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\begin{document}
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\markboth{\sc \centerline{Proceedings of the IWOP'96} }{\sc \centerline{Proceedings of the IWOP'96} }


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\begin{center}
{\large \bf{ORTHOGONAL POLYNOMIALS AND QUASY-EXACTLY SOLVABLE POTENTIALS ON THE LINE}}
%%Orthogonal Polynomials and Quasi-Exactly Solvable Potentials on the Line} }

\vskip .4cm 

{\bf {F. Finkel \qquad A. Gonz\'alez-L\'opez \qquad M.A. Rodr\'\i guez}}
\end{center}

%%%\date{\small July 3, 1996}


\begin{abstract}
In this paper, we survey recent results relating \qes\ systems with the theory of
orthogonal polynomials. The necessary and sufficient conditions for a \qes\ potential
on the line to generate a family of weakly orthogonal polynomials are presented. We also
introduce a new characterization theorem for weakly orthogonal polynomials.
\end{abstract}

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\section{Introduction}\label{sec.intro}

In a recent paper, \cite{BD96}, Bender and Dunne found a new connection between
{\em\qes} (QES) models in quantum mechanics and the theory of orthogonal
polynomials. They considered the following well known class of QES Hamiltonians:
\begin{equation}\label{BDH}
H=-\Dx^2+\frac{(4s-1)(4s-3)}{4x^2}-(4s+4n+2)x^2+x^6\,,
\end{equation}
with $n$ a positive integer and $s\in\Bbb R$. Inserting the asymptotic
expansion
$$
\psi_E(x) = e^{-\frac{x^4}4}x^{2s-\frac12}
\sum_{k=0}^\infty \frac{(-1)^k}{k!}
\frac{P_k(E)}{\Ga(k+2s)}\left(\frac{x^2}4\right)^k
$$
in the time-independent \sch\ equation, the coefficients $P_k(E)$ 
are easily shown to satisfy the three-term recursion relation
\begin{equation}\label{BDrec}
P_k(E)=E\,P_{k-1}(E)+16(k-1)(k-n)(k+2s-2)\,P_{k-2}(E)\,,
\end{equation}
for $k\geq 1$. Imposing the initial condition $P_0(E)=1$, each coefficient $P_k(E)$
becomes a monic polynomial of the $k$-th degree. The form of the recursion
relation \eqref{BDrec} has some remarkable implications. In the first place,
the polynomials $P_k(E)$ are orthogonal with respect to some Stieltjes
measure $d\om(E)$, \cite{Ch78}:
$$
{\cal L}(P_k\,P_l)=\int_\Bbb R P_k(E)P_l(E)d\om(E)=0\qquad \text{if }k\neq l\,.
$$
Secondly, the norms of the polynomials vanish for $k\geq n+1$, and
therefore $\{P_k\}_{k=0}^\infty$ defines what is known as a {\em weakly orthogonal} polynomial
system, \cite{Ch78}. Another interesting consequence of \eqref{BDrec} is the
factorization property:
$$
P_{k+n+1}(E)=Q_k(E)\,P_{n+1}(E)\,,\qquad\text{for}\quad k\geq 0\,,
$$
where $Q_k$ is a (monic) $k$-th degree polynomial. Finally, the roots $E_l$ of the
{\em critical polynomial} $P_{n+1}$ belong to the point spectrum of the
Hamiltonian \eqref{BDH}, the corresponding eigenfunctions being $\psi_{E_l}$.

Unlike some previously known results relating classical orthogonal polynomials and
QES models, \cite{Tu94}, \cite{Za94}, in Bender and Dunne's approach the polynomials
$P_k(E)$ are the coefficients of a certain expansion of a solution to a QES equation,
but are {\em not} solutions themselves.

It was shown in \cite{FGR96} that a family of weakly orthogonal polynomials
satisfying properties analogous to those obtained by Bender and Dunne
can indeed be constructed for {\em any} QES Hamiltonian on the line, provided some
very mild conditions are satisfied. Some additional properties
of these weakly-orthogonal polynomials were also studied in \cite{FGR96}.
In particular, it was rigorously proved that
their associated Stieltjes measure $d\om(E)$ is supported on a finite set, as claimed
in \cite{KUW96}, and is unique. Finally, the moments of $d\om(E)$ were shown to
satisfy a recursion relation with constant coefficients. In this paper, we
survey these results and also introduce a theorem characterizing weakly orthogonal
polynomials independently of their connection to QES spectral problems.

The paper is organized as follows. In Section~\ref{sec.QESpot} we make a concise
introduction to QES potentials on the line. In Section~\ref{sec.rec},
we construct the family of weakly orthogonal polynomials associated to each
of the canonical forms of a QES Hamiltonian discussed in the previous
section. Finally, in Section~\ref{sec.wop} we review some properties of
weakly orthogonal polynomials including the above mentioned theorem.

\section{QES Potentials on the Line}\label{sec.QESpot}

In this section we introduce the elements of the theory of \qes\ potentials
on the line that we need in the sequel, referring the reader to \cite{Sh89} and
\cite{GKO93a} for an in-depth review of the field. We start with the following

\begin{dfn}
A \fd\ Lie algebra $\fg$ of first order \dfo s on the line
$$
\fg=\Span\{\,T_a\,;\quad 1\leq a\leq r\}\:,\qquad T_a=\xi_a(x)\Dx+\eta_a(x)\,,
$$
is called \qes\ \(QES\) if it preserves a \fd\ subspace $\n$ of $C^\infty(\Bbb R)$
functions. A Hamiltonian \(or \sch\ operator\)
\begin{equation}\label{H}
H=-\Dx^2+V(x)
\end{equation}
is QES if it lies in the universal enveloping algebra of a QES Lie algebra $\fg$.
\end{dfn}
In other words, a QES Hamiltonian is a polynomial (of the second degree, as it
turns out) in the generators of a QES Lie algebra. It follows from this definition
that the subspace $\n$ is also preserved by $H$, and therefore a number of its
eigenvalues and eigenfunctions may be computed by diagonalizing the \fd\ matrix
which represents it in $\End(\n)$, the set of endomorphisms of $\n$. We shall use the term {\em
algebraic} to refer to these eigenfunctions and eigenvalues.  If we impose appropriate boundary
conditions on the functions spanning $\n$, ---like square integrability if we are looking for
bound states of
$H$, periodicity, etc.--- which amounts to requiring $H$ to be a self-adjoint operator, then the
matrix representing $H$ in $\End(\n)$ will be Hermitian. In this case, there are exactly
$\dim\n$ algebraic eigenfunctions of $H$ whose corresponding {\em real} eigenvalues may be
computed algebraically, and $H$ is said to be {\em fully algebraic}.

The classification of QES Hamiltonians involves (roughly speaking) two steps. In the first
place, one needs to classify QES Lie algebras under a suitable notion of equivalence.
In the second step, these Lie algebras are used to construct QES \sch\ operators, and appropriate
boundary conditions are imposed. Both of these problems have been solved in the case
of QES Hamiltonians acting on the line. It is worth mentioning that only the first step
has been carried out for QES Hamiltonians in two variables, \cite{GKO93a}, and not even this
for QES Hamiltonians in more than two variables. The classification of QES Lie algebras
on the line is provided by the following well known result, \cite{Tu88}, \cite{KO90}:

\begin{thm}
Every QES Lie algebra $\fg$ is equivalent under a local diffeomorphism
and a gauge transformation by a non-vanishing function to \(a subalgebra of\) one of
the Lie algebras $\fg_n=\fh_n\oplus\Bbb R$, where $\fh_n=\Span\{\Jp n,\J0 n,\Jm\}$, and
\begin{equation}\label{Js}
\Jp n=z^2 \Dz - n z\: ,\qquad \J0 n=z \Dz - \frac n2\: ,\qquad \Jm =\Dz\:,
\end{equation}
with $n\in\Bbb Z^+$. The invariant $\fg_n$-module is $\p n=\Span\{1,z,\ldots,z^n\}$.
\end{thm}
%
It follows from this theorem that every QES Hamiltonian $H$ on the line may always
be written in ``gauge'' form as a second degree polynomial in the generators of $\fh_n$:
\begin{align*}
-H_{\text{gauge}}(z) & = -\left.\frac1{\mu(z)}\cdot H(x)\cdot\mu(z)\right|_{x=\ze^{-1}(z)}\\
	             & = \sum_{a,b} c_{ab}\,J^a_n\,J^b_n+\sum_a c_a\,J^a_n + c_*\,,
\end{align*}
where the indices $a$ and $b$ run through $+,0,-$. Using the explicit expressions \eqref{Js},
we obtain:
\begin{equation}\label{Hgauge}
-H_{\text{gauge}}(z) = P\,\Dz^2+\left\{Q-\frac{n-1}2P'\right\}\,\Dz
                +\left\{R-\frac n2 Q'+\frac{n(n-1)}{12}P''\right\},
\end{equation}
where $P$, $Q$, and $R$ are polynomials in $z$ of degrees 4, 2, and 0,
respectively, given by
\begin{align}\label{PQR}
P & = c_{++}\,z^4+2c_{+0}\,z^3+c_{00}\,z^2+2c_{0-}\,z+c_{--}\,,\notag\\
Q & = c_+\,z^2+c_0\,z + c_-\,,\notag\\
R & = \frac{n(n+2)}{12}c_{00}+c_*\,.
\end{align}

Let us remark that gauge transformations and changes of variables respect the Lie bracket between
\dfo s. They also preserve the \sch\ eigenvalue equation. In fact, if $\chi(z)$ is an eigenfunction
of $H_{\text{gauge}}$ with eigenvalue $E$, $\psi(x)=\mu(z)\chi(z)|_{z=\ze(x)}$ is an eigenfunction
of $H$ with the same eigenvalue. Note that we do not require $\mu(z)$ to be unimodular, so
square-integrability is {\em not} preserved. Regarding the second step in the construction
of QES \sch\ operators, there are explicit formulas for the change of variables and the
gauge factor needed to transform the \dfo\ \eqref{Hgauge} into \sch\ form \eqref{H},
\cite{GKO93b}:
$$
x=\ze^{-1}(z)=\!\int^z\!\!\!\frac{dy}{\sqrt{P(y)}},\qquad
\mu(z) = P(z)^{-n/4}\exp\left\{\int^z\!\frac{Q(y)}{2P(y)}dy\right\}\,,
$$
where $z$ belongs to an interval $I$ in which $P(z)>0$. The potential is given by:
$$
V(x)=-R-\frac 1{12 P}\bigg[ n(n+2)(P P''-\frac 34 {P'}^2)
                           +3(n+1)(Q P'-2 P Q')-3Q^2 \bigg]_{z=\ze(x)}\,,
$$
where the prime denotes derivative with respect to $z$. Therefore, any second degree polynomial
in the generators \eqref{Js} may be written in \sch\ form up to sign (this property is no longer true
in more than one variable, \cite{GKO93a}). If $V(x)$ turns out to be independent of the
``spin'' parameter $n$, we can compute an arbitrary large number of its eigenvalues and
eigenfunctions algebraically (leaving boundary conditions aside). In this case we shall say that
$V$ is {\em exactly-solvable}.

The ``gauge'' form \eqref{Hgauge} of the QES Hamiltonian $H$ is {\em not} unique (the
physical form \eqref{H} is unique up to a traslation in $x$). There is a residual
group of transformations which preserve the Lie algebras $\fh_n$. These
transformations are the combination of a change of variables given
by the projective action of $\GL(2,\Bbb R)$ on $\RP$:
$$
z=\ze(w)=\frac{\al w+\be}{\ga w+\de}\,,\qquad \De=\al\de-\be\ga\neq 0\,,
$$
and a gauge transformation with gauge factor
$$
\mu(z)=(\ga w+\de)^n\,.
$$
The space $\p n$ is preserved under these transformations, whose action
$$
p(z)\in\p n\mapsto \hat p(w) = (\ga w+\de)^n p\left(\frac{\al
w+\be}{\ga w + \de}\right)\,,
$$
defines the (irreducible) multiplier representation $\rho_n$ of $\GL(2,\Bbb R)$ on $\p n$.
We also introduce the tensor product representations $\rho_{n,i}=\rho_n\otimes\det^i$,
where $\det$ is the usual determinantal representation of $\GL(2,\Bbb R)$. The generators
\eqref{Js} then transform according to the representation $\rho_{2,-1}$, and the polynomials
$P$, $Q$, and $R$ defining $H_{\text{gauge}}$ do so under $\rho_{4,-2}$, $\rho_{2,-1}$, and
$\rho_0$, respectively, \cite{GKO93b}. Note that the eigenfunctions of $H_{\text{gauge}}$
transform (by definition) under $\rho_n$. We may use the  $\GL(2,\Bbb R)$ action to reduce
the polynomial $P$ to one of the following canonical forms, \cite{GKO93a}:
\begin{xalignat}{2}\label{canP}
& 1)\quad\nu(z^2+1)\,,\quad	& 6) &\quad\nu(1-z^2)(1-\ka^2z^2)\,,\notag\\
& 2)\quad\nu(z^2-1)\,,\quad	& 7) &\quad\nu(1-z^2)\big(1-\ka^2(1-z^2)\big)\,,\notag\\
& 3)\quad\nu z^2\,,\quad	& 8) &\quad\nu(1+z^2)\big(1+(1-\ka^2)z^2\big)\,,\notag\\
& 4)\quad z\,,\quad		& 9) &\quad\nu(1+z^2)^2\,,\notag\\
& 5)\quad 1\,,\quad		& 10)&\quad\nu(1-z^2)\,,
\end{xalignat}
where $\nu>0$ and $0<\ka<1$. The first five canonical forms lead to {\em normalizable} QES
Hamiltonians (that is, the algebraic eigenfunctions are square-integrable after imposing
suitable constraints in the remaining parameters), whereas the
remaining five lead to periodic potentials. It is readily seen that the Hamiltonian \eqref{BDH}
corresponds to the fourth canonical form. Finally, let us mention that this invariance
of $H_{\text{gauge}}$ was used in \cite{GKO93b} to classify all normalizable and fully algebraic QES
Hamiltonians on the line.

\section{The Recursion Relation}\label{sec.rec}

In this section, we obtain the generating function and the recursion relation satisfied
by the system of weakly orthogonal polynomials associated to each class of QES Hamiltonians
determined by the canonical forms \eqref{canP}. Let $H$ be a QES Hamiltonian, and let
$H_{\text{gauge}}$ be one of its gauge forms. We consider the ``gauged'' \sch\ equation:
\begin{equation}\label{eqgauge}
(H_{\text{gauge}}-E)\,\chi_E=0\,,
\end{equation}
and we try a formal (no boundary conditions) solution of the form
$$
\chi_E(z) = \sum_{k=0}^\infty P_k(E)\frac{z^k}{k!}\\.
$$
A straightforward computation leads to the following five-term recursion relation for
the coefficients $P_k(E)$:
\begin{align}\label{5term}
-c_{--}P_{k+2} & = \left[(2k-n+1)c_{0-}+c_-\right] P_{k+1}\notag\\
& \quad +\left[ E+c_*+c_0\Big( k-\frac n2\Big)+c_{00}\Big( k-\frac n2\Big)^2\right] P_k\notag\\
& \quad +k(k-1-n)\left[(2k-n-1)c_{+0}+c_+\right] P_{k-1}\notag\\
& \quad +k(k-1)(k-1-n)(k-2-n)\,c_{++}P_{k-2}, \qquad k\geq 0.
\end{align}
If $c_{--}$ does not vanish, then $z=0$ is a regular point of the equation \eqref{eqgauge},
and $\chi_E(z)$ depends on two arbitrary functions, $P_0(E)$ and $P_1(E)$. If we
choose them as polynomials in $E$, each $P_k(E)$ will also be a polynomial, but
not a $k$-th degree polynomial unless $c_{--}=0$. Moreover, we need $c_{++}=0$ if we want
\eqref{5term} to be three-term (so that the polynomials $P_k(E)$ can be orthogonal
with respect to some nonzero Stieltjes measure). Note that $c_{--}=c_{++}=0$ simply
means that $P(z)$ in \eqref{PQR} vanishes at $z=0\,,\infty$ when $z$ takes values on
the complex projective line. The idea is trying to use (complex) projective
transformations analogous to those discussed in the previous section to map $P(z)$ into
another polynomial $\hat P(w)$ with roots at $0,\infty$. We assume that $P(z)$ is
one of the canonical forms \eqref{canP}. We have the following possibilities:

\ni{\em Case 1:} $P$ has two different roots, $z_1\neq z_2=\infty$ (canonical
forms 1--4,10).\newline
The translation $w=z-z_1$ maps $P(z)$ into $\hat P(w)$ vanishing at $0, \infty$. The generating
function in the canonical coordinate $z$ is:
\begin{equation}\label{chi1}
\chi_E(z) = \sum_{k=0}^\infty P_k(E)\frac{(z-z_1)^k}{k!}\,,
\end{equation}
and the polynomials $P_k(E)$ are determined from eq.~\eqref{eqgauge}. Note that
we are expanding around $z=z_1$, which is a singular point of the differential equation.
We could also have written the generating function in the ``physical'' coordinate $x$ as:
$$
\psi_E(x)=\mu(\ze(x))\chi_E(\ze(x)).
$$

\ni{\em Case 2:} $P$ has two different {\em finite} roots, $z_1\neq z_2$ (canonical
forms 6--9).\newline
The appropriate transformation is now $w=(z-z_1)/(z-z_2)$, and the generating function is
given by:
\begin{equation}\label{chi2}
\chi_E(z) = (z-z_2)^n\,\sum_{k=0}^\infty\frac1{k!} P_k(E)\left(\frac{z-z_1}{z-z_2}\right)^k\,.
\end{equation}

\ni{\em Case 3:} $P$ has a quadruple root. This corresponds to the fifth canonical form ($P=1$).
In this case, the recursion relation \eqref{5term} reduces to a pair of two-term recursion
relations for the even and odd coefficients $P^0_j=P_{2j}$ and  $P^1_j=P_{2j+1}$,
$$
-P^\ep_{j+1} =\left[ E+c_*+c_0\big(2j+\ep-\frac n2\big)\right] P^\ep_{j},\qquad j\geq 0;
\quad\ep=0,1\,,
$$
after we impose normalizability of the algebraic wavefunctions. The (normalizable) potentials
associated to this canonical form, $V(x)=\frac 14 c_0^2 x^2-c_*$, are  exactly solvable.
It may be shown that all normalizable exactly solvable potentials lead to a two-term recursion
relation, \cite{FGR96}.

In conclusion, for every canonical form but the fifth one, we can construct a family of
polynomials $P_k(E)$ satisfying a three-term recursion relation of the form $(k\geq 1)$:
\begin{multline}\label{3term}
-\left[ (2k-n-1)\hat c_{0-}+\hat c_-\right] P_{k}\\
\begin{split}
&=\left[ E+c_*+\hat c_0 \big( k-\frac n2-1\big)+\hat c_{00}\big( k-\frac n2-1\big)^2\right] P_{k-1}\\
&\quad +(k-1)(k-2-n)\left[(2k-n-3)\hat c_{+0}+\hat c_+\right] P_{k-2}\,,
\end{split}
\end{multline}
where $\hat c_{ab}$ denote the coefficients of $\hat P$ and $\hat Q$, the transforms of $P$ and
$Q$ under the projective transformation $z\mapsto w$ defined above. There is still an
important point to be addressed: one needs to make sure that the coefficient
in front of $P_k$ in~\eqref{3term} does not vanish for any positive $k$:
\begin{equation}\label{coeff}
 (2k-n-1)\hat c_{0-}+\hat c_-\neq 0\,,\qquad k\geq 1\,.
\end{equation}
If $H$ does {\em not} correspond to the canonical forms 1, 8, and 9, this is the case
provided $H$ is {\em fully algebraic} and the eigenfunctions are all regular. For the
canonical forms 1, 8, and 9, the condition \eqref{coeff} must be imposed {\em ad hoc};
see \cite{FGR96} for more details. Finally, if \eqref{coeff} holds, we introduce new
(normalized) polynomials $\hat P_k$ by
\begin{equation}\label{hatP}
P_k =
\begin{cases}
\frac{(-1)^k}{(2\hat c_{0-})^k}\,{\displaystyle
\frac{\hat P_k}{\Ga\left(\frac{\hat c_-}{2\hat c_{0-}}+k-\frac n2+\frac12\right)}},&
\qquad \text{if}\quad\hat c_{0-}\neq 0;\\
\frac{(-1)^k}{\hat c_-^k}\,\hat P_k,&\qquad \text{if}\quad\hat c_{0-}=0\,,
\end{cases}
\end{equation}
and arrive at our main result:

\begin{thm}
Let $H$ be a {\em fully algebraic} QES Hamiltonian whose eigenfunctions are all regular. Assume
also that eq.~\eqref{coeff} holds, if $H$ corresponds to the canonical forms 1, 8 or 9. Then
$H$ defines a family $\{\hat P_k\}_{k=0}^\infty$ of \(monic\) {\em weakly orthogonal} polynomials
via equations \eqref{chi1}, \eqref{chi2}, and \eqref{hatP} satisfying
\begin{equation}\label{3termOK}
\hat P_{k+1} = (E-b_k)\,\hat P_k-a_k\,\hat P_{k-1},\qquad k\geq 0,
\end{equation}
where
\begin{align*}
a_k  &= k(k-n-1)\left[\hat c_{+0}(2k-n-1)+\hat c_+\right]\left[\hat c_{0-}(2k-n-1)
+\hat c_-\right]\\
-b_k &= c_*+\hat c_0\left( k-\frac n2\right)+\hat c_{00}\left( k-\frac n2\right)^2\,.
\end{align*}
\end{thm}

\section{Weakly Orthogonal Polynomials}\label{sec.wop}

In the previous section we have shown that ``almost'' every QES Hamiltonian generates a family
of weakly orthogonal polynomials, that is a set of polynomials $\{\hat P_k\}_{k=0}^\infty$
verifying a recursion relation of the form \eqref{3termOK} with $P_0=1$ and $a_{n+1}=0$ (for a
certain $n$). In this final section, we summarize some properties of weakly-orthogonal polynomials
without making  reference to their relation with QES spectral problems. Let us first recall a pair of
standard results of the theory of orthogonal polynomials, \cite{Ch78}. According to Favard's Theorem
there is a {\em unique} moment functional $\cal L$ defined on $\Bbb C[E]$ such that the polynomials
are orthogonal under it:
\begin{equation}\label{orth}
{\cal L}(\hat P_k\,\hat P_l)=\ga_k\,\de_{kl}\,.
\end{equation}
Moreover (Boas's Theorem), there is a (not necessarily unique) function of bounded variation $\om$
such that
$$
{\cal L}(p)=\int_{-\infty}^\infty p(E)\,dw(E)\,,
$$
for an arbitrary polynomial $p$. We make use of these results to state the following
characterization theorem for weakly orthogonal polynomials:

\begin{thm}
Let $\{\hat P_k\}_{k=0}^\infty$ be a familiy of polynomials satisfying a three-term recursion
relation of the form \eqref{3termOK}, with $\hat P_{-1}=0$ and $\hat P_0=1$. Let $\cal L$ be the
associated moment functional. The following statements are then equivalent:

\begin{enumerate}
\item The coefficient $a_k$ is nonzero for $1\leq k\leq n$, and $a_{n+1}=0$.
\item The norms $\ga_k$ in \eqref{orth} are nonzero for $0\leq k\leq n$ and vanish for $k>n$.
\item $\hat P_{n+2}$ \(and hence $\hat P_{n+1+l}$ for $l\geq 1$\) is divisible
by $\hat P_{n+1}$, and $\hat P_{k+1}$ is not divisible by $\hat P_k$ for all
$k\leq n$.
\item Let $c_i$ be the coefficient of $E^{n+1-i}$ in $\hat P_{n+1}$. Then, the moments of
$\cal L$ satisfy the following recursion relation:
$$
\mu_k = \sum_{i=1}^{n+1} c_i\,\mu_{k-i},\qquad k\geq n+1\,.
$$
\item If, in addition, $a_k>0$ and $b_k$ is real, the moment functional $\cal L$ is given by
\begin{equation}\label{calL}
{\cal L}=\sum_{l=0}^n\om_l\,\de(E-E_l)\,,
\end{equation}
where the numbers $\om_l$ satisfy
\begin{equation}\label{omega}
\sum_{l=0}^n\hat P_k(E_l)\,\om_l=\de_{k0}\,,\qquad 0\leq k\leq n\,,
\end{equation}
and $E_l$ $(0\leq l\leq n)$ are the roots of $\hat P_{n+1}$, which are all simple and real.
\end{enumerate}
\end{thm}

\ni{\em Proof}. The proof was given in \cite{FGR96}, except for the following statements:
\begin{mylist}
\item $1\Leftrightarrow 3$: Clearly $a_k=0$ implies that $\hat P_{k+1}$ is divisible by $\hat P_k$.
Conversely, if $\hat P_{k+1}=(E-\tilde b_k)\hat P_k$, the recursion relation~\eqref{3termOK}
implies that $\tilde b_k=b_k$ and $a_k=0$.
\item $1\Rightarrow 5$: By uniqueness, we only need to show that $\cal L$ in \eqref{calL} is
consistent with \eqref{orth}. This was proved in \cite{FGR96} using the fact that the matrix
of coefficients $M_{kl}=\hat P_k(E_l)$ is non-singular if the polynomials are associated to a fully
algebraic QES Hamiltonian. However, since
$$
\det M=\prod_{i<j}(E_j-E_i)\,,
$$
and it can be shown that the roots of $P_{n+1}$ are real and simple if $a_k>0$
and $b_k$ is real, the same is true in this case.\QED
\end{mylist}

\begin{thebibliography}{50}

\bibitem{BD96}
{\sf C.M. Bender and G.V. Dunne}: {\it Quasi-exactly solvable systems
and orthogonal polynomials}. {\sc J. Math. Phys.} {\bf 37}. (1996). 6-11.

\bibitem{Ch78}
{\sf T.S. Chihara}: {\sc An Introduction to Orthogonal Polynomials}. (Gordon and
Breach, New York, 1978.)

\bibitem{FGR96}
{\sf F. Finkel, A. Gonz\'alez--L\'opez and M.A. Rodr\'\i guez}:
{\it Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials}.
{\sc J. Math. Phys.} {\bf 37}. (1996). 3954-3972.

\bibitem{GKO93a}
{\sf A. Gonz\'alez--L\'opez, N. Kamran and P.J. Olver}:
{\it Quasi-exact solvability}. {\sc Contemp. Math.} {\bf 160}. (1993). 113-140.

\bibitem{GKO93b}
{\sf A. Gonz\'alez--L\'opez, N. Kamran and P.J. Olver}:
{\it Normalizability of one-dimensional quasi-exactly solvable \sch{}
operators}. {\sc Commun. Math. Phys.} {\bf 153}. (1993). 117-146.

\bibitem{KO90}
{\sf N. Kamran and P.J. Olver}:
{\it Lie algebras of differential operators and Lie-algebraic
potentials}. {\sc J. Math. Anal. Appl.} {\bf 145}. (1990). 342-356.

\bibitem{KUW96}
{\sf A. Krajewska, A. Ushveridze and Z. Walczak}:
{\it Bender--Dunne polynomials and quasi-exact solvabiltity}. Preprint
hep-th 9601088. (1996).

\bibitem{Sh89}
{\sf M.A. Shifman}: {\it New findings in quantum mechanics (partial
algebraization of the spectral problem)}. {\sc Int. J. Mod. Phys.} {\bf A126}.
(1989). 2897-2952.

\bibitem{Tu88}
{\sf A.V. Turbiner}: {\it Quasi-exactly solvable problems and $\frak{sl}(2)$ algebra}.
{\sc Commun. Math. Phys.} {\bf 118}. (1988). 467-474.

\bibitem{Tu94}
{\sf A. Turbiner}: {\it Quasi-exactly solvable differential equations}. Preprint
IFUNAM FT 94-57. (1994).

\bibitem{Za94}
{\sf O.B. Zaslavskii}: {\it Two-dimensional quasi-exactly solvable models and classical
orthogonal polynomials}. {\sc J. Phys. A} {\bf 27}. (1994). 323-328.  

\end{thebibliography}

{\sc 
\begin{flushleft}
{{ F. Finkel${}^{\dagger}$ \qquad A. Gonz\'alez-L\'opez \qquad M.A. Rodr\'\i guez :} \\
Departamento de F\'\i sica Te\'orica II, 
Universidad Complutense, Madrid,  SPAIN.} \\
${}^{\dagger}${\sf E-Mail} { \small\tt: federico@@ciruelo.fis.ucm.es }
\end{flushleft}
}

\end{document}

