sf M}( \lambda) \left. \right|  p_{\sf M}(\lambda)$, or equivalently,
\begin{equation}\label{dividecharpol}
p_{\sf M}( \lambda)
   = d_{\sf M}(\lambda) m_{\sf M}( \lambda) \, ,
\ee
where $d_{\sf M}(\lambda)$ is a non-zero polynomial of degree less than
$N$. The characteristic and the minimal polynomial of the matrix
{\sf M} coincide in the case where $d_{\sf M}(\lambda) \equiv 1$.

In general, the minimal polynomial has $\nu_0$ roots $M_\nu$,
\begin{equation}\label{genminpoly}
m_{\sf M} (\lambda)
 = \prod_{\nu=1}^{\nu_0} (\lambda - M_\nu)^{\mu_\nu} \, ,
   \quad \nu_0 \leq N \, ,
\ee
with multiplicities $\mu_\nu$ summing to $N_0 = \mu_1 + \mu_2 + \dots + \mu_{\nu_0}$. Here is the important property of the polynomial $m_{\sf M} (\lambda)$:  the matrix  {\sf M} is diagonalizable if and only if each root $M_\nu$ in (\ref{genminpoly}) has multiplicity one, $\mu_\nu
\equiv 1, \nu =1, \dots, \nu_0$, that is,
\begin{equation}\label{genminpolydiag}
m_{\sf M}( \lambda)
 = \prod_{\nu=1}^{\nu_0} (\lambda - M_\nu)\, ,
   \quad \mbox{all } \, M_\nu \mbox{ distinct} \, .
\ee
No polynomial of degree less than $m_{\sf M}( \lambda)$  annihilates the matrix ${\sf M}$.

Let us illustrate the properties of minimal polynomials using
low-dimensional matrices. Consider the matrix {\sf A}
with entries $(1,1,2)$ on the diagonal, and zero elsewhere. Its characteristic polynomial is given by
\begin{equation}\label{charpolA}
p_{\sf A}(\lambda)= (\lambda-1)^2 (\lambda-2) \, ,
\ee
while its minimal polynomial reads
\begin{equation}\label{minpolA}
m_{\sf A}(\lambda)= (\lambda-1) (\lambda-2) \, ,
\ee
being of the form (\ref{genminpoly}), with $N_0 = \nu_0 = 2$.
This is easy to verify since $m_{\sf A}(\sf A) = {\sf A}^2 -3 {\sf A}+2{\sf E}=0 $ holds, while none of its factors annihilates ${\sf M}$: both $({\sf M} - {\sf E}) $ and $({\sf M} - 2{\sf E})$ are different from zero. Thus, the minimal polynomial divides the characteristic one, $p_{\sf A}(\lambda) = (\lambda - 1) m_{\sf A}(\lambda)$, leading to $d_{\sf A}(\lambda) = (\lambda -1)$. Due to (\ref{minpolA}), the matrix ${\sf A}$ {\em is} diagonalizable--- a correct but hardly surprising result since the matrix {\sf A} has been diagonal from the outset.

Here is the instructive part of the example: consider the matrix
\begin{equation}\label{defB}
{\sf B} = \left(
          \begin{array}{ccc}
                1 & b & 0 \\
                0 & 1 & 0 \\
                0 & 0 & 2
          \end{array}
          \right) \, , \qquad b \in \mathbb C  \, ,
\ee
which is different from {\sf A} as long as $b$ is different from
zero. The characteristic polynomial of {\sf B} equals that of {\sf A} but the matrix {\sf B} must have a different
minimal polynomial since $m_{\sf A}({\sf B}) \neq 0$. It is not difficult to verify that no linear or quadratic polynomial annihilates {\sf B} as long as $b \neq 0$. This implies that its minimal polynomial coincides with its characteristic polynomial,
\begin{equation}
p_{\sf B}(\lambda)
 = m_{\sf B}(\lambda) \, , \quad d_{\sf B}(\lambda) \equiv 1 \, .
\label{Bpolys}
\ee
Consequently, the minimal polynomial of {\sf B} does {\em not} have the form specified in (\ref{genminpolydiag}), and the matrix ${\sf B}$ is {\em not} similar to a diagonal matrix. Inspection shows that {\sf B} indeed contains a $(2 \times2)$ Jordan block for any nonzero value of $b$.

For \pti\ matrices, both the polynomials $d_{\mathsf{H}}(\lambda)$ and $m_{\mathsf{H}}(\lambda)$ have real coefficients only, just as the characteristic polynomial. This will be shown once the function $d_{\mathsf{M}}(\lambda)$ in (\ref{dividecharpol}) has been defined in general (cf. Sec. {\sf IV.2}).
\subsection*{\sf IV. An algorithmic test for diagonalizability}
A square matrix ${\sf M}$ of dimension $N$ is diagonalizable  if its minimal polynomial is a product of factors $(\lambda - M_\nu)$ with all numbers $M_\nu, \nu = 1,2, \ldots , n \leq \nu_0$, distinct, as shown in Eq. (\ref{genminpolydiag}). Consequently, to test for diagonalizability of given a matrix {\sf M}, one needs to
\begin{enumerate}
\item[($\imath$)] find the minimal polynomial $m_{\sf M}(\lambda)$ of the matrix {\sf M};
\item[($\imath \imath$)] determine whether the polynomial $m_{\sf M}(\lambda)$ has single roots only.
\end{enumerate}
To calculate numerically the roots of either the characteristic or the minimal polynomial is not a valid approach since, in general, the {\em exact} roots of a polynomial cannot be specified in a finite procedure. Any algorithmic implementation must generate answers to ($\imath$) and ($\imath \imath$) in a {\em finite} number of steps. Note that even if the first step has been implemented, it is unlikely that the minimal polynomial will emerge in factorized form.

Interestingly, it is possible to construct the minimal polynomial of a matrix and to check for degenerate roots in a finite number of steps. In both cases one searches for common factors of polynomials, which is achieved algorithmically by the {\em Euclidean division algorithm}. These results seem to have been put together for the first time in \cite{abate97} in order to decide algorithmically whether a given matrix is diagonalizable. As it stands, it could be applied to the non-hermitean matrix govering the motion of two coupled damped classical oscillators studied in \cite{heiss04}.

In the following, a slightly simplified approach to the problem of diagonalizability is presented, adapted to matrices with \pty. Before
implementing the steps ($\imath$)  and ($\imath\imath$), the Euclidean algorithm for polynomials will be presented briefly to establish notation.
\subsubsection*{\sf IV.1 Euclidean division algorithm for polynomials}
Given two integer numbers $p_0 > p_1$, say, the Euclidean division
algorithm outputs their greatest common divisor, denoted by
$\mbox{gcd}(p_0,p_1) \in {\mathbb N}_0$, after a finite number of
steps. It works as follows: first, you need to express the larger
number as $q_1$-fold multiple of the smaller number plus a
remainder $p_2$,
\begin{equation}\label{eucl}
p_0 = q_1 p_1 + p_2 \, , \qquad q_1,p_2 \in {\mathbb N}_0\, ,
                 \quad   p_1 > p_2 \leq 0 \, .
\ee
This relation implies that any common divisor of $p_0$ and $p_1$ divides $p_2$ as well, hence $\mbox{gcd}(p_0,p_1) = \mbox{gcd}(p_1,p_2)$. Thus, it is sufficient to search for the greatest common divisor of the pair $(p_1,p_2)$. This can be achieved by increasing each index in (\ref{eucl}) and feeding in the pair $(p_1,p_2)$ instead of $(p_0,p_1)$, etc. Since $p_0>p_1$ and $p_1>p_2$, the algorithm will stop after a finite number of iterations and produce a remainder equal to zero, $p_{k+1}=0$, say.  The {\em non-zero} remainder $p_k$ generated in the penultimate step is the desired result, $\mbox{gcd}(p_0,p_1) = p_k$. If $\mbox{gcd}(p_0,p_1)= 1$, the numbers $p_0$ and $p_1$ are relatively prime, otherwise a common divisor different from one has been identified.

A polynomial in the variable $\lambda$ can be written as a unique
product of linear factors $(\lambda - \lambda_n)$ where the
numbers $\lambda_n \in \mathbb C$ are its roots. This
representation makes polynomials similar to integer numbers in
some respects. The equivalent of the Euclidean algorithm, when
applied to two polynomials, outputs their greatest common divisor,
which is a polynomial itself. This result is based on the fact
that any two polynomials $p_0(\lambda)$ and $p_1(\lambda)$, with
$\deg p_0(\lambda) > \deg p_1(\lambda)$ are related by
\begin{equation}\label{euclpoly}
p_0(\lambda) = q_1(\lambda) p_1(\lambda) + p_2(\lambda) \, ,
              \quad \deg p_1(\lambda) > \deg p_2(\lambda) \geq 0 \, ,
\ee
which is the equivalent of (\ref{eucl}). The polynomials $q_1(\lambda)$, with $\deg q_1(\lambda) = (\deg p_0(\lambda) - \deg p_1(\lambda))$, and hence $p_2(\lambda)$, are  found from long division. If $p_0(\lambda)$ and $p_1(\lambda)$ have
a common factor, then $p_2(\lambda)$ must have this factor as well. Thus, it is sufficient to search for $\mbox{gcd}(p_1(\lambda),p_2(\lambda))$ instead of $\mbox{gcd} (p_0(\lambda),p_1(\lambda))$ but the degrees of the polynomials involved have effectively been reduced. Consequently, this procedure can be repeated all over again and it halts once a {\em vanishing} remainder has been obtained, $p_{k+1}(\lambda) = 0$, say. Then, the greatest common factor of the polynomials $p_0(\lambda)$ and $p_1(\lambda)$ is
given by the last non-zero remainder polynomial, $p_k (\lambda)$, calculated  in the next-to-last application of the algorithm. If $\deg p_k (\lambda) = 0$, the initial polynomials are ``relatively prime,'' otherwise their greatest common divisor is a polynomial of degree at least one.
\subsubsection*{\sf IV.2 Step ($\imath$): Finding the minimal polynomial of a matrix}
The function $d_{\sf M} (\lambda)$ relates the minimal polynomial $m_{\sf M}(\lambda)$ of the matrix {\sf M} to its characteristic polynomial $p_{\sf M}(\lambda)$ according to Eq.
(\ref{dividecharpol}). Hence, the minimal polynomial associated with ${\sf M}$ is known once the characteristic polynomial and the function $d_{\sf M} (\lambda)$ have been determined.

Two steps are required to construct the function $d_{\sf M} (\lambda)$ \cite{lancaster+85}. First, you need to calculate the matrix ${\sf D}_{\sf M} = \mbox{adj } (\lambda {\sf E} - {\sf M})$, given by the transposed cofactors---or signed minors---of the matrix $(\lambda {\sf E} - {\sf M})$. The adjoint of a matrix, {\sf C} say, always exists, and it satisfies the relation
 %
 \begin{equation}\label{adjoint}
 %
 {\sf C} \,  \mbox{adj } {\sf C} = (\det {\sf C}) {\sf E} \, .
 %
\end{equation}
 %
For $\det \sf C \neq 0$, Eq. (\ref{adjoint}) leads to the familiar expression of the inverse matrix of ${\sf C}$.

According to \cite{lancaster+85} the polynomial $d_{\sf M}(\lambda)$ is given by the  {\em greatest} (monic) {\em common divisor} of the $N^2$ elements of $\mbox{adj } (\lambda {\sf E} - {\sf M})$,

 \begin{equation}
 %
 d_{\mathsf{M}} (\lambda)
  = \mbox{gcd} \left\{ (\mathsf{D}_{\sf M})_{nm} |
                                       n,m=1,\ldots,N\right\}
   \, ,
 %
 \label{dhldefinition}
 \end{equation}
 %
Thus, in a second step, you need to apply the Euclidean algorithm to all pairs of entries of the matrix $\mbox{adj } (\lambda {\sf E} - {\sf M})$. Having thus identified the function $d_{\sf M} (\lambda)$, the minimum polynomial of {\sf M} follows from (\ref{dividecharpol}),
 %
 \begin{equation}\label{findminpoly}
 %
 m_{\sf M} (\lambda)
   = \frac{p_{\sf M} (\lambda)}{d_{\sf M}(\lambda)} \, .
 %
\end{equation}
 %

Now it is possible to show that, for a \pti\ matrix, the polynomials $d_{\sf M}(\lambda)$ and $m_{\sf M} (\lambda)$ have real coefficients only, just as the characteristic polynomial. Using a basis in which all elements of {\sf H} are real, leads to
 %
 \begin{equation}
 %
 ({\sf D}_{\sf H} (\lambda))^*
  = \left( \mbox{adj}(\lambda \mathsf{E} - \mathsf{H} ) \right)^{*}
  = \mbox{adj} (\lambda^{*} \mathsf{E}-\mathsf{H)}
  = {\sf D}_{\sf H} (\lambda^*) \, .
 %
 \label{realadjoint}
 %
 \end{equation}
 %
 which states that in this basis the adjoint of {\sf H} has only real matrix elements (except for the unknown $\lambda$). Taking (\ref{dhldefinition}) into account this leads to
 %
 \begin{equation}
 %
 \left( d_{\mathsf{H}} (\lambda) \right)^{*}
 =\mbox{gcd} \left\{ (\mathsf{D}_{\mathsf{H}})_{nm}(\lambda^{*})|n,m=1,\ldots,N\right\}
 = d_{\mathsf{H}} (\lambda^{*}) \, ,
 %
 \label{dhlreaL}
 \end{equation}
 %
which, in conjunction with $\left( p_{\mathsf{H}} (\lambda) \right)^{*}    = p_{\mathsf{H}} (\lambda^{*})$ and Eq. (\ref{findminpoly}) implies indeed $\left( m_{\mathsf{H}}(\lambda) \right)^{*}=m_{\mathsf{H}}(\lambda^{*})$.

Let us verify that this procedure outputs the correct minimal polynomials for the matrices {\sf A} and {\sf B} introduced in Section 3. The adjoint of the matrix $(\lambda {\sf E} - {\sf B})$ reads
 %
 \bea \label{cofB}
 %
 \mbox{adj } (\lambda {\sf E} - {\sf B})
       &=& \mbox{adj } \left(
         \begin{array}{ccc}
            \lambda - 1 & - b & 0 \\
            0 & \lambda -1 & 0 \\
            0 & 0 & \lambda -2
          \end{array}
         \right) \nonumber \\
        &=& \left(
         \begin{array}{ccc}
            (\lambda - 1) (\lambda -2) & b(\lambda -2) & 0 \\
            0 &  (\lambda - 1) (\lambda -2) & 0 \\
            0 & 0 &   (\lambda - 1)^2
          \end{array}
         \right) \, .
 %
\eea
 %
Due to the simplicity of the matrices involved, the Euclidean algorithm can be run ``by inspection:'' for $b\neq 0$, the only common factor  among the entries in (\ref{cofB}) is given by $d_{\sf B}(\lambda) = 1$. Consequently, the minimal and the characteristic polynomial of {\sf B} coincide as stated in Eq. (\ref{Bpolys}). If the parameter $b$ takes the value zero, {\sf B} turns into {\sf A}, and and a non-constant greatest common divisor emerges, $d_{\sf A}(\lambda) = (\lambda - 1)$. Using Eq. (\ref{dividecharpol}), one obtains the minimal polynomial $m_{\sf A}(\lambda) = (\lambda -1)(\lambda-2)$, agreeing with Eq. (\ref{minpolA}).

In \cite{abate97}, a different approach to determine the minimal
polynomial of a matrix {\sf M} has been presented which, ultimately, is also based on finding the greatest common divisor of specific  polynomials. According to \cite{horn+99}, any method to determine whether the matrices ${\sf M}^0 \equiv {\sf E}, {\sf M}, {\sf M}^2, \ldots , {\sf M}^{N-1}$, are linearly dependent, can be used to construct the minimal polynomial of {\sf M}; two such methods are described in this reference, and a third one can be found in \cite{horn+85}. The latter approaches have in common that they are {\em not} based on the Euclidean algorithm. For actual calculations, it is convenient to resort to a {\tt Mathematica} program  \cite{weisstein+05} to find the minimal polynomial of a matrix ${\sf M}$.
\subsubsection*{\sf 4.3 Step ($\imath \imath$): Identifying degenerate
roots of a polynomial}
 %
Once the minimal polynomial $m_{\sf M} (\lambda)$ has been found,
one needs an algorithm to decide whether it has single roots only
\cite{abate97}. Imagine a polynomial $m(\lambda)$ to have an
$s$-fold root $\lambda_0$, $2 \leq s \leq N$. Its factorization
reads
 %
 \begin{equation}\label{sfold}
 %
 m(\lambda) = (\lambda - \lambda_0)^s \ldots \, ,
 %
\end{equation}
 %
where the dots indicate a polynomial of degree $(N-s)$. Its
derivative takes the form
 %
 \begin{equation}\label{sfoldprime}
 %
 \frac{dm}{d\lambda} = (\lambda - \lambda_0)^{s-1} \dots  \, ,
 %
\end{equation}
 %
 the dots standing again for some polynomial of degree $(N-s)$.
Obviously, the polynomial and its  derivative are not relatively
prime: $m(\lambda)$ and $m^\prime(\lambda)$ have a factor
$(\lambda - \lambda_0)^{s-1}$ of order at least one in common.
Thus, applying the division algorithm to the pair $(m_{\sf
M}(\lambda), m_{\sf M}^{\prime} (\lambda))$ checks whether the
polynomial $m_{\sf M} (\lambda)$ has the form
(\ref{genminpolydiag}). If the procedure outputs $\mbox{gcd}
(m_{\sf M}(\lambda), m_{\sf M}^{\prime} (\lambda)) \propto 1$, all
roots of $m_{\sf M}$ are distinct and the associated matrix {\sf
M} is diagonalizable, otherwise it is not.

This concludes the description of an algorithm to test for
diagonalizability of a given \ptc\ matrix ${\sf M}$. No
fundamental changes are necessary if one studies a
parameter-dependent family of matrices ${\sf M} (\varepsilon)$. However, the algorithm will output conditions polynomial
in the parameter $\varepsilon$, indicating specific parameter
values where diagonalizability breaks down. It is convenient to
study the resulting modifications by working out some simple
examples, illustrating at the same time the proposed algorithm.

 %
 \subsection*{\sf V. Examples}
 %
 \subsubsection*{\sf V.1 Matrices of dimension ($2\times 2$)}
 %

Let us apply the algorithm described above to the matrix ${\sf H}$
in (\ref{2by2}}) assuming the numbers $a$ and $b$ to be different
from zero. Its characteristic polynomial reads
 %
 \begin{equation}\label{charpolH}
 %
 p_{\sf H} (\lambda)
    = \det (\lambda {\sf E} - {\sf H})
    = \lambda^2 - 2 (\Re a) \lambda + | a |^2  - |b|^2 \, ,
 %
\end{equation}
 %
while its minimal polynomial is found via the function $d_{\sf
H}(\lambda)$ equal to the highest common factor of the matrix
 %
 \begin{equation} \label{hcfH?}
 %
 {\sf D}_{\sf H} (\lambda) =  \mbox{adj } (\lambda {\sf E} - {\sf H} )
        = \left(
          \begin{array}{cc}
          \lambda - a^* &  -b \\
          -b^* & \lambda- a
          \end{array}
          \right) \, .
 %
\end{equation}
 %
By inspection, a non-constant factor only exists among the four entries of  ${\sf D}_{\sf H} (\lambda)$ if $b=\Im a =0$. In this case, ${\sf H}$ turns into a real multiple of the identity, hence it is diagonalizable. This observation illustrates a fine point of the construction of the minimal polynomial: even upon identifying a non-constant function $d_{\sf H} (\lambda)$, the minimal polynomial $m_{\sf H}(\lambda)$ may still be of the form (\ref{genminpolydiag}). For $b=\Im a =0$, the characteristic polynomial turns into $(\lambda - \Re a)^2$, implying indeed $m_{\sf H} (\lambda) = (\lambda - \Re a)$. Here, the function $d_{\sf H}(\lambda) = (\lambda - \Re a)$ removes factors of the characteristic polynomial which stem from the {\em degeneracy} of an eigenvalue of ${\sf H}$.

From now on, either $b$ or $\Im a$ are assumed to be different from zero, hence $d_{\sf H} (\lambda) =1$, and the minimal polynomial $m_{\sf H}(\lambda)$ is given by Eq. (\ref{charpolH}),
 %
 \begin{equation}\label{char=miniH}
 %
 m_{\sf H} (\lambda) =  p_{\sf H} (\lambda) \, ,
 %
\end{equation}
 %
which concludes the first step of the algorithm.

In the second step of the algorithm, the search for multiple roots of $m_{\sf H}(\lambda) \equiv p_0(\lambda)$, one needs to determine the highest common factor of the minimal polynomial and its derivative,
 %
 \begin{equation}\label{derminpoly}
 %
 m^\prime_{\sf H} (\lambda)
    = 2 (\lambda -  \Re a) \equiv p_1 (\lambda) \, .
 %
\end{equation}
 %
Applying the Euclidean algorithm to the pair $(m_{\sf H}
(\lambda), m^\prime_{\sf H} (\lambda))$ means to solve for a
polynomial $q_1 (\lambda)= A \lambda + B$ and for $p_2 (\lambda)$
such that
 %
 \begin{equation}\label{mmprimeeuclid}
 %
 m_{\sf H} (\lambda)
      = (A \lambda +B) m_{\sf H}^\prime (\lambda) + p_2(\lambda)   \, ,
      \quad A, B \in \mathbb R \, .
 %
\end{equation}
 %
The unknowns are easily obtained as
 %
 \begin{equation}\label{euclappl1}
 %
 A = \frac{1}{2} \, , \quad B= - \frac{1}{2} \Re a \, ,
 \quad p_2(\lambda) = (\Im a)^2 - | b |^2  \simeq \lambda^0 \, .
 %
\end{equation}
 %
Two possibilities now arise: either $p_2(\lambda)$ equals zero or
it does not. The first case occurs if
 %
 \begin{equation}\label{condp2}
 %
 (\Im a)^2 = | b |^2 \, ,
 %
\end{equation}
 %
and the algorithm comes to a halt. As mentioned above, the
greatest common divisor of the initial polynomials is then given
by the penultimate (monic) remainder polynomial, i.e. $p_1
(\lambda) = ( \lambda - \Re a)$. It follows that $m_{\sf H}
(\lambda)$ and its derivative do have a common non-constant
divisor, so that ${\sf H}$ cannot be brought to diagonal form. It
is easy to verify that $m_{\sf H} (\lambda) = ( \lambda - \Re
a)^2$ when (\ref{condp2}) holds, confirming that the minimal
polynomial of ${\sf H}$ has a {\em double} root $\Re a$.
Furthermore, a simple calculation shows that the matrix $\sf H$
has indeed a {\em single} eigenstate only if the relation $\Im a =
\pm |b|$ holds. Note that this result covers the example of a
non-diagonalizable ${\sf H}$ of Section 2, where $a=i$ and $b=\pm
1$ had been considered.

Finally, if $a$ and $b$ do not satisfy (\ref{condp2}), the
remainder polynomial $p_2(\lambda)$ does not vanish. Being a constant, the algorithm is bound to stop after the next iteration. Determine $q_2(\lambda)=(C\lambda + D)$ and $p_3
(\lambda)$ such that
  %
 \begin{equation}\label{mmprimeeuclid2}
 %
  (\lambda -  \Re a)
      = (C \lambda + D) ((\Im a)^2 - | b |^2) + p_3 (\lambda) \, ,
      \quad C, D \in \mathbb R \, ,
 %
\end{equation}
 %
holds, i.e.,
 %
 \begin{equation}\label{euclappl2}
 %
       C = \frac{1}{(\Im a)^2 - | b |^2} \, ,
 \quad D = \frac{- \Re a}{(\Im a)^2 - | b |^2} \, ,
 \quad p_3(\lambda) = 0 \, .
 %
\end{equation}
 %
The algorithm halts indeed due to $p_3(\lambda) = 0$, and the
penultimate remainder is $p_2 (\lambda) \propto 1$, indicating
that the minimal polynomial does not have any degenerate roots,
and ${\sf H}$ is diagonalizable.

In summary, the matrix {\sf H} is diagonalizable for all parameter
values except when $\Im a = \pm |b|$. In this case the algebraic
multiplicity of its eigenvalue is two, while its geometric
multiplicity is one; otherwise the multiplicities both equal two.
It is important to note that it was not necessary at any stage to
determine the eigenvalues of ${\sf H}$.
 %
 \subsubsection*{\sf V.2 Matrices of dimension ($4 \times 4$)}
 %
It is instructive to apply the algorithm to a \pti\ matrix of dimension $4$,
 %
 \begin{equation}\label{4by4}
 %
 {\sf H} = \left(
          \begin{array}{cccc}
          i \varepsilon & s & 0 & 0\\
          s & -i \varepsilon & \delta & 0 \\
          0 & \delta & i \varepsilon & s  \\
          0 & 0 & s & -i \varepsilon \\
          \end{array} \right) \, , \quad
                       s,\delta > 0 \, ,
 %
\end{equation}
 %
which depends on a perturbation parameter $\varepsilon$. As before, the action of ${\sf T}$ on a matrix effects complex conjugation of its entries, while ${\sf P}$ is now given by a $(4 \times 4)$ matrix with entries equal to one along its minor
diagonal and zero elsewhere. Eq. (\ref{ptinv}) is then readily
verified. The characteristic polynomial of ${\sf H}$ is given by
 %
 \begin{equation}\label{charpoldim4}
 %
 p_{\sf H} (\lambda) = \lambda^4 + \alpha_\varepsilon \lambda^2 + \beta_\varepsilon \, ,
 %
\end{equation}
 %
with $\alpha_\varepsilon = 2 \varepsilon^2 - 2 s^2 - \delta^2$ and
$ \beta_\varepsilon = \varepsilon^4 - (2s^2  +
\delta^2)\varepsilon^2 + s^4$. The minimal polynomial of ${\sf H}$
coincides with the characteristic one, $m_{\sf H} (\lambda) \equiv
p_{\sf H} (\lambda)$, since the only common factor of the matrix elements
of ${\sf D}_{\sf H}$ is equal to one, $d_{\sf H} (\lambda) = 1$. To see this, it is sufficient to calculate the two matrix elements $[{\sf D}_{\sf H}]_{14} = -s^2 \delta$, and $[{\sf D}_{\sf H}]_{23} = - (\lambda^2 + \varepsilon^2) \delta$, for
example. Whatever the value of $\varepsilon$, for nonzero $s$ and
$\delta$ the only common divisor is one, so that $d_{\sf H}
(\lambda) =1$.

Let us now determine $\mbox{gcd} (m_{\sf H} (\lambda),
m^\prime_{\sf H} (\lambda ))$ by the Euclidean algorithm, where
$m_{\sf H} (\lambda) \equiv p_0(\lambda) $ is given in Eq.
(\ref{charpoldim4}) and $m^\prime_{\sf H} (\lambda ) \equiv
p_1(\lambda) = 4 \lambda^3 + 2\alpha_\varepsilon\lambda$. Comparing powers of
$\lambda$ in Eq. (\ref{mmprimeeuclid}) with the polynomials just
defined, one obtains
 %
 \begin{equation}\label{edastep2}
 %
 A= \frac{1}{4}\, , \quad B= 0\, ,
 \quad p_2(\lambda) = \frac{\alpha_\varepsilon}{2}\lambda^2 + \beta_\varepsilon \, .
 %
\end{equation}
 %
The algorithm only stops here if $\alpha_\varepsilon =
\beta_\varepsilon = 0$ which would require $s=\delta = 0$, contrary to both $s$ and $\delta$ being different from zero. If $\alpha_\varepsilon=0$ is assumed, ${\sf H}$ is diagonalizable for all $\varepsilon$ since $\beta_\varepsilon$ cannot take the value zero, and the algorithm stops after the next step, outputting $\beta_\varepsilon \propto 1$ as greatest common factor. Assume now $\alpha_\varepsilon \neq 0$ and apply the division algorithm to the pair $(p_1(\lambda),p_2
(\lambda))$. The unknown constants in $q_2(\lambda) = C\lambda +
D$, and the remainder polynomial $p_3(\lambda)$ are found to be
 %
 \begin{equation}\label{edastep3}
 %
 C = \frac{8}{\alpha_\varepsilon}, \quad D = 0 \, ,
     \quad p_3 (\lambda) = \frac{2}{\alpha_\varepsilon}
             (\alpha_\varepsilon^2 - 4 \beta_\varepsilon)\lambda \, .
 %
\end{equation}
 %
For the algorithm to stop, one must have $p_3(\lambda) = 0$. This, however, does not happen whatever the value of $\varepsilon$ since $\alpha_\varepsilon^2 - 4 \beta_\varepsilon = \delta^2 (4s^2 + \delta^2) >0$. The next iteration of the algorithm leads to
 %
 \begin{equation}\label{edastep4}
 %
 E = \frac{\alpha^2_\varepsilon}{4(\alpha_\varepsilon^2 - 4 \beta_\varepsilon)}, \quad F = 0 \, ,
     \quad p_4 (\lambda) = \beta_\varepsilon \, ,
 %
\end{equation}
 %
where $q_3(\lambda) = E\lambda + F$. Producing a remainder polynomial of degree zero in $\lambda$, the condition for the minimal polynomial to have multiple roots is finally given by
 %
 \begin{equation}\label{remaindercond}
 %
 \beta_\varepsilon =  \varepsilon^4 - (2s^2  + \delta^2)\varepsilon^2 + s^4 = 0 \, .
 %
\end{equation}
 %
This fourth-order polynomial in $\varepsilon$ has roots
 %
 \begin{equation}\label{roots}
 %
 \pm \varepsilon_\pm = \pm \sqrt{ \sigma \pm \Delta \sigma} \, ,
  \quad \sigma = s^2 +\frac{\delta^2}{2} > 0 \, ,
  \quad \Delta \sigma = \sqrt{\sigma^2 -s^4} \in (0,\sigma) \, .
 %
\end{equation}
 %
For each of these four real values of the parameter $\varepsilon$,
the matrix ${\sf H}$ is {\em not} diagonalizable. In other words,
the algebraic and geometric multiplicity of the eigenvalues of
${\sf H}$ do not coincide, and its eigenstates do not span the
space ${\mathbb C}^4$. It is important to note that the
eigenvalues of the matrix $\sf H$ are not known at this stage.
\subsubsection*{\sf V.3 The global structure of ${\sf H}$}
\label{sec:53SfGlobalStructure}
Let us now determine the global properties of ${\sf H}$. This is easily done upon combining (\ref{roots}) with the characteristic polynomial (\ref{charpoldim4}) in its factorized form,
 %
 \begin{equation}\label{charpoldim4factorized}
 %
 p_{\sf H} (\lambda) =
 (\lambda - \lambda_{+})(\lambda + \lambda_{+})
 (\lambda - \lambda_{-})(\lambda + \lambda_{-}) \,
 %
\end{equation}
with roots
 %
 \begin{equation}\label{fourevs}
 %
 \pm \lambda_{\pm}
   = \pm \sqrt{\sigma \pm \Delta \sigma - \varepsilon^2} \, ,
 %
\end{equation}
 %
expressed directly in terms of $\sigma$ and $\Delta \sigma$.

To graphically represent the parameter space of {\sf H} and its properties, it is convenient to
eliminate the parameter $s$ by the scaling $\varepsilon \to s
\varepsilon$ and $\delta \to s \delta$. This effectively amounts
to sending $s \to 1$, and Eq. (\ref{remaindercond}) simplifies to
$\varepsilon^4 - 2(1 + \delta^2/2)\varepsilon^2 + 1 = 0$, plotted in
Fig. 1.

\begin{figure}[t]
 \begin{center}
	\includegraphics{paramspace.eps}
 \end{center}
	\caption{Parameter Space of the matrix {\sf H} defined in (\ref{4by4}); region I: four real eigenvalues; regions II: two real and one pair of complex-conjugate eigenvalues; region III: a two pairs of complex-conjugate eigenvalues; the matrix {\sf H} is not diagonalizable on the full lines separating the regions I,II, and III}
	\label{parameterspace}
\end{figure}

Imagine to move along the dashed vertical line, determined by a fixed positive value of $\delta>0$ and variable $\varepsilon$. For $\varepsilon = 0$, the matrix ${\sf H}$ is hermitean, hence it has four distinct real eigenvalues and four orthonormal eigenstates. In region I, where $0 < |\varepsilon| < \sqrt{\sigma - \Delta \sigma}$, Eq. (\ref{fourevs}) says that the eigenvalues remain real and distinct; a complete, not necessarily orthonormal set of four eigenstates continues to exist since (\ref{remaindercond}) does not hold. When $\varepsilon = \pm\sqrt{(\sigma - \Delta \sigma)}$, two eigenvalues coincide numerically, and the corresponding two eigenstates merge into a single one, leaving ${\sf H}$ with an incomplete basis. Then, for $\sqrt{\sigma - \Delta \sigma} < |\varepsilon|
< \sqrt{\sigma + \Delta \sigma}$, in region II, two real eigenvalues and a pair of complex-conjugate ones exist, with $\sf H$ being diagonalizable
throughout since (\ref{remaindercond}) does not hold. At
$\varepsilon = \pm\sqrt{(\sigma + \Delta  \sigma)}$, the remaining two
real eigenvalues degenerate to a single one, leaving ${\sf H}$
non-diagonalizable again with only three eigenstates. Finally, in region III, defined by $\sqrt{\sigma + \Delta \sigma} <
|\varepsilon|$, the matrix $\sf H$ is diagonalizable and it comes with two  pairs of complex-conjugate eigenvalues.

Finally, for $\delta=0$, the matrix ${\sf H}$ in (\ref{4by4}) decouples into an pair of identical two-dimensional matrices. The left boundary of region I sees the real eigenvalues of ${\sf H}$ degenerate pairwise which is consistent with the observations made earlier. At $\varepsilon =\pm 1$, only two eigenstates exist while all four eigenvalues coincide numerically. Beyond this value of $\varepsilon$, there are two pairs of identical complex-conjugate eigenvalues, and the associated basis is complete.

For \ptc\ systems described by matrices of higher dimensions it is, in general, not possible to find the roots of the characteristic polynomial. Nevertheless, a discussion of the parameter space can still be given: to this end one needs to detect the number of real and complex eigenvalues for each set of parameter values; an algorithm capable of doing this will be presented in \cite{weigert05b}.

\subsection*{\sf VI. Discussion and Outlook}\label{sec:Discussion-and-Outlook}

An algorithm has been presented which allows one to determine
whether a given PT-invariant matrix ${\sf M}$ is diagonalizable. To do so, it is not necessary to determine the roots of its characteristic
polynomial. In terms of Linear Algebra, the algorithm decides whether the given matrix is {\em similar} \cite{lancaster+85} to a diagonal matrix or to a matrix containing at least one Jordan block of dimension two or more. Somewhat surprisingly, this question seems to have been addressed only recently from an algorithmic point of view.

It seems worthwhile to point out that the test for multiple
roots of a polynomials can, in fact, be used without any change to
determine whether the eigenvalues of a given hermitean matrix are
degenerate or not. The present author is not aware that this
observation has been made before.

When applied to a family of PT-symmetric matrices, the algorithm outputs polynomial conditions on the perturbation parameter. These conditions are satisfied
for sets of matrices all of which are not diagonalizable, and they
divide the full parameter space into regions of diagonalizable matrices
with qualitatively different spectra. When combined with an algorithm to identify the number of real and complex eigenvalues of {\sf M}, a complete picture of the system's properties in the entire parameter space can be established.

Many PT-symmetric systems--including the first one studied from this perspective \cite{bender+98}--have been defined on Hilbert spaces with countably infinite dimension. Various concepts such
as eigenvalues and eigenstates, or the difference between algebraic and geometric multiplicities of degenerate eigenvalues continue to
exist in the more general case \cite{kato84}. In spite of a close similarity of hermitean operators in finite- and infinite-dimensional spaces, many concepts of the matrix case are not easily carried over to the more general situation. For any \emph{algorithm}, finiteness is a crucial feature: the number of steps required to identify a potential common factor of two polynomials
is always finite, no matter what their degree.  It will be interesting to see whether algorithmic tests for diagonalizability of operators acting on spaces with countably infinite dimension can be found.