On the Extended-Hamiltonian Structure

On the Extended-Hamiltonian Structure of Certain
Superintegrable Systems on Constant-Curvature
Riemannian and Pseudo-Riemannian Surfaces

Claudia Maria CHANU and Giovanni RASTELLI

C.M. Chanu and G. Rastelli

Dipartimento di Matematica, Università di Torino, Torino, Italia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received March 21, 2020, in final form May 20, 2020; Published online June 11,
2020

We prove the integrability and superintegrability of a family of natural
Hamiltonians which includes and generalises those studied in some literature,
originally defined on the 2D Minkowski space. Some of the new Hamiltonians are
a perfect analogy of the well-known superintegrable system on the Euclidean
plane proposed by Tremblay–Turbiner–Winternitz and they are defined on
Minkowski space, as well as on all other 2D manifolds of constant curvature,
Riemannian or pseudo-Riemannian. We show also how the application of the
coupling-constant-metamorphosis technique allows us to obtain new
superintegrable Hamiltonians from the previous ones. Moreover, for the
Minkowski case, we show the quantum superintegrability of the corresponding
quantum Hamiltonian operator.Our results are obtained by applying the theory
of extended Hamiltonian systems, which is strictly connected with the geometry
of warped manifolds.

extended-Hamiltonian; superintegrable systems; constant curvature

37J35; 70H33

##  1 Introduction

In [16] it is proved that the Hamiltonian

$\displaystyle H_{0}=p_{1}p_{2}-\alpha q_{2}^{2k+1}q_{1}^{-2k-3},$ (1.1)

$\alpha\in\mathbb{R}$, is superintegrable for any $k\in\mathbb{Q}$. The proof
is obtained by introducing an irregular bi-Hamiltonian structure, in analogy
with the rational Calogero–Moser system.

A partial generalization of this Hamiltonian has been later proved to be again
superintegrable in [2]. This is the Hamiltonian

$\displaystyle H=p_{1}p_{2}-\alpha
q_{2}^{2k+1}q_{1}^{-2k-3}-\frac{\beta}{2}q^{k}_{2}q_{1}^{-k-2},$ (1.2)

with $k\in\mathbb{N}$, $\alpha,\beta\in\mathbb{R}$. In both the previous
articles, the superintegrability of the Hamiltonians is proved via algebraic
methods. In this article, we prove that $H$ is indeed superintegrable for any
$k\in\mathbb{Q}$ and that its Hamilton–Jacobi equation is separable, implying
the integrability of the system, for any $k\in\mathbb{R}$. Moreover, we
determine a new class of Hamiltonians which generalizes $H$ and is again
superintegrable. We obtain these results by showing that $H$ can be written in
form of an extended Hamiltonian. We introduced extended Hamiltonian systems in
[4] and in [3, 5, 7, 8] we generalized the idea of extended systems. An
extended Hamiltonian is a Hamiltonian $H$ on a cotangent bundle, with $n+1$
degrees of freedom, which is obtained from a Hamiltonian $L$ with $n$ degrees
of freedom, depending on a rational parameter $k$ and admitting a non-trivial
characteristic and explicitly determined first integral $K$. When $L$ is
polynomial in the momenta, the same is for $H$ and for $K$. Moreover, if $L$
is a natural Hamiltonian on the cotangent bundle of a Riemannian (or pseudo-
Riemannian) manifold, i.e., $L$ is a Hamiltonian of mechanical type, then its
extensions (if any) are natural Hamiltonians too. Even if in the most general
setting the Hamiltonians involved in the extension procedure are not
necessarily natural (see [11] for examples of extensions of non-natural
Hamiltonians), in this article, however, we apply the general theory to
natural Hamiltonians only, because we are dealing with the natural Hamiltonian
(1.2) and some generalizations of it. The term extension is motivated by the
fact that our procedure increases the degrees of freedom of the Hamiltonian
system, this is the only relation with other existing extension procedures,
for example the Eisenhart extension. Since the characteristic first integral
is generated by the recursive application of a particular operator, our
extension procedure is essentially algebraic, but it admits a geometric
characterization in the case of natural Hamiltonians. The class of extended
Hamiltonians, parametrized by $k\in\mathbb{Q}$, includes many (but not all,
see Remark 3.3) of the known superintegrable Hamiltonians admitting a
polynomial in the momenta first integral of degree dependent on any chosen
rational number $k$. Examples which belong to the above class are, for
instance, the isotropic and non-isotropic harmonic oscillators, the three-
particle Jacobi–Calogero system, the Wolfes system, the
Tremblay–Turbiner–Winternitz system, the Post–Winternitz system. In
particular, this last system is studied in [9] by applying the technique of
coupling-constant-metamorphosis. We apply here the same technique to obtain
from $H$, and our generalizations of it, new superintegrable systems. In
Section 2 we recall the basic elements of the theory of extended systems. In
Section 3 we show that $H$ is indeed an extended Hamiltonian. In Section 4 we
apply the theories of extended Hamiltonians and coupling-constant-
metamorphosis to obtain new superintegrable systems. Moreover we show that the
same family of extended Hamiltonians of $H$ includes Hamiltonians on other
constant-curvature two-dimensional manifolds: Euclidean planes, spheres,
pseudo-spheres, de Sitter and anti-de Sitter spaces. These Hamiltonians can be
considered as generalisations of $H$ to constant-curvature manifolds. In
Section 5, the Laplace–Beltrami quantization of $H$ and its characteristic
first integral, together with their generalisations on flat manifolds, is
obtained via the shift-ladder approach proposed by S. Kuru and J. Negro in
[15] and adapted to extended Hamiltonians in [10].

##  2 Extensions of Hamiltonian systems

Let $L\big{(}q^{i},p_{i}\big{)}$ be any Hamiltonian function with $N$ degrees
of freedom, defined on the cotangent bundle of an $N$-dimensional manifold
with coordinates $\big{(}q^{i}\big{)}$ and conjugate momenta $(p_{i})$.

We say that $L$ admits extensions, if there exists
$(c,c_{0})\in\mathbb{R}^{2}\setminus\\{(0,0)\\}$ such that there exists a non
null solution $G\big{(}q^{i},p_{i}\big{)}$ of

$\displaystyle X_{L}^{2}(G)=-2(cL+c_{0})G,$ (2.1)

where $X_{L}$ is the Hamiltonian vector field of $L$.

If $L$ admits extensions, then, for any $\gamma(u)$ solution of the ODE

$\displaystyle\gamma^{\prime}+c\gamma^{2}+C=0,$ (2.2)

depending on the arbitrary constant parameter $C$, we say that any Hamiltonian
$H\big{(}u,q^{i},p_{u},p_{i}\big{)}$ with $N+1$ degrees of freedom of the form

$\displaystyle
H=\frac{1}{2}p_{u}^{2}-\left(\frac{m}{n}\right)^{2}\gamma^{\prime}L+\left(\frac{m}{n}\right)^{2}c_{0}\gamma^{2}+\frac{\Omega}{\gamma^{2}},\qquad(m,n)\in\mathbb{N}\setminus\\{0\\},\quad\Omega\in\mathbb{R}$
(2.3)

is an extension of $L$.

Extensions of Hamiltonians where introduced in [3] and studied because, when
$L$ is a polynomial in the momenta Hamiltonian and in particular when $L$ is a
natural Hamiltonian, they admit a polynomial in the momenta first integral
generated via a recursive algorithm; its degree in $(p_{u},p_{i})$ depends on
the value of $m,n\in\mathbb{N}\setminus\\{0\\}$. For a generic Hamiltonian
$L$, given any $m,n\in\mathbb{N}\setminus\\{0\\}$, let us consider the
operator

$\displaystyle U_{m,n}=p_{u}+\frac{m}{n^{2}}\gamma X_{L}.$ (2.4)

######  Proposition 2.1 ([7]).

For $\Omega=0$, the Hamiltonian (2.3) is in involution with the function

$\displaystyle
K_{m,n}=U_{m,n}^{m}(G_{n})=\left(p_{u}+\frac{m}{n^{2}}\gamma(u)X_{L}\right)^{m}(G_{n}),$
(2.5)

where $G_{n}$ is the $n$-th term of the recursion

$\displaystyle G_{1}=G,\qquad G_{n+1}=X_{L}(G)G_{n}+\frac{1}{n}GX_{L}(G_{n}),$
(2.6)

starting from any solution $G$ of (2.1).

For $\Omega\neq 0$, the recursive construction of a first integral is more
complicated: we construct the following function, depending on two strictly
positive integers $s$, $r$

$\displaystyle\bar{K}_{2s,r}=\left(U_{2s,r}^{2}+2\Omega\gamma^{-2}\right)^{s}(G_{r}),$
(2.7)

where the operator $U^{2}_{2s,r}$ is defined according to (2.4) as

$\displaystyle
U^{2}_{2s,r}=\left(p_{u}+\frac{2s}{r^{2}}\gamma(u)X_{L}\right)^{2},$

and $G_{r}$ is, as in (2.5), the $r$-th term of the recursion (2.6), with
$G_{1}=G$ solution of (2.1). For $\Omega=0$ the functions (2.7) reduce to
(2.5) and can be computed also when the first of the indices is odd.

######  Theorem 2.2 ([8]).

For any $\Omega\in\mathbb{R}$, the Hamiltonian (2.3) satisfies, for $m=2s$,

$\displaystyle\big{\\{}H,\bar{K}_{m,n}\big{\\}}=0,$

for $m=2s+1$,

$\displaystyle\big{\\{}H,\bar{K}_{2m,2n}\big{\\}}=0.$

We call $K$ and $\bar{K}$, of the form (2.5) and (2.7) respectively,
_characteristic first integrals_ of the corresponding extensions. It is proved
in [3, 8] that the characteristic first integrals $K$ or $\bar{K}$ are
functionally independent from $H$, $L$, and from any first integral
$I\big{(}p_{i},q^{i}\big{)}$ of $L$. This means that the extensions of
(maximally) superintegrable Hamiltonians are (maximally) superintegrable
Hamiltonians with one additional degree of freedom (see also [6]). In
particular, any extension of a one-dimensional Hamiltonian is maximally
superintegrable.

The explicit expression of the characteristic first integrals is given as
follows [8]. For $r\leq m$, we have

$\displaystyle U_{m,n}^{r}(G_{n})=P_{m,n,r}G_{n}+D_{m,n,r}X_{L}(G_{n}),$

with

$\displaystyle
G_{n}=\sum_{j=0}^{[(n-1)/2]}\binom{n}{2j+1}G^{2j+1}(X_{L}G)^{n-2j-1}(-2)^{j}(cL+c_{0})^{j},$
$\displaystyle
P_{m,n,r}=\sum_{j=0}^{[r/2]}\binom{r}{2j}\left(\frac{m}{n}\gamma\right)^{2j}p_{u}^{r-2j}(-2)^{j}(cL+c_{0})^{j},$
$\displaystyle
D_{m,n,r}=\frac{1}{n}\sum_{j=0}^{[(r-1)/2]}\binom{r}{2j+1}\left(\frac{m}{n}\gamma\right)^{2j+1}p_{u}^{r-2j-1}(-2)^{j}(cL+c_{0})^{j},\qquad
m>1,$

where $[\cdot]$ denotes the integer part and
$D_{1,n,1}=\frac{1}{n^{2}}\gamma$.

The expansion of the first integral (2.7) is

$\displaystyle\bar{K}_{2m,n}=\sum_{j=0}^{m}\binom{m}{j}\left(\frac{2\Omega}{\gamma^{2}}\right)^{j}U_{2m,n}^{2(m-j)}(G_{n}),$

with $U^{0}_{2m,n}(G_{n})=G_{n}$.

######  Remark 2.3.

If $L$ is a natural Hamiltonian with a metric and a potential which are
algebraic (resp. rational) functions of the $\big{(}q^{i}\big{)}$ and if $L$
admits an extension associated with a function $G\big{(}q^{i},p_{i}\big{)}$
which is polynomial in the momenta and algebraic (resp. rational) in the
$\big{(}q^{i}\big{)}$, then the extension of $L$ with $\gamma=(cu)^{-1}$ has
first integrals which are polynomial in the momenta and algebraic (resp.
rational) in the $\big{(}q^{i}\big{)}$. The first integrals $\bar{K}_{2m,n}$
are polynomial w.r.t. $G$, $L$, $X_{L}G$, $\gamma$ and $p_{u}$. Indeed, by
(2.1), it is easy to check that $X_{L}(G_{n})$ is a polynomial in $G$,
$X_{L}G$, $L$, as well as $G_{n}$.

######  Remark 2.4.

In [5] it is proven that the ODE (2.2) defining $\gamma$ is a necessary
condition in order to get a characteristic first integral of the form (2.5) or
(2.7). According to the value of $c$ and $C$, the explicit form of $\gamma(u)$
is given (up to constant translations of $u$) by

$\displaystyle\gamma=\begin{cases}-Cu,&c=0,\\\
\dfrac{1}{T_{\kappa}(cu)}=\dfrac{C_{\kappa}(cu)}{S_{\kappa}(cu)},&c\neq
0,\end{cases}$

where $\kappa=C/c$ is the ratio of the constant parameters appearing in (2.2)
and $T_{\kappa}$, $S_{\kappa}$ and $C_{\kappa}$ are the trigonometric tagged
functions [19] (see also [7] for a summary of their properties)

$\displaystyle
S_{\kappa}(x)=\begin{cases}\dfrac{\sin\sqrt{\kappa}x}{\sqrt{\kappa}},&\kappa>0,\\\
x,&\kappa=0,\\\
\dfrac{\sinh\sqrt{|\kappa|}x}{\sqrt{|\kappa|}}&\kappa<0\end{cases}\qquad
C_{\kappa}(x)=\begin{cases}\cos\sqrt{\kappa}x,&\kappa>0,\\\ 1,&\kappa=0,\\\
\cosh\sqrt{|\kappa|}x,&\kappa<0,\end{cases}\qquad
T_{\kappa}(x)=\frac{S_{\kappa}(x)}{C_{\kappa}(x)}.$

Therefore, we have

$\displaystyle\gamma^{\prime}=\begin{cases}-C,&c=0,\\\
\dfrac{-c}{S_{\kappa}^{2}(cu)},&c\neq 0.\end{cases}$

######  Remark 2.5.

When $c\neq 0$ and $\kappa\neq 0$, a translation of the variable $u$ allows
not so evident changes in $\gamma$ and $\gamma^{\prime}$. Indeed, by using
standard formulas, we can write for example, for $c\neq 0$,

$\displaystyle\gamma^{\prime}(cu)=\frac{4\kappa c}{({\rm
e}^{\sqrt{-\kappa}cu}-{\rm
e}^{-\sqrt{-\kappa}cu})^{2}},\qquad\gamma^{2}(cu)=-\kappa\frac{({\rm
e}^{\sqrt{-\kappa}cu}+{\rm e}^{-\sqrt{-\kappa}cu})^{2}}{({\rm
e}^{\sqrt{-\kappa}cu}-{\rm e}^{-\sqrt{-\kappa}cu})^{2}}$

so that, for $c=\pm 1$, $\kappa=\pm 1$ we can easily compute, in
correspondence with the indicated translations of $u$, the expressions of
$\gamma^{\prime}$

Table of $\gamma^{\prime}$ | $\kappa=1$ |  $\kappa=-1$ 2pt
---|---|---
$c=1$ | $-\sin^{-2}u$ |  $-\sinh^{-2}u$2pt
$c=-1$ | $\sin^{-2}u$ |  $\sinh^{-2}u$ 2pt
translation: | $u\rightarrow u+\pi/2$ |  $u\rightarrow u+{\rm i}\pi/2$ 2pt
$c=1$ | $-\cos^{-2}u$ |  $\cosh^{-2}u$ 2pt
$c=-1$ | $\cos^{-2}u$ |  $-\cosh^{-2}u$ 2pt

and of $\gamma^{2}$

Table of $\gamma^{2}$ | $\kappa=1$ |  $\kappa=-1$ 2pt
---|---|---
$c=\pm 1$ | $\tan^{-2}u$ |  $\tanh^{-2}u$ 2pt
translation: | $u\rightarrow u+\pi/2$ |  $u\rightarrow u+{\rm i}\pi/2$ 2pt
$c=\pm 1$ | $\tan^{2}u$ |  $\tanh^{2}u$ 2pt

where the complex translation of the real variable $u$ leaves (2.3) real. The
table shows the remarkable fact that, for both positive and negative values of
$c$, we can always choose positive or negative values of $\gamma^{\prime}$, by
selecting suitable translations and values of $\kappa$, i.e., we can obtain
extensions with different signature. This is particularly important since, as
pointed out in [5], if $L$ is a natural Hamiltonian with more than one degree
of freedom, then the integrability conditions of (2.1) require that $c$
satisfies certain conditions (related with the sectional curvatures of the
metric of $L$), therefore, the sign of $c$ is fixed by $L$, while $\kappa$
remains free. Hence, without taking in account translations in $u$, the
signature of the metric of the extended manifold (i.e., the sign of
$\gamma^{\prime}$ in (2.3)) will be constrained by $L$. Actually, that sign
can be freely chosen.

In order to check if any $(N+1)$-dimensional Hamiltonian $H$ is an extension
of a $N$-dimensional Hamiltonian $L$, we must check if

  1. (i)

There exist canonical coordinates $(u,p_{u})$ such that $H$ can be written as
a _warped Hamiltonian_ , that is

$\displaystyle
H=\frac{1}{2}p_{u}^{2}+f(u)+\left(\frac{m}{n}\right)^{2}\alpha(u)L,\qquad
m,n\in\mathbb{N}\setminus\\{0\\},$ (2.8)

and the Hamiltonian $L$ is independent of $(u,p_{u})$. As we see in the next
section, these canonical coordinates can be determined by following an
invariant geometrical procedure, when $H$ is a natural Hamiltonian.

  2. (ii)

For some constants $c$ and $c_{0}$ not both vanishing, the equation (2.1)
admits a non null solution $G$.

  3. (iii)

The functions $\alpha$ and $f$ in (2.8) can be written as in (2.3) for a
$\gamma$ satisfying (2.2).

We remark that if (2.1) has a solution for $c\neq 0$, then we may assume
without loss of generality $c_{0}=0$, because $L$ is determined up to additive
constants.

Examples of the procedure of extension applied on Hamiltonians $L$ which are
not of mechanical type, defined possibly on Poisson manifolds, are given in
[11]. In the following, we assume that all Hamiltonian functions are natural
and defined on cotangent bundles of Riemannian or pseudo-Riemannian manifolds.

##  3 Determination of the extended Hamiltonian structure

In order to show that the natural Hamiltonian (1.2) is an extended
Hamiltonian, we have first to check if it can be written, by a point-
transformation of coordinates, in the form of a warped Hamiltonian (2.8). We
will use the intrinsic characterization of such natural Hamiltonians making a
simple generalization of the result given in [5, Theorem 9]

Let us consider a natural Hamiltonian

$\displaystyle H=\frac{1}{2}\tilde{g}^{ab}p_{a}p_{b}+\tilde{V}$ (3.1)

on a $(n+1)$-dimensional Riemannian manifold
$\big{(}\tilde{Q},\tilde{\mathbf{g}}\big{)}$ and let $X$ be a conformal
Killing vector of $\tilde{\mathbf{g}}$, that is a vector field satisfying

$\displaystyle[X,\tilde{\mathbf{g}}]=\mathcal{L}_{X}\tilde{\mathbf{g}}=\phi\tilde{\mathbf{g}},$

where $\phi$ is a function on $\tilde{Q}$, called conformal factor, and
$[\cdot,\cdot]$ denotes the Schouten–Nijenhuis bracket. We define $X^{\flat}$
as the 1-form obtained by lowering the indices of the vector field $X$ with
the metric tensor $\tilde{\mathbf{g}}$.

######  Theorem 3.1.

If on $\tilde{Q}$ there exists a conformal Killing vector field $X$ with
conformal factor $\phi$ such that

$\displaystyle{\rm d}X^{\flat}\wedge X^{\flat}=0,$ (3.2) $\displaystyle{\rm
d}\phi\wedge X^{\flat}=0,$ (3.3) $\displaystyle{\rm d}\|X\|\wedge
X^{\flat}=0,$ (3.4) $\displaystyle{\rm
d}\big{(}X\big{(}\tilde{V}\big{)}+\phi\tilde{V}\big{)}\wedge X^{\flat}=0,$
(3.5)

then, there exist on $\tilde{Q}$ coordinates $\big{(}u,q^{i}\big{)}$ such that
$\partial_{u}$ coincides up to a rescaling with $X$ and the natural
Hamiltonian (3.1) has the form (2.8). Moreover, if

$\displaystyle\tilde{R}(X)=aX,\qquad a\in\mathbb{R},$ (3.6)

where $\tilde{R}$ is the Ricci tensor of the Riemannian manifold
$\big{(}\tilde{Q},\tilde{\mathbf{g}}\big{)}$, then the function $\alpha(u)$ in
(2.8) satisfies the condition $\alpha=-\gamma^{\prime}$ with $\gamma$ solution
of (2.2).

###### Proof.

Following [5], conditions (3.2), (3.3), (3.4) imply that the metric
$\tilde{g}^{ab}$ is in the warped form (2.8) in suitable canonical coordinates
$(u,q^{i})$ such that in these coordinates $X$ is of the form

$\displaystyle X=F(u)\partial_{u}$

and

$\displaystyle\tilde{g}^{ij}\big{(}u,q^{h}\big{)}=F^{-2}g^{ij}\big{(}q^{h}\big{)},\qquad\phi=2\partial_{u}F.$

Condition (3.5) means that $X\big{(}\tilde{V}\big{)}+\phi\tilde{V}$ is
constant on the leaves orthogonal to $X$; in other words,
$X\big{(}\tilde{V}\big{)}+\phi\tilde{V}$ is a function of $u$ only. Being
$X=F(u)\partial_{u}$, $\phi=2F^{\prime}$ and
$X\big{(}\tilde{V}\big{)}=F(u)\partial_{u}\tilde{V}$, last condition reads as
$F(u)\partial^{2}_{ui}\tilde{V}+2F^{\prime}\partial_{i}\tilde{V}=0$ for all
$i$, that is $\partial^{2}_{ui}\tilde{V}-\partial_{u}\ln
F^{2}\partial_{i}\tilde{V}=0$. Hence,
$\partial_{u}\big{(}\ln\big{(}F^{2}\partial_{i}\tilde{V}\big{)}\big{)}=0$ and
therefore $F^{2}\partial_{i}\tilde{V}$ is independent of $u$, that is
$\partial_{i}\tilde{V}\big{(}u,q^{h}\big{)}=\partial_{i}V\big{(}q^{h}\big{)}F^{-2}$.
It follows that

$\displaystyle\tilde{V}\big{(}u,q^{h}\big{)}=V\big{(}q^{h}\big{)}F^{-2}+f(u)=\alpha(u)V\big{(}q^{h}\big{)}+f(u)$

and thus it has the form of the potential in (2.8) where
$V\big{(}q^{h}\big{)}$ is the potential of $L$. The second part of the
statement is proved in [5]. ∎

######  Remark 3.2.

Conditions (3.2), (3.3), (3.4) are the already known intrinsic
characterization of a warped metric, while condition (3.5) intrinsically
assures that the scalar part of the Hamiltonian is compatible with the warped
structure. It is remarkable the fact that an intrinsic condition – equation
3.6 – characterizes also the subset of the warped metrics which are involved
in extended Hamiltonians. The extended potentials with $\Omega=0$ are
intrinsically characterised by $X\big{(}\tilde{V}\big{)}+\phi\tilde{V}=0$.

We apply the above statement to the Hamiltonian (1.2) defined on Minkowski
plane. First, we remark that on $\mathbb{M}^{2}$ the Ricci tensor is zero,
therefore, the condition (3.6) is verified for any vector $X$ with $a=0$. A
conformal Killing vector satisfying the remaining conditions is

$\displaystyle
X=C\big{(}q^{1}\partial_{1}+q^{2}\partial_{2}\big{)},\qquad\phi=2C,$

where $C$ is any non zero real constant.

The vector $X$ is the gradient of the function $2Cq^{1}q^{2}$. Performing the
coordinate transformation

$\displaystyle q^{1}=\frac{1}{\sqrt{2}}\big{(}x^{1}+x^{2}\big{)},\qquad
q^{2}=\frac{1}{\sqrt{2}}\big{(}x^{1}-x^{2}\big{)},$

we see that $\big{(}x^{i}\big{)}$ are the standard pseudo-Cartesian
coordinates such that the geodesic term of the Hamiltonian $H$ becomes

$\displaystyle\frac{1}{2}\big{(}p_{1}^{2}-p_{2}^{2}\big{)}.$

In these coordinates, we have

$\displaystyle 2q^{1}q^{2}=\big{(}x^{1}\big{)}^{2}-\big{(}x^{2}\big{)}^{2},$

thus, $2q^{1}q^{2}=C$ is the equation verified by points having fixed distance
from the origin in the pseudo-polar coordinates of $\mathbb{M}^{2}$. Hence, a
coordinate system $(u,\chi)$ such that $X$ is parallel to $\partial_{u}$ is
the pseudo-polar one, with $u$ coinciding with the hyperbolic radius of the
hyperbolic circles. Indeed, by applying the transformations

$\displaystyle x^{1}=u\cosh\chi,\qquad$ $\displaystyle x^{2}=u\sinh\chi,$
$\displaystyle
u=\sqrt{\big{(}x^{1}\big{)}^{2}-\big{(}x^{2}\big{)}^{2}},\qquad$
$\displaystyle\chi=\tanh^{-1}\frac{x^{2}}{x^{1}},$ $\displaystyle
q^{1}=\frac{1}{\sqrt{2}}u{\rm e}^{\chi},\qquad$ $\displaystyle
q^{2}=\frac{1}{\sqrt{2}}u{\rm e}^{-\chi},$

where $\chi$ is the pseudo-angle, we get after simple computations that (1.2)
becomes

$\displaystyle
H=\frac{1}{2}\left(p_{u}^{2}-\frac{1}{u^{2}}p_{\chi}^{2}\right)-\frac{1}{u^{2}}\left(2\alpha\big{(}{\rm
e}^{-2\chi}\big{)}^{2(k+1)}+\beta\big{(}{\rm e}^{-2\chi}\big{)}^{k+1}\right).$

Finally, we apply a further coordinate transformation, by setting

$\displaystyle\psi=(k+1)\chi,$

and we get

$\displaystyle
H=\frac{1}{2}p_{u}^{2}-\frac{(k+1)^{2}}{u^{2}}\left(\frac{1}{2}p_{\psi}^{2}+\tilde{\alpha}{\rm
e}^{-4\psi}+\tilde{\beta}{\rm e}^{-2\psi}\right),$ (3.7)

with $\tilde{\alpha}=\frac{2}{(k+1)^{2}}\alpha$,
$\tilde{\beta}=\frac{\beta}{(k+1)^{2}}$, which is an extended Hamiltonian on
$\mathbb{M}^{2}$ with

$\displaystyle(k+1)^{2}=-\frac{m^{2}}{n^{2}c}$

of the one-dimensional Hamiltonian

$\displaystyle L=\frac{1}{2}p_{\psi}^{2}+V,$ (3.8)

where

$\displaystyle V=\tilde{\alpha}{\rm e}^{-4\psi}+\tilde{\beta}{\rm
e}^{-2\psi},$

and $\gamma=\frac{1}{cu}$ satisfying (2.2) for $c_{0}=0$, $C=0$, hence with
$\kappa=0$, and $c$ satisfying the condition

$\displaystyle c=-\frac{m^{2}}{n^{2}(k+1)^{2}},$

i.e., $c=-\eta^{2}$ with $\eta\in\mathbb{Q}$.

As a second step, we have to find a non-trivial solution $G(p_{\psi},\psi)$ of
the equation

$\displaystyle X_{L}^{2}G=-2cLG=2\eta^{2}LG.$ (3.9)

If we assume

$\displaystyle G=g(\psi)p_{\psi},$

by expanding equation (3.9) we obtain

$\displaystyle\big{(}g^{\prime\prime}-\eta^{2}g\big{)}p_{\psi}^{3}+\big{[}\tilde{\beta}\big{(}6g^{\prime}-\big{(}4+2\eta^{2}\big{)}g\big{)}{\rm
e}^{-2\psi}+\tilde{\alpha}\big{(}12g^{\prime}-\big{(}16+2\eta^{2}\big{)}g\big{)}{\rm
e}^{-4\psi}\big{]}p_{\psi}=0.$

The coefficients of $p_{\psi}^{3}$ and $p_{\psi}$ vanish identically if and
only if the following conditions hold

$\displaystyle g=a_{1}{\rm e}^{-\eta\psi}+a_{2}{\rm e}^{\eta\psi},$
$\displaystyle g=a_{3}\big{(}\tilde{\beta}{\rm
e}^{2\psi}+2\tilde{\alpha}\big{)}^{\frac{\eta^{2}}{12}-\frac{1}{3}}{\rm
e}^{(\frac{\eta^{2}}{6}+\frac{4}{3})\psi},$

where $a_{i}$ are arbitrary real constants. A common solution $g$ exists for

$\displaystyle\eta=2,\qquad a_{1}=0,\qquad a_{2}=a_{3}.$

Indeed, we get the non-trivial solution of (3.9)

$\displaystyle G={\rm e}^{2\psi}p_{\psi},\qquad c=-4,$

that corresponds to the extended Hamiltonian

$\displaystyle H=\frac{1}{2}p_{u}^{2}-\frac{m^{2}}{4n^{2}u^{2}}L,\qquad c=-4.$

Hence, by solving

$\displaystyle\frac{m}{n}=2(k+1),$ (3.10)

(for any $k\in\mathbb{Q}$, the equation admits positive integer solutions $m$,
$n$) we can write the Hamiltonian (3.7) as an extended Hamiltonian.

As immediate corollaries of the fact that the Hamiltonian (1.2) is an
extension of (3.8), we get additional information about (1.2)

  1. (i)

the function $L$ is a quadratic first-integral of $H$, in accordance with the
results of [2];

  2. (ii)

as a consequence of (i), the Hamilton–Jacobi equation associated with $H$ is
separable in a Stäckel coordinates system for any $k\in\mathbb{R}$. It is
evident from (3.7) that this coordinate system is $(u,\psi)$ (or,
equivalently, $(u,\chi)$), the pseudo-polar coordinates of $\mathbb{M}^{2}$.
The separability of the Hamiltonian (1.2) was not noticed until now, while the
separability of (1.1) with $\beta=0$, $k\in\mathbb{R}$ was found in [16] in
the same pseudo-polar coordinates.

######  Remark 3.3.

In [16] it is proved the superintegrability of other two Hamiltonians in
$\mathbb{M}^{2}$,

$\displaystyle H_{1}=2p_{1}p_{2}+\frac{q_{2}^{d}}{\sqrt{q_{1}}},\qquad
d=p,\qquad\text{or}\qquad d=\frac{1-2p}{2},\qquad p\in\mathbb{N},$

with the quadratic in the momenta first integral

$\displaystyle
I_{1}=2p_{1}(q_{2}p_{2}-p_{1}q_{1})+\frac{q_{2}^{d+1}}{\sqrt{q_{1}}},$

and

$\displaystyle H_{2}=2p_{1}p_{2}+q_{1}q_{2}^{d},\qquad
d=\frac{1-p}{p},\qquad\text{or}\qquad d=\frac{1+2p}{1-2p},\qquad
p\in\mathbb{N},$

with first integral

$\displaystyle I_{2}=p_{1}^{2}+\frac{q_{2}^{d+1}}{d+1}.$

These Hamiltonians cannot be written as extensions by using point
transformations of coordinates. Indeed, due to the form of the extended
Hamiltonians, in dimension two it is necessary that the extended Hamiltonian
is diagonalized in the coordinates adapted to the warped product
characterizing the extension. Necessarily, also the first integral $L$,
quadratic in the momenta, must be diagonalized in these coordinates, since the
Hamiltonian is Stäckel separable. It is easy to check that $I_{1}$ and $I_{2}$
are the unique non-trivial quadratic first integrals of the Hamiltonians
$H_{1}$ and $H_{2}$ respectively, and it is impossible to find coordinates,
real or complex, simultaneously diagonalizing $H_{1}$ and $I_{1}$, or $H_{2}$
and $I_{2}$. It is possible to see that the Killing tensors associated to
$I_{1}$ and $I_{2}$ have an eigenvalue with algebraic multiplicity two, but
with one-dimensional eigenspace (in the case of indefinite metric there exist
symmetric tensors which do not admit a basis of eigenvectors). Therefore, the
Killing tensors cannot determine separable coordinates [1].

##  4 Generalizations

Since the theory of extended Hamiltonians can be applied to Hamiltonian (1.2),
which is the extension of (3.8) as it is proved in Section 3, we can obtain
from (1.2) some new families of extended Hamiltonians, which are maximally
superintegrable systems on $\mathbb{M}^{2}$ and other two-dimensional
Riemannian manifolds of constant curvature.

(i) The first evident generalization is that, from (3.10), we have that (1.2)
is an extended Hamiltonian for any

$\displaystyle k=\frac{m}{2n}-1,$

and, therefore, for any rational number $k\in\mathbb{Q}$ instead of
$k\in\mathbb{N}$. In [16] the superintegrability of (3.7) with $\beta=0$ was
proved for $k\in\mathbb{Q}$.

From Section 2, it is straightforward to find another generalization of the
Hamiltonian (1.2) just by adding to it the scalar

$\displaystyle\Omega u^{2}=2\Omega q^{1}q^{2},\qquad\Omega\in\mathbb{R},$

so that we get a family of superintegrable potentials on $\mathbb{M}_{2}$
depending on three real parameters $(\alpha,\beta,\Omega)$, in analogy with
the Tremblay–Turbiner–Winternitz potential on $\mathbb{E}_{2}$. Indeed, the
general theory of extended Hamiltonians resumed in Section 2 assures the
existence of a constant of motion of the form (2.7) for these new
Hamiltonians.

(ii) We can try to find more general scalar potentials $V(\psi)$ in
$L=\frac{1}{2}p_{\psi}^{2}+V$ allowing the existence of functions $G$
solutions of (3.9), where $c=-\eta^{2}$ in order to stay on a Lorentzian
manifold, without restrictions on $\eta\in\mathbb{R}$.

We start by looking for $G=G(\psi)$. By solving equation (3.9) for both $G$
and $V$ as in the previous section, we find

$\displaystyle G=C_{1}{\rm e}^{\eta\psi}-C_{2}{\rm e}^{-\eta\psi},$ (4.1)
$\displaystyle V=C_{3}\big{(}C_{1}{\rm e}^{\eta\psi}+C_{2}{\rm
e}^{-\eta\psi}\big{)}^{-2},$ (4.2)

where $C_{1}$, $C_{2}$, $C_{3}$ are arbitrary parameters (with the obvious
constraints $C_{3}\neq 0$, $C_{1}$ and $C_{2}$ not both zero). These general
expressions of $G$ and $V$ were obtained already in [3]. Instead, by looking
for $G$ of the form $G=g(\psi)p_{\psi}$, we obtain

$\displaystyle g=C_{1}{\rm e}^{\eta\psi}+C_{2}{\rm e}^{-\eta\psi},$ (4.3)
$\displaystyle V=(C_{1}{\rm e}^{\eta\psi}+C_{2}{\rm
e}^{-\eta\psi})^{-2}\big{(}C_{3}+C_{4}\big{(}C_{1}{\rm e}^{\eta\psi}-C_{2}{\rm
e}^{-\eta\psi}\big{)}\big{)}=\left(C_{3}+\frac{C_{4}}{\eta}g^{\prime}\right)g^{-2},$
(4.4)

where $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$ are arbitrary parameters (with the
obvious constraints $C_{1}$, $C_{2}$ not both zero, as well as $C_{3}$,
$C_{4}$).

It is immediate to see that (4.2) is a particular case of (4.4), obtained for
$C_{4}=0$, while the functions $G$ and $g$ (given in (4.1) and (4.3),
respectively) are equal up to the sign of $C_{2}$. By setting

$\displaystyle\eta=2,\qquad C_{1}=1,\qquad C_{2}=0,\qquad
C_{3}=\tilde{\alpha},\qquad C_{4}=\tilde{\beta},$

in (4.3), (4.4), we exactly find the potential of the Hamiltonian (3.8), with
the corresponding function $G$ solution of (3.9). In general, the change of
coordinates

$\displaystyle q^{1}=\frac{u}{\sqrt{2}}{\rm e}^{\frac{\eta}{k}\psi},\qquad
q^{2}=\frac{u}{\sqrt{2}}{\rm e}^{-\frac{\eta}{k}\psi},$

transforms the extension

$\displaystyle
H=\frac{1}{2}p_{u}^{2}-k^{2}\frac{1}{\eta^{2}u^{2}}\left(\frac{1}{2}p_{\psi}^{2}+V\right)+\eta^{4}\Omega
u^{2},\qquad k\in\mathbb{Q},$

into

$\displaystyle
H=p_{1}p_{2}-\frac{4\tilde{k}^{2}}{2\eta^{2}q^{1}q^{2}}\big{(}C_{1}z^{\tilde{k}}+C_{2}z^{-\tilde{k}}\big{)}^{-2}\big{(}C_{3}+C_{4}\big{(}C_{1}z^{\tilde{k}}-C_{2}z^{-\tilde{k}}\big{)}\big{)}+2\eta^{4}\Omega
q^{1}q^{2},$

with

$\displaystyle z=\frac{q^{1}}{q^{2}},\qquad\tilde{k}=\frac{k}{2}.$

It is now evident that, since $\eta$ is now an inessential multiplicative
parameter of the potential, the extensions with a general $\eta\in\mathbb{R}$
are Hamiltonians essentially equivalent to those obtained for $\eta=2$.

We can write the functions (4.3) and (4.4) in a simpler form, focusing on
essential parameters (with $\eta$ fixed equal to 2). According to the sign of
$C_{1}C_{2}$ we can write (4.3) as

$\displaystyle g=a_{0}{\rm e}^{2\epsilon\psi},\qquad\epsilon=\pm 1,\qquad
V=\alpha{\rm e}^{-2\epsilon\psi}+\beta{\rm e}^{-4\epsilon\psi},\qquad
C_{1}C_{2}=0,$

where $a_{0}\neq 0$ is $C_{1}$ or $C_{2}$, $\epsilon=1$ if $C_{2}=0$ and
$\epsilon=-1$ if $C_{1}=0$, $\alpha=C_{4}\epsilon a_{0}^{-1}$,
$\beta=C_{3}a_{0}^{-2}$,

$\displaystyle g=A\cosh(2\psi+\psi_{0}),\qquad
V=\dfrac{\alpha+\beta\sinh(2\psi+\psi_{0})}{\cosh^{2}(2\psi+\psi_{0})},\qquad
C_{1}C_{2}>0,$ (4.5) $\displaystyle g=A\sinh(2\psi+\psi_{0}),\qquad
V=\dfrac{\alpha+\beta\cosh(2\psi+\psi_{0})}{\sinh^{2}(2\psi+\psi_{0})},\qquad
C_{1}C_{2}<0,$ (4.6)

where $C_{1}=A{\rm e}^{\psi_{0}}$, $C_{2}=\frac{|C_{1}C_{2}|}{C_{1}C_{2}}A{\rm
e}^{-\psi_{0}}$, $\psi_{0}=\tfrac{1}{2}\ln|C_{1}/C_{2}|$,
$\alpha=C_{3}/A^{2}$, $\beta=C_{4}/A$.

The form (4.6) appeared already in [8] and it is mapped to (4.5) by an
imaginary translation of ${\rm i}\pi/2$. Moreover, the transformation

$\displaystyle q^{1}=\frac{u}{\sqrt{2}}{\rm e}^{\frac{2}{k}\psi},\qquad
q^{2}=\frac{u}{\sqrt{2}}{\rm e}^{-\frac{2}{k}\psi},$

maps the extended Hamiltonians

$\displaystyle\frac{1}{2}p_{u}^{2}-k^{2}\frac{1}{4u^{2}}\left(\frac{1}{2}p_{\psi}^{2}+V\right)+16\Omega
u^{2},\qquad k\in\mathbb{Q},$

into

$\displaystyle
H=p_{1}p_{2}-\frac{\tilde{k}^{2}}{2q^{1}q^{2}}\big{(}C_{1}z^{\tilde{k}}+C_{2}z^{-\tilde{k}}\big{)}^{-2}\big{(}C_{3}+C_{4}\big{(}C_{1}z^{\tilde{k}}-C_{2}z^{-\tilde{k}}\big{)}\big{)}+32\Omega
q^{1}q^{2}.$

######  Remark 4.1.

The potentials $V$ described in (4.4) are always algebraic functions of
$q^{1}$, $q^{2}$, provided $\eta\in\mathbb{Q}$ (which in this case is absorbed
by the rational $k$).

The generalized extended Hamiltonian obtained from (1.2) is therefore

$\displaystyle\tilde{H}=\frac{1}{2}p_{u}^{2}-\frac{m^{2}}{\eta^{2}n^{2}u^{2}}\left(\frac{1}{2}p_{\psi}^{2}+V\right)+\eta^{4}\Omega
u^{2},$ (4.7)

where $V$ is (4.4) and $\frac{m}{n}=\eta(k+1)$, $c=-\eta^{2}$.

######  Remark 4.2.

The Hamiltonian (4.7) becomes (3.7) when $C_{2}=0$, $C_{1}=1$,
$C_{3}=\tilde{\alpha}$, $C_{4}=\tilde{\beta}$. When $C_{2}=1$, $C_{1}=0$, the
potentials correspond up to the exchange between $q^{1}$ and $q^{2}$. For
$C_{1}$ and $C_{2}$ both $\neq 0$, we obtain new superintegrable potentials.
Of the four parameters $C_{i}$ only two are essential, the remaining two being
reducible to a single multiplicative factor.

One could search for other superintegrable potentials $V$ by assuming
different expressions for $G=G(\psi,p_{\psi})$, but the resulting differential
equations are much less easy to solve than for $G$ constant or linear in the
momentum $p_{\psi}$.

(iii) From the Hamiltonian (4.7) we can obtain another family of
superintegrable Hamiltonians depending on a rational parameter $k$ by applying
to it the coupling-constant metamorphosis (CCM) as described in [9, 12, 14,
18].

The CCM transforms integrable or superintegrable systems in new integrable or
superintegrable ones, by mapping first integrals in first integrals. It is
characterized by the following theorem

Let us consider a Hamiltonian $H=\hat{H}-\tilde{E}U$ in canonical coordinates
$\big{(}x^{i},p_{i}\big{)}$, where $\hat{H}\big{(}x^{i},p_{i}\big{)}$ is
independent of the arbitrary parameter $\tilde{E}$ and $U\big{(}x^{i}\big{)}$,
with an integral of the motion $K$ $($depending on $\tilde{E})$. If we define
the CCM of $H$ and $K$ as $H^{\prime}=U^{-1}\big{(}\hat{H}-E\big{)}$ and
$K^{\prime}=K|_{\tilde{E}=\tilde{H}}$ then $K^{\prime}$ is an integral of the
motion for $H^{\prime}$.

In our case, the configuration manifold of the resulting system is again
$\mathbb{M}^{2}$. Indeed, from (4.7), by applying the CCM based on
$\tilde{E}=-\eta^{4}\Omega$, and recalling that $k+1=\frac{m}{n}$, we get

$\displaystyle
H^{\prime}=\frac{1}{2u^{2}}p_{u}^{2}-\frac{m^{2}}{\eta^{2}n^{2}u^{4}}\left(\frac{1}{2}p_{\psi}^{2}+V\right)-\frac{E}{u^{2}},$

where $V$ is given by (4.4). By performing the rescaling $u^{2}=2v$, we obtain

$\displaystyle
H^{\prime}=\frac{1}{2}p_{v}^{2}-\frac{m^{2}}{4\eta^{2}n^{2}v^{2}}\left(\frac{1}{2}p_{\psi}^{2}+V\right)-\frac{E}{2v},$
(4.8)

or, in coordinates

$\displaystyle q^{1}=\frac{v}{\sqrt{2}}{\rm e}^{\frac{\eta}{k}\psi},\qquad
q^{2}=\frac{v}{\sqrt{2}}{\rm e}^{-\frac{\eta}{k}\psi},$ $\displaystyle
H^{\prime}=p_{1}p_{2}-\frac{4\tilde{k}^{2}}{2\eta^{2}q^{1}q^{2}}\big{(}C_{1}z^{\tilde{k}}+C_{2}z^{-\tilde{k}}\big{)}^{-2}\big{(}C_{3}+C_{4}\big{(}C_{1}z^{\tilde{k}}-C_{2}z^{-\tilde{k}}\big{)}\big{)}-\frac{E}{2\sqrt{2q^{1}q^{2}}},$

with

$\displaystyle z=\frac{q^{1}}{q^{2}},\qquad\tilde{k}=\frac{m}{4n}.$

The configuration manifold of $H^{\prime}$, is again $\mathbb{~{}}M^{2}$. The
form of the characteristic first integral of (4.8) can be found in [9].

(iv) The choice $c=-\eta^{2}$, for $\eta=2$, and $C=\kappa=0$ in Section 3 has
been made in order to obtain the Hamiltonian (1.2) in $\mathbb{M}^{2}$ as an
extension of (3.8). However, from Section 1 we know that for different values
of $c$ and $C$ we can obtain extensions of the same $L$ of (3.8) which are
defined on manifolds different from $\mathbb{M}^{2}$, but which still belong
to the same family of extensions of $L$ and, therefore, which can be seen as
generalizations of the Hamiltonian (1.2) in these new manifolds. A first
possible generalization is the following: we consider the expressions (4.3),
(4.4) for $G$ and $V$ and we allow that $\eta$ takes imaginary values. Such
values of $\eta$ imply the appearance of trigonometric functions in $G$ and
$V$, which remain real provided $C_{1}=\bar{C}_{2}$ and $C_{4}$ are imaginary.
In this way we see that the extension

$\displaystyle
H_{0}=\frac{1}{2}p_{u}^{2}+\frac{m^{2}}{|\eta|^{2}n^{2}u^{2}}L+\eta^{4}u^{2}\Omega$
(4.9)

is now defined in the Euclidean plane, since $c=|\eta|^{2}>0$. It follows that
$H_{0}$ is nothing but the Tremblay–Turbiner–Winternitz Hamiltonian discussed
in [8].

Moreover, we can consider values of $\kappa$ different from 0, according to
the freedom of choice of the parameter $C$ in (2.2). Consequently, for $\eta$
real or imaginary (i.e., for $c<0$ or $c>0$), we have the possible extensions
of $L$

$\displaystyle
H_{1}=\frac{1}{2}p_{u}^{2}+\frac{(k+1)^{2}c}{\sin^{2}cu}L+\Omega\tan^{2}cu,\qquad\kappa=1,$
(4.10) $\displaystyle
H_{2}=\frac{1}{2}p_{u}^{2}+\frac{(k+1)^{2}c}{\sinh^{2}cu}L+\Omega\tanh^{2}cu,\qquad\kappa=-1,$
(4.11)

corresponding, for $c>0$, to Hamiltonians on the sphere $\mathbb{S}^{2}$ and
on the pseudosphere $\mathbb{H}^{2}$, respectively, and, for $c<0$, to
Hamiltonians defined on the de Sitter space ${\rm dS}^{2}$ and on the anti-de
Sitter space ${\rm AdS}^{2}$, respectively. We recall that, in this case, the
value of $c$ is related to the value of the curvature of the configuration
manifold of $H_{1}$, $H_{2}$.

In this way, we obtained some kind of generalizations of the Hamiltonian (1.2)
in manifolds of constant positive and negative curvature, on Riemannian and
pseudo-Riemannian manifolds, since they are all obtained as extensions of the
same one-dimensional Hamiltonian.

Finally, we consider the case $c=0$, which cannot be deduced from (4.3),
(4.4), since we assumed there that $c_{0}=0$ in (2.1). For the case $c=0$,
$c_{0}\neq 0$, we have from [8] either

$\displaystyle g=(a_{1}\psi+a_{2}),\qquad
V=\frac{c_{0}}{4a_{1}^{2}}g^{2}+c_{1}g+c_{2},$

or

$\displaystyle g=a_{2},\qquad V=c_{0}\psi^{2}+c_{1}\psi+c_{2}.$

Therefore, we have that the function $L$ (3.8) is not included in this case.

The Hamiltonians (4.10), (4.11), defined on conformally flat Riemannian
manifolds, can be again transformed into new superintegrable systems by
applying to them the CCM.

The results of above are summarized in the following statement

######  Theorem 4.3.

The Hamiltonians (4.7), (4.8), (4.9), (4.10), (4.11) are superintegrable with
a first integral polynomial in the momenta computed through the procedure
described in Section 2.

######  Remark 4.4.

The Hamiltonian

$\displaystyle\tilde{H}=p_{1}p_{2}-\alpha
q_{2}^{2k+1}q_{1}^{-2k-3}-\frac{\beta}{2}q^{k}_{2}q_{1}^{-k-2}+2\Omega
q^{1}q^{2},\qquad k\in\mathbb{R},$

as well as any Hamiltonian if the form

$\displaystyle
H=p_{1}p_{2}+\frac{1}{q^{1}q^{2}}F\left(\frac{q^{1}}{q^{2}}\right)+G\big{(}q^{1}q^{2}\big{)},$

remain separable in pseudo-polar coordinates for any $k\in\mathbb{R}$, and
therefore it is Liouville integrable.

######  Remark 4.5.

All the Hamiltonians of Theorem 4.3 correspond essentially to the
Tremblay–Turbiner–Winternitz system, its generalizations on constant-curvature
manifolds and its CCM transformed Hamiltonians. This becomes evident by
comparing the Hamiltonians of above with those given in [8, 13, 17] and
references therein.

##  5 Quantization

We consider the Hamiltonian (4.7) with $V$ given by (4.4), thus including the
Hamiltonian (1.2). We indicate how to quantize it together with its
characteristic first integral corresponding to any rational parameter, proving
the quantum superintegrability of this two-dimensional system with a potential
involving three arbitrary parameters. The quantization is made possible by
applying the method described in [10], where the Kuru–Negro quantization
introduced in [15], based on factorization in shift and ladder operators, is
adapted to extended Hamiltonians on flat manifolds. Indeed, in order to apply
the method for $H$ defined on a two-dimensional manifold, it is necessary to
find functions of the form

$\displaystyle F^{\pm}=\pm
F^{\prime}(\psi)p_{\psi}+F(\psi)f+\frac{1}{f}c_{1},\qquad
f=\sqrt{2\big{(}\eta^{2}L+c_{0}\big{)}},$

solutions of

$\displaystyle X_{L}^{2}F^{\pm}=fF^{\pm}.$ (5.1)

The functions $F^{\pm}$ are known as ladder functions of $L$ and they are
determined by solutions $F(\psi)$ of the system

$\displaystyle F^{\prime\prime}-\eta^{2}F=0,$ $\displaystyle
V^{\prime}F^{\prime}+2\eta^{2}VF+c_{1}=0,$

equivalent to (5.1) and corresponding to (20) and (21) of [10], for a suitable
constant $c_{1}$. We have in our case

$\displaystyle F=g^{\prime},\qquad c_{1}=-C_{4}\eta^{3},$

where $g$ is given in (4.3). The construction of the quantum Hamiltonian and
of the characteristic symmetry operator is then explicitely described in [10,
Theorem 10]. To the function $L$ we associate the quantum operator

$\displaystyle\hat{L}=-\frac{\hbar^{2}}{2}\partial^{2}_{\psi\psi}+V,$

which can be extended to the quantum Hamiltonian

$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2}\left(\partial^{2}_{uu}+\frac{1}{u}\partial_{u}-\frac{k^{2}}{\eta^{2}u^{2}}\partial^{2}_{\psi\psi}\right)-\frac{k^{2}}{\eta^{2}u^{2}}V+\frac{\Omega\eta^{4}}{2}u^{2},\qquad
k=\frac{m}{n},\qquad m,n\in\mathbb{N}\setminus\\{0\\},$

which commutes with $\hat{L}$. The operator $\tilde{H}$ coincides with the
Laplace–Beltrami operator of the metric

$\displaystyle{\rm d}s^{2}={\rm d}u^{2}-\frac{\eta^{2}u^{2}}{k^{2}}{\rm
d}\psi^{2},$

which, for $\eta^{2}=1$, is the metric of the Minkowski plane in pseudo-polar
coordinates $(u,k\psi)$, plus a scalar term. Then, the operator

$\displaystyle\hat{X}^{k}_{\epsilon,E}=\big{(}\hat{G}^{+}_{\epsilon}\big{)}^{2n}\circ\big{(}\hat{A}^{1,1}_{k\epsilon}\big{)}^{2m}\circ\big{(}\hat{D}^{+}_{E}\big{)}^{m},$

is such that

$\displaystyle\hat{H}\big{(}\hat{X}^{k}_{\epsilon,E}f_{E}\big{)}=E\hat{X}^{k}_{\epsilon,E}(f_{E}),$

for all multiplicatively separated eigenfunctions
$f_{E}=\phi_{E}^{M}(u)\chi_{\lambda}(\psi)$ of $\hat{H}$ such that
$\chi_{\lambda}$ is an eigenfunction of $\hat{L}$ with eigenvalue $\lambda$,
and

$\displaystyle
M=k^{2}\lambda,\qquad\epsilon=\sqrt{|\lambda|},\qquad\eta^{2}\lambda\leq 0,$

where $\phi^{M}_{E}(u)$ is (not restrictively) any eigenfunction of the
“radial” operator

$\displaystyle\hat{H}^{M}=-\frac{\hbar^{2}}{2}\left(\partial^{2}_{uu}+\frac{1}{u}\partial_{u}\right)-\frac{M}{\eta^{2}u^{2}},$

corresponding to the eigenvalue $E$, and where

$\displaystyle\hat{A}^{1,1}_{k\epsilon}=\partial_{u}-\frac{1}{\hbar}\left(\eta^{2}\sqrt{\Omega}u+\frac{k\epsilon\sqrt{2}}{2\eta
u}\right),$
$\displaystyle\hat{D}^{\pm}_{E}=\frac{\hbar}{\sqrt{2}}u\partial_{u}+\frac{\hbar}{\sqrt{2}}\pm\left(-\frac{E}{2\eta^{2}\sqrt{\Omega}}+\sqrt{\frac{\Omega}{2}}\eta^{2}u^{2}\right),$
$\displaystyle\hat{G}^{\pm}_{\epsilon}=\eta^{2}g\partial_{\psi}\pm\epsilon\eta\frac{\sqrt{2}}{\hbar}g^{\prime}+\frac{2\eta^{2}C_{4}}{\hbar\big{(}\hbar\eta\pm
2\sqrt{2}\epsilon\big{)}}.$

The operator $\hat{X}^{k}_{\epsilon,E}$ is therefore a symmetry of $\hat{H}$,
called “warped symmetry” in [10], corresponding to the characteristic first
integral of the classical extended Hamiltonian $H$.

##  6 Conclusions

In this article we show that the theory of extended Hamiltonians allows not
only to prove the superintegrability of the Hamiltonian (1.2) for any non-zero
rational $k$, but also gives immediate generalizations of $H$, both on flat
and curved manifolds, which are still superintegrable Hamiltonians and include
the superposition of a harmonic oscillator term.

Moreover, the application of the coupling-constant-metamorphosis produces
further superintegrable Hamiltonians, when adapted to the structure of
extended Hamiltonians.

The possible generalizations are not exhausted by the examples given in the
present paper. For example, the methods employed in [6] allow to extend
$n$-dimensional superintegrable Hamiltonians into $(n+1)$-dimensional ones,
again superintegrable. The application of those methods to the present case
has not been considered yet.

The procedure of Laplace–Beltrami quantization of extended Hamiltonians and
their associated first integrals of high-degree developed in [10] on flat
manifolds is applied to the present case, so the quantum superintegrability of
(1.2) and its generalisations on flat manifolds is proved. The problem of the
quantum superintegrability of (1.2) was left open in [2]. The quantization of
extended Hamiltonians on curved manifolds, such as (4.10) and (4.11), is still
an open problem.

The search for extended Hamiltonian structures is limited for the moment to
point-coordinate transformations. A possible future direction of research is
the search for extended Hamiltonian structures under more general canonical
transformations, for example concerning the systems described in Remark 3.3.

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#
Probabilistic Dynamic Logic of Phenomena and Cognition ††thanks: Evgenii
Vityaev is with the Department of Mathematical Logic, Sobolev Institute of
Mathematics of the Russian Academy of Sciences and with the Department of
Discrete mathematics and Informatics of the Novosibirsk State University,
630090, Novosibirsk, Russia, email<EMAIL_ADDRESS>††thanks: Boris
Kovalerchuk is with the Department of Computer Science, Central Washington
University, Ellensburg, WA 98926-7520, e-mail<EMAIL_ADDRESS>††thanks: Leonid
Perlovsky is with the Harvard University and the Air Force Research
Laboratory, Sensors Directorate, Hanscom AFB<EMAIL_ADDRESS>††thanks: Stanislav Smerdov, Novosibirsk State University, Sobolev Institute
of Mathematics of the Russian Academy of Sciences, 630090, Novosibirsk,
Russia, email<EMAIL_ADDRESS>
Evgenii Vityaev, Boris Kovalerchuk, Leonid Perlovsky, Stanislav Smerdov

###### Abstract

The purpose of this paper is to develop further the main concepts of Phenomena
Dynamic Logic (P-DL) and Cognitive Dynamic Logic (C-DL), presented in the
previous paper. The specific character of these logics is in matching
vagueness or fuzziness of similarity measures to the uncertainty of models.
These logics are based on the following fundamental notions: _generality
relation, uncertainty relation, simplicity relation, similarity maximization
problem with empirical content and enhancement (learning) operator_. We
develop these notions in terms of logic and probability and developed a
Probabilistic Dynamic Logic of Phenomena and Cognition (P-DL-PC) that relates
to the scope of probabilistic models of brain. In our research the
effectiveness of suggested formalization is demonstrated by approximation of
the expert model of breast cancer diagnostic decisions. The P-DL-PC logic was
previously successfully applied to solving many practical tasks and also for
modelling of some cognitive processes.

##  I Introduction

In the paper [1] there was introduced a Phenomena Dynamic Logic (P-DL) and
Cognitive Dynamic Logic (C-DL) as a generalization of the Dynamic Logic and
Neural Modelling Fields theory (NMF) introduced in the previous papers [2, 3].
Logics P-DL, C-DL provide the most general description of Dynamic Logic in the
following fundamental notions _generality relation, uncertainty relation,
simplicity relation, similarity maximization problem with empirical content
and enhancement (learning) operator_. This generalization provide
interpretation of P-DL, C-DL logics in the frame of other approaches.

In this paper we interpret logics P-DL, C-DL in terms of logic and
probability: uncertainty we interpret as probability, while the process of
learning as a semantic probabilistic inference [4, 9, 6, 5]. We also interpret
mentioned fundamental notions. The resulting Probabilistic Dynamic Logic of
Phenomena and Cognition (P-DL-PC) belong to the scope of the probabilistic
models of brain [19, 20]. Thus, through logics P-DL, C-DL we extend the
interpretation of Dynamic Logic and Neural Modelling Fields theory to
probabilistic models of brain. The P-DL-PC logic as probabilistic model of
brain was previously applied to modelling of some cognitive process [7, 8, 9,
21]. The effectiveness of P-DL-PC logic demonstrated in this paper by
approximation of the expert model of breast cancer diagnostic decisions.

##  II Universal productions. Data for prediction

In our study learning models will be generated as sets of _universal
productions_ (_u-productions_), which are introduced in this section. Note
that every set of _universal formulas_ is logically equivalent to a certain
set of u-productions.

Consider a fixed first-order language $\mathfrak{L}$ in a countable signature.
Hereafter denote $\mathbf{A}_{\mathfrak{L}}$ the set of all atoms;
$\mathbf{L}_{\mathfrak{L}}$ – the set of all literals;
$\mathbf{S}_{\mathfrak{L}}^{0}$ – the set of ground sentences. The set of
ground atoms and the set of ground literals are denoted
$\mathbf{A}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{A}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$
and
$\mathbf{L}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{L}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$
correspondingly. Following examples of atoms and literals are given in the
section VIII for the task of approximation of the expert model of breast
cancer diagnostic decisions: ‘number of calcifications per $cm^{3}$ less than
20‘, ‘volume of calcifications in $cm^{3}$ not less or equal to 5‘, ‘total
number of calcifications more than 30 and etc.

Let $\Theta$ be the set of all substitutions and $\Theta^{0}\subseteq\Theta$
the set of ground substitutions, that are mappings variables to ground terms.
All necessary notions from model theory and logic programming are elementary
and can be easily found in books [12], [13, 14].

######  Definition.

A record of the type

$\mathrm{R}\leftrightharpoons\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right),$

where
$\mathrm{A}_{1},\cdots\mathrm{A}_{m},\mathrm{B}_{1},\cdots,\mathrm{B}_{n}$ are
literals, and $\tilde{\forall}$ stands for a bloc of quantifiers over all free
variables of the formulae in brackets (universal closure), is called a _u-
production_. _A variant of u-production $\mathrm{R}$_ is

$\mathrm{R}\theta\rightleftharpoons\tilde{\forall}\left(\mathrm{A}_{1}\theta\wedge\cdots\mathrm{A}_{m}\theta\Leftarrow\mathrm{B}_{1}\theta\wedge\cdots\wedge\mathrm{B}_{n}\theta\right),$

where $\theta$ is an arbitrary one-to-one correspondence over the set of
variables. Let ${\tt Prod}$ be the set of all u-productions.

For example in section X presented the following u-production that was
discovered by the learning model:

> IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5
> $cm^{3}$, and DENSITY of calcifications is moderate,
> THEN Malignant.

Let $\mathtt{Fact}_{v}\subset\mathbf{A}_{\mathfrak{L}}$ be a set of atoms from
A that are valid for verification in algebraic system ${\mathfrak{B}}$
appearing in practice. Our aim is to investigate as much “extra” facts about
$\mathfrak{L}$ as possible, i.e., to predict or explain them. A natural
assumption is that we can verify (falsify) each element of

$\mathtt{Fact}_{\rm o}\leftrightharpoons\left\\{{\rm
A}\theta\mid\theta\in\Theta^{0},~{}{\rm A}\in\mathtt{Fact}_{v}\right\\}.$

Certainly we may postulate our ability to check any literal of
$\mathtt{Fact}_{v}^{\ast}\leftrightharpoons\mathtt{Fact}_{v}\cup\left\\{\neg{\rm
A}\mid{\rm A}\in\mathtt{Fact}_{v}\right\\}$. For the rest of the literals (and
their conjunctions) the machinery of _probabilistic prediction_ will be
defined later on. Note that ${\tt Fact}_{\rm o}^{\ast}={\tt Fact}_{\rm
o}\cup{\tt Fact}_{\rm o}^{\neg}$, where ${\tt Fact}_{\rm
o}^{\neg}\rightleftharpoons\left\\{\neg\mathrm{A}\mid\mathrm{A}\in{\tt
Fact}_{\rm o}\right\\}$ is _the complete set of alternatives_ allowing a real
test.

The _data_ are defined as a maximal (logically) consistent subset of the
complete set of alternatives, i.e., being given a mapping
$\zeta_{\mathfrak{B}}:{\tt Fact}_{\rm o}\mapsto\left\\{{\bot,\top}\right\\}$
(here _$\bot$ – “false”, $\top$ – “true”_) we conclude that

${\tt Data}\left[\mathfrak{B}\right]\rightleftharpoons\left\\{{{\rm A}\mid{\rm
A}\in{\tt Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm
A}\right)=\top}\right\\}\cup$ $\left\\{{\neg{\rm A}\mid{\rm A}\in{\tt
Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm
A}\right)=\bot}\right\\}.$

##  III Generality relation between theories

The idea of a _generality relation_ between theories can be viewed, for
example, as a reduction of the set of properties predicted by the use of these
theories. A more general theory (potentially) predicts a greater number of
formal features. We start with a generality relation between one-element
specifications, i.e., between u-productions.

######  Definition.

For two productions
$\mathrm{R}_{1}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m_{1}}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n_{1}}\right)$
and
$\mathrm{R}_{2}\equiv\tilde{\forall}\left(\mathrm{C}_{1}\wedge\cdots\wedge\mathrm{C}_{m_{2}}\Leftarrow\mathrm{D}_{1}\wedge\cdots\wedge\mathrm{D}_{n_{2}}\right)$
a relation $\mathrm{R}_{1}\succ\mathrm{R}_{2}$ (“more general than”) takes
place if and only if there exists $\theta\in\Theta$ such that
$\left\\{{\mathrm{B}_{1}\theta,\cdots,\mathrm{B}_{n_{1}}\theta}\right\\}\subseteq\left\\{{\mathrm{D}_{1},\cdots,\mathrm{D}_{n_{2}}}\right\\}$,
$\left\\{{\mathrm{A}_{1}\theta,\cdots,\mathrm{A}_{m_{1}}\theta}\right\\}\supseteq\left\\{{\mathrm{C}_{1},\cdots,\mathrm{C}_{m_{2}}}\right\\}$,
and $n_{1}\leqslant n_{2}$, $m_{1}\geqslant m_{2}$, $\not\vdash{\rm
R}_{1}\equiv{\rm R}_{2}$.

The inclusion of the sets of premises designates that the more general
u-production is, then the wider its field of application. The inverse
inclusion (for conclusions) says that $\mathrm{R}_{1}$ predicts a greater
number of properties using a smaller premise.

Let $S\subseteq{\tt Prod}$. Denote ${\tt Fact}\left[S;\mathfrak{B}\right]$ the
set of all $\mathrm{A}\in\mathbf{L}_{\mathfrak{L}}^{0}$ such that for some
${\rm R}\in S$ and $\theta\in\Theta^{0}$,

$\mathrm{R}\theta\equiv\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right)$,
holds

$\left\\{\mathrm{B}_{1},\cdots,\mathrm{B}_{n}\right\\}\subseteq{\tt
Data}\left[\mathfrak{B}\right]$ and
$\mathrm{A}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$.

Thus, ${\tt Fact}\left[S;\mathfrak{B}\right]$ is the set of ground literals
predicted according to available data (about the model $\mathfrak{B}$)
together with u-productions in $S$.

In the sequel let $\succcurlyeq$ be a reflexive closure of $\succ$. One should
pay attention to the fact: $\mathrm{R}_{1}\succcurlyeq\mathrm{R}_{2}$ entails
that ${\tt Fact}\left[\left\\{\mathrm{R}_{1}\right\\};\mathfrak{B}\right]$
contains ${\tt
Fact}\left[\left\\{\mathrm{R}_{2}\right\\};\mathfrak{B}\right]$.

Thereafter it isn’t difficult to extend the domain of our generality relation
to subsets of ${\tt Prod}$.

######  Definition.

Let $S,S^{\prime}\subseteq{\tt Prod}$, and for any ${\rm R}^{\prime}\in
S^{\prime}$ we find ${\rm R}\in S$ such that
$\mathrm{R}\succcurlyeq\mathrm{R}^{\prime}$. In this case we say ‘$S$ is not
less general than $S^{\prime}$’ ($S\vartriangleright S^{\prime}$).

It’s straightforward to notice that ${\tt
Fact}\left[S^{\prime};\mathfrak{B}\right]\subseteq{\tt
Fact}\left[S;\mathfrak{B}\right]$ for $S$ and $S^{\prime}$ from the definition
above. Remark that $S$ may include u-productions apart from those, which are
generalizations of elements of $S^{\prime}$.

##  IV Probability/degree of belief

The topic of distributing probability over formulas of propositional logic (as
well as over ground statements in a first order language) being widely
discussed in a literature and meets Kolmogorov’s understanding of probability
measure [11]. The following definition is given on the basis of analysis cited
in [10].

######  Definition.

A probability over $F\subseteq{\mathbf{S}}_{\mathfrak{L}}^{0}$ closed with
respect to $\wedge$, $\vee$ and $\neg$, is a function
$\mu:F\mapsto\left[{0,1}\right]$ satisfying the following conditions:

  1. 1.

if $\vdash\Phi$ (“$\Phi$ is a tautology”), then $\mu\left(\Phi\right)=1$;

  2. 2.

if $\vdash\neg\left({\Phi\wedge\Psi}\right)$, then
$\mu\left({\Phi\vee\Psi}\right)=\mu\left(\Phi\right)+\mu\left(\Psi\right)$.

For any ground instance of a u-production its probability is defined as
conditional, i.e.,

$\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\Leftarrow{\rm
B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=$

$=\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\mid{\rm
B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=\frac{\mu\left({{\rm
A}_{1}\wedge\cdots{\rm A}_{m}\wedge{\rm B}_{1}\wedge\cdots\wedge{\rm
B}_{n}}\right)}{\mu\left({{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}}\right)}$

Let ${\rm R}\in{\tt Prod}$. Denote as ${\tt Sub}\left[{\rm R}\right]^{\mu}$
those substitutions $\theta\in\Theta^{0}$, for which the premise of
u-production ${\rm R}\theta$ has a non-zero probability.

${\tt Prod}^{\mu}\rightleftharpoons\left\\{{\rm R}\in{\tt Prod}\mid{\tt
Sub}\left[{\rm R}\right]^{\mu}\neq\varnothing\right\\}$;

$\underline{\mu}\left({\rm R}\right)\rightleftharpoons{\rm
inf}\left\\{\mu\left({{\rm R}\theta}\right)\mid\theta\in{\tt Sub}\left[{\rm
R}\right]^{\mu}\right\\}$, where ${\rm R}\in{\tt Prod}^{\mu}$.

A value of conditional probability serves to characterize our _degree of
belief_ (and responsible for an _uncertainty relation_) in reliability of
different causal connections included in temporary specification. Note that
two productions are not necessary comparable with respect to generality
relation $\succcurlyeq$; moreover, their premisses may not be contained in the
complete set of alternatives (and so these productions will be not valid for a
direct check in a real structure $\mathfrak{B}$).

##  V Simplicity of probabilistic theories

Adding comparison of lower probabilistic estimations to the definition of
generality relation we obtain the following definition.

######  Definition.

Let $S,S^{\prime}\subseteq{\tt Prod}^{\mu}$. We say that $S$ is _more
$\mu$-general than_ $S^{\prime}$ iff for every $\mathrm{C}^{\prime}\in
S^{\prime}$ there exists $\mathrm{C}\in S$ such that
$\mathrm{C}\succcurlyeq\mathrm{C}^{\prime}$ and
$\underline{\mu}(\mathrm{C})\geqslant\underline{\mu}(\mathrm{C}^{\prime})$,
and in at least one of the cases the strong relation $\succ$ takes place.

Hence, $\mu$-generalization allows us to define a more general set $S$ in such
a way that the lower estimations of probabilities is not declined. When our
belief to the elements of $S$ is no less than that of $S^{\prime}$, then we
have a _simplicity relation_ – the set $S$ is simpler than $S^{\prime}$ in
order to describe/predict the properties.

##  VI Similarity measure with the empirical content

By elaboration of u-productions we mean the gain of its conditional
probability.

######  Definition.

A relation ${\rm R}_{1}\sqsubset{\rm R}_{2}$ (‘probabilistic inference’) for
${\rm R}_{1},{\rm R}_{2}\in{\tt Prod}^{\mu}$ means that ${\rm R}_{1}\succ{\rm
R}_{2}$ and $\underline{\mu}\left({{\rm
R}_{1}}\right)<\underline{\mu}\left({{\rm R}_{2}}\right)$.

######  Definition.

Let $\pi$ be some requirements to be applied to elements of ${\tt
Prod}^{\mu}$, i.e. $\pi:{\tt Prod}^{\mu}\mapsto\left\\{\bot,\top\right\\}$
(value is equal to $\bot$, if u-production satisfies $\pi$, and $\top$ –
otherwise); $\mathsf{\Pi}\leftrightharpoons\left\\{{\rm R}\in{\tt
Prod}^{\mu}\mid\pi\left({\rm R}\right)=\top\right\\}$. We say that
$\mathrm{R}_{2}\in\mathsf{\Pi}$ is a _minimal follower of
$\mathrm{R}_{1}\in{\tt Prod}^{\mu}$ relative to $\sqsubset$ in $\mathsf{\Pi}$_
(denoted as $\mathrm{R}_{1}\sqsubset_{\pi}\mathrm{R}_{2}$), iff
$\mathrm{R}_{1}\sqsubset\mathrm{R}_{2}$ and there is no intermediate
u-production $\mathrm{R}_{3/2}\in\Pi$ such that
$\mathrm{R}_{1}\sqsubset\mathrm{R}_{3/2}\sqsubset\mathrm{R}_{2}$.

In the prediction of a literal ${\rm H}$ the _similarity measure_ for
u-productions, which are valid for verification and applicable to the goal
${\rm H}$, is equal to conditional probability
$\underline{\mu}\left(\cdot\right)$. Thus we deal with a uniform measure of
similarity.

##  VII Learning operator

######  Definition.

A production

$\mathrm{R}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\leftarrow\mathrm{B}_{1}\wedge\dots\wedge\mathrm{B}_{n}\right)$

is called a _maximal specific u-production (ums-production) for prediction of
a conjunction
$\mathrm{H}\equiv\left(\mathrm{H}_{1}\wedge\cdots\wedge\mathrm{H}_{k}\right)$,_
where
$\left\\{\mathrm{H}_{1},\cdots,\mathrm{H}_{k}\right\\}\subset\mathbf{L}_{\mathfrak{L}}$
and $m\leqslant k$, iff the following conditions are satisfied:

  1. 1.

there is a substitution $\theta$ (not necessary ground) such that
$\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}\subseteq\left\\{\mathrm{H}_{1}\theta,\cdots,\mathrm{H}_{k}\theta\right\\}$,
$\left\\{{\mathrm{B}_{1},\dots,\mathrm{B}_{n}}\right\\}\subseteq\left\\{\mathrm{B}\theta\mid\mathrm{B}\in\mathtt{Fact}_{v}^{\ast}\right\\}$;

  2. 2.

if $\mathrm{D}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$ and
$\theta_{\rm o}\in{\tt Sub}\left[{\rm R}\right]^{\mu}$, then
$\mu\left({\mathrm{A}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm o}\right)<\\\
\mu\left({\mathrm{A}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm
o}\mid{\mathrm{B}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$
and $\mu\left(\mathrm{D}\theta_{\rm o}\right)<\mu\left(\mathrm{D}\theta_{\rm
o}\mid{\mathrm{B}_{1}\theta_{\rm
o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$;

  3. 3.

there is no ${\rm R^{\prime}}\in\mathtt{Prod}^{\mu}$, for which points (1–2)
are hold along with ${\rm R}\sqsubset{\rm R^{\prime}}$;

  4. 4.

the u-production ${\rm R}$ can’t be generalized up to some ${\rm
R^{\prime}}\in\mathtt{Prod}^{\mu}$ satisfying all the previous points (1–3)
without decreasing its estimation $\underline{\mu}\left(\cdot\right)$.

The conditions above (for corresponding ums-productions) are denoted as
‘point.i’, $1\leqslant i\leqslant 4$.

######  Remark.

Though condition point.4 emphasizes the nature of definition, but it isn’t
necessary for indication. Indeed, if $\mathrm{R}$ may be generalized up to
$\mathrm{R}^{\prime}$ under preserving point.1–3, then
$\underline{\mu}\left(\mathrm{R}\right)\leqslant\underline{\mu}\left(\mathrm{R}^{\prime}\right)$
(otherwise we get ${\rm R^{\prime}}\sqsubset{\rm R}$ – that contradicts
point.3 for $\mathrm{R}$.

Let $\pi\left({\rm R}\right)=\top$ be fulfilled for ${\rm
R}\in\mathtt{Prod}^{\mu}$ iff conditions points.1–2 are satisfied for ${\rm
R}$ and ${\rm H}$ (the last one is fixed from this moment); denote
$\mathsf{\Pi}\leftrightharpoons\pi^{-1}\left(\top\right)$.

Define the _probabilistic fix-point operator_
$\mathrm{T}_{\pi}:2^{\mathtt{Prod}^{\mu}}\mapsto 2^{\mathtt{Prod}^{\mu}}$ as
follows: for a set $S\subseteq\mathtt{Prod}^{\mu}$ it produces

$S^{\prime}\leftrightharpoons\left\\{\mathrm{R}^{\prime}\mid\mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\
\mbox{\emph{for some}}\ \mathrm{R}\in S\right\\}\cup$

$\cup\left\\{\mathrm{R}\mid\mathrm{R}\in S\cap\mathsf{\Pi}\ \mbox{\emph{and
there is no}}\ \mathrm{R}^{\prime}\ \mbox{\emph{such that}}\
\mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\right\\}$.

Therefore the operator $\mathrm{T}_{\pi}:S\mapsto S^{\prime}$ possess
important properties:

  1. 1.

the set $S^{\prime}$ is always more precise than $S$ (relative to
$\succcurlyeq$);

  2. 2.

the conditional probabilities $\underline{\mu}\left(\cdot\right)$ increase
during the conversion to more particular cases (and so fuzziness decreases);

  3. 3.

the similarity measure with the empirical content becomes greater for at least
one u-production (in $S$) when the operator converts $S$ to $S^{\prime}$ (if
not $S=S^{\prime}$, of course);

As a result the operator $\mathrm{T}_{\pi}$ is the _enhancement, or learning,
operator_ in the sense of [1].

######  Definition.

A fix-point (f.p., for short) $S$ of $\mathrm{T}_{\pi}$ is _optimal_ iff there
is no other f.p. $S^{\prime}$ of considered operator, which is more
$\mu$-general than $S$.

######  Statement.

A subset $S\subseteq\mathtt{Prod}^{\mu}$ is a fix-point of the operator
$\mathrm{T}_{\pi}$ iff every element of $S$ satisfies points.1–3 for
$\mathrm{H}$.

######  Corrolary.

A subset $S\subseteq\mathtt{Prod}^{\mu}$ is an optimal fix-point of the
operator $\mathrm{T}_{\pi}$ iff every element of $S$ is a ums-production for
prediction of $\mathrm{H}$.

Ums-productions may be viewed as a result of performing generalized scheme of
the _semantic probabilistic inference_ [4, 5], which is realized by the fix-
point operator described above. The program system ‘Discovery’ (see [16, 17,
9, 21]) was developed: it carries out the propositional version of the
probabilistic fix-point (learning) operator and was successfully applied to
solving many practical tasks [21].

##  VIII Extraction of the expert model of breast cancer diagnostic decisions

We applied our method to approximation of the expert model of breast cancer
diagnostic decisions that was obtained from the radiologist J.Ruiz [17]. At
first we extract this model from the expert by the special procedure using
monotone boolean functions [17] and then apply the program system ‘Discovery’
[16] to approximate this model.

###  VIII-A Hierarchical Approach

At first we ask an expert to describe particular cases using the binary
features. Then we ask a radiologist to evaluate a particular cases, when
features take on specific values. A typical query will have the following
format: ”If feature 1 has value $v_{1}$, feature 2 has value $v_{2}$, …,
feature n has value $v_{n}$, then is a case suspicious of cancer or not?”

Each set of values ($v_{1},v_{2},...,v_{n}$) represent a possible clinical
case. It is practically impossible to ask a radiologist to generate diagnosis
for thousands of possible cases. A hierarchical approach combined with the use
of the property of monotonicity makes the problem manageable. We construct a
hierarchy of medically interpretable features from a very generalized level to
a less generalized level. This hierarchy follows from the definition of the 11
medically oriented binary attributes. The medical expert indicate that the
original 11 binary attributes
$w_{1},w_{2},w_{3},y_{1},y_{2},y_{3},y_{4},y_{5},x_{3},x_{4},x_{5}$ could be
organized in terms of a hierarchy with development of two new generalized
attributes $x_{1}$, depending on attributes $w_{1},w_{2},w_{3}$, and $x_{2}$,
depending on attributes $y_{1},y_{2},y_{3},y_{4},y_{5}$.

A new generalized feature, $x_{1}$ – ‘Amount and volume of calcifications’
with grades (0 - ‘benign’ and 1 - ‘cancer’) was introduced based on features:
$w_{1}$ – number of calcifications/cm3, $w_{2}$ – volume of calcification, cm3
and $w_{3}$ – total number of calcifications. We view $x_{1}$ as a function
$g(w_{1},w_{2},w_{3})$ to be identified. Similarly a new feature $x_{2}$ –
‘Shape and density of calcification’ with grades: (1) for ‘cancer’ and
(0)-‘benign’ generalizes features: $y_{1}$ – ‘irregularity in shape of
individual calcifications’ $y_{2}$ – ‘variation in shape of calcifications’
$y_{3}$ – ‘variation in size of calcifications’ $y_{4}$ – ‘variation in
density of calcifications’ $y_{5}$ – ‘density of calcifications’. We view
$x_{2}$ as a function $x_{2}=h(y_{1},y_{2},y_{3},y_{4},y_{5})$ to be
identified for cancer diagnosis.

As result we have a decomposition of our task as follows:

$f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)=$
$f\left(g\left(w_{1},w_{2},w_{3}\right),h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right),x_{3},x_{4},x_{5}\right).$

###  VIII-B Monotonicity

Giving the above definitions we can represent clinical cases in terms of
binary vectors with five generalized features as:
$(x_{1},x_{2},x_{3},x_{4},x_{5})$. Let us consider two clinical cases that are
represented by the two binary sequences: (10110) and (10100). If radiologist
correctly diagnose (10100) as cancer, then, by utilizing the property of
monotonicity, we can also conclude that the clinical case (10110) should also
be cancer. Medical expert agreed with presupposition about monotonicity of the
functions $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and
$h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$.

Let us describe the interview with an expert using minimal sequence of
questions to completely infer a diagnostic function using monotonicity. This
sequence is based on fundamental Hansel lemma [15]. We omit a detailed
description of the specific mathematical steps. They can be found in [18].
Table 1 illustrates this.

###  VIII-C Expert model extraction

Columns 2 and 3 present values of above defined functions $f$ and $h$. We omit
a restoration of function $g\left(w_{1},w_{2},w_{3}\right)$ because few
questions are needed to restore this function. All 32 possible cases with five
binary features $\langle x_{1},x_{2},x_{3},x_{4},x_{5}\rangle$ are presented
in column 1 in table 1. They are grouped and the groups are called Hansel
chains [17]. The sequence of chains begins with the shortest chain 1 –
$(01100)<(11100)$ for five binary features. Then largest chain 10 consists of
6 ordered cases: $(00000)<(00001)<(00011)<(00111)<(01111)<(11111)$. The chains
are numbered there from 1 to 10 and each case has its number in the chain,
e.g., 1.2 means the second case in the first chain. Asterisks in columns 2 and
3 mark answers obtained from an expert, e.g., 1* for case (01100) in column 3
means that the expert answered ‘yes’. The answers for some other chains in
column 3 are automatically obtained using monotonicity. The value f(01100) = 1
for case 1.1 is extended for cases 1.2, 6.3. and 7.3 in this way. Similarly
values of the monotone Boolean functions h are computed using the table 1. The
attributes in the sequence (10010) are interpreted as
$y_{1},y_{2},y_{3},y_{4},y_{5}$ for the function h instead of
$x_{1},x_{2},x_{3},x_{4},x_{5}$. The Hansel chains are the same if the number
of attributes is the same five in this case.

Column 5 and 6 list cases for extending functions’ values without asking an
expert. Column 5 is for extending functions’ values from 1 to 1 and column 6
is for extending them from 0 to 0. If an expert gave an answer opposite
(f(01100) = 0) to that presented in table 1 for function $f$ in the case 1.1,
then this 0 value could be extended in column 2 for cases 7.1 (00100) and 8.1
(01000). These cases are listed in column 5 for case (01100). There is no need
to ask an expert about cases 7.1 (00100) and 8.1 (01000). Monotonicity
provides the answer. The negative answer f(01100) = 0 can not be extended for
f(11100). An expert should be queried regarding f(11100). If his/her answer is
negative f(11100) = 0 then this value can be extended for cases 5.1. and 3.1
listed in column 5 for case 1.2. Relying on monotonicity, the value of f for
them will also be 0.

The total number of cases with asterisk (*) in columns 2 and 3 are equal to 13
and 12. These numbers show that 13 questions are needed to restore the
function $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and 12 questions are
needed to restore the function $h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$.
This is only 37.5% of 32 possible questions. The full number of questions for
the expert without monotonicity and hierarchy is $2^{11}=2048$.

##  IX Approximation of the expert model by learning operator

For the Approximation of the expert model we used the program system
‘Discovery’ [16], that realizes the propositional case of the probabilistic
fix-point learning operator. We discovered several dozens diagnostic rules
that were statistically significant on the 0.01, 0.05 and 0.1 levels of
(F-criterion). Rules were extracted using 156 cases (73 malignant, 77 benign,
2 highly suspicious and 4 with mixed diagnosis). In the Round-Robin test our
rules diagnosed 134 cases and refused to diagnose 22 cases. The total accuracy
of diagnosis is 86%. Incorrect diagnoses were obtained in 19 cases (14% of
diagnosed cases). The false-negative rate was 5.2% (7 malignant cases were
diagnosed as benign) and the false-positive rate was 8.9% (12 benign cases
were diagnosed as malignant). Some of the rules are shown in table 2. This
table presents examples of discovered rules with their statistical
significance. In this table:

  * •

‘NUM’ – number of calcifications per $cm^{3}$;

  * •

‘VOL’ – volume in $cm^{3}$;

  * •

‘TOT’ – total number of calcifications;

  * •

‘DEN’ – density of calcifications;

  * •

‘VAR’ – variation in shape of calcifications;

  * •

‘SIZE’ – variation in size of calcifications;

  * •

‘IRR’ – irregularity in shape of calcifications;

  * •

‘SHAPE’ – shape of calcifications.

We studied three levels of similarity measure: 0.7, 0.85 and 0.95. A higher
level of conditional probability decreases the number of rules and diagnosed
patients, but increases accuracy of diagnosis.

Table 1. Dynamic sequence of questions to expert
---
1 | 2 | 3 | 4 | 5 | 6 | 7
Number | $f$ | $h$ | Monotonic extrapolation | Chain | Case
| Diagnose | Form and V | $1\mapsto 1$ | $0\mapsto 0$ |  |
$(01100)$ | 1* | 1* | 1.2, 6.3, 7.3 | 7.1, 8.1 | Chain 1 | 1.1
$(11100)$ | 1 | 1 | 6.4, 7.4 | 5.1, 3.1 |  | 1.2
$(01010)$ | 0* | 1* | 2.2, 6.3, 8.3 | 6.1, 8.1 | Chain 2 | 2.1
$(11010)$ | 1* | 1 | 6.4, 8.4 | 3.1, 6.1 |  | 2.2
$(11000)$ | 1* | 1* | 3.2 | 8.1, 9.1 | Chain 3 | 3.1
$(11001)$ | 1 | 1 | 7.4, 8.4 | 8.2, 9.2 |  | 3.2
$(10010)$ | 0* | 1* | 4.2, 9.3 | 6.1, 9.1 | Chain 4 | 4.1
$(10110)$ | 1* | 1 | 6.4, 9.4 | 6.2, 5.1 |  | 4.2
$(10100)$ | 1* | 1* | 5.2 | 7.1, 9.1 | Chain 5 | 5.1
$(10101)$ | 1 | 1 | 7.4, 9.4 | 7.2, 9.2 |  | 5.2
$(00010)$ | 0 | 0* | 6.2, 10.3 | 10.1 | Chain 6 | 6.1
$(00110)$ | 1* | 0* | 6.3, 10.4 | 7.1 |  | 6.2
$(01110)$ | 1 | 1 | 6.4, 10.5 |  |  | 6.3
$(11110)$ | 1 | 1 | 10.6 |  |  | 6.4
$(00100)$ | 1* | 0* | 7.2, 10.4 | 10.1 | Chain 7 | 7.1
$(00101)$ | 1 | 0* | 7.3, 10.4 | 10.2 |  | 7.2
$(01101)$ | 1 | 1* | 7.4, 10.5 | 8.2, 10.2 |  | 7.3
$(11101)$ | 1 | 1 | 5.6 |  |  | 7.4
$(01000)$ | 0 | 1* | 8.2 | 10.1 | Chain 8 | 8.1
$(01001)$ | 1* | 1 | 8.3 | 10.2 |  | 8.2
$(01011)$ | 1 | 1 | 8.4 | 10.3 |  | 8.3
$(11011)$ | 1 | 1 | 10.6 | 9.3 |  | 8.4
$(10000)$ | 0 | 1* | 9.2 | 10.1 | Chain 9 | 9.1
$(10001)$ | 1* | 1 | 9.3 | 10.2 |  | 9.2
$(10011)$ | 1 | 1 | 9.4 | 10.3 |  | 9.3
$(10111)$ | 1 | 1 | 10.6 | 10.4 |  | 9.4
$(00000)$ | 0 | 0 | 10.2 |  | Chain 10 | 10.1
$(00001)$ | 0* | 0 | 10.3 |  |  | 10.2
$(00011)$ | 1* | 0 | 10.4 |  |  | 10.3
$(00111)$ | 1 | 1* | 10.5 |  |  | 10.4
$(01111)$ | 1 | 1 | 10.6 |  |  | 10.5
$(11111)$ | 1 | 1 |  |  |  | 10.6
Questions | 13 | 12 |  |  |  |
Table 2. Examples of discovered diagnostic rules
---
Diagnosis |  $f$-criteria | Value. $f$-criteria | Precision
rule |  | 0.01 | 0.05 | 0.1 | on control
If $10<{\rm NUM}<20$ | NUM | 0.0029 | + | + | + | 93.3%
and ${\rm VOL}>5$ | VOL | 0.0040 | + | + | + |
then malignant |  |  |  |  |  |
If ${\rm TOT}>30$ | TOT | 0.0229 | - | + | + | 100.0%
and ${\rm VOL}>5$ | VOL | 0.0124 | - | + | + |
and DEN is moderate | DEN | 0.0325 | - | + | + |
then malignant |  |  |  |  |  |
If VAR is marked | VAR | 0.0044 | + | + | + | 100.0%
and $10<{\rm NUM}<20$ | NUM | 0.0039 | + | + | + |
and IRR is moderate | IRR | 0.0254 | - | + | + |
then malignant |  |  |  |  |  |
If SIZE is moderate | SIZE | 0.0150 | - | + | + | 92.86%
and SHAPE is mild | SHAPE | 0.0114 | - | + | + |
and IRR is mild | IRR | 0.0878 | - | - | + |
then benign |  |  |  |  |  |

Results for them are marked as Discovery1, Discovery2 and Discovery3. We
extracted 44 statistically significant diagnostic rules for 0.05 level of F
-criterion with a conditional probability no less than 0.75 (Discovery1).
There were 30 rules with a conditional probability no less than 0.85
(Discovery2) and 18 rules with a conditional probability no less than 0.95
(Discovery3). The most reliable 30 rules delivered a total accuracy of 90%,
and the 18 most reliable rules performed with 96.6% accuracy with only 3 false
positive cases (3.4%).

##  X Decision rule (model) extracted from the expert through monotone Boolean
functions

We obtained Boolean expressions for function
$h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$ (‘shape and density of
calcification’) from the information depicted in table 1 with the following
steps:

  1. -

Find all the maximal lower units for all chains as elementary conjunctions;

  2. -

Take the disjunction of obtained conjunctions;

  3. -

Exclude the redundant terms (conjunctions) from the end formula.

Using 1 and 3 columns we have

$x_{2}=h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)=y_{2}y_{3}\vee
y_{2}y_{4}\vee y_{1}y_{2}\vee y_{1}y_{4}\vee y_{1}y_{3}\vee$ $\vee
y_{2}y_{3}y_{5}\vee y_{2}\vee y_{1}\vee y_{3}y_{4}y_{5}\equiv y_{2}\vee
y_{1}\vee y_{3}y_{4}y_{5}.$

Function $g\left(w_{1},w_{2},w_{3}\right)=w_{2}\vee w_{1}w_{3}$ we may obtain
by direct $2^{3}=8$ questions for the expert.

Using 1 and 2 columns we have

$f\left(\overline{x}\right)=x_{2}x_{3}\vee x_{1}x_{2}x_{4}\vee x_{1}x_{2}\vee
x_{1}x_{3}x_{4}\vee x_{1}x_{3}\vee x_{3}x_{4}\vee x_{3}$ $\vee x_{2}x_{5}\vee
x_{1}x_{5}\vee x_{4}x_{5}\equiv x_{1}x_{2}\vee
x_{3}\vee\left(x_{2}x_{1}x_{4}\right)x_{5}\equiv$ $\left(w_{2}\vee
w_{1}w_{3}\right)\left(y_{1}\vee y_{2}\vee y_{3}y_{4}y_{5}\right)\vee
x_{3}\vee$ $\vee\left(y_{1}\vee y_{2}\vee
y_{3}y_{4}y_{5}\right)\left(w_{2}\vee w_{1}w_{3}\right)x_{4}x_{5}.$

##  XI Comparison of the expert model with its approximation by learning
operator

For compare rules discovered by the learning operator (Discovery system) with
the expert model we asked the expert to evaluate this rules. Below we present
some rules, discovered by Discovery system, and radiologists comments
regarding these rules as approximation of his model.

> IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5
> $cm^{3}$, and DENSITY of calcifications is moderate,
> THEN Malignant.

$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$100\%$. Radiologist’s comment - this rule might have promise, but I would
consider it risky.

> IF VARIATION in shape of calcifications is marked, and NUMBER of
> calcifications is between 10 and 20, and IRREGULARITY in shape of
> calcifications is moderate,
> THEN Malignant.

$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$100\%$. Radiologist’s comment - I would trust this rule.

> IF variation in SIZE of calcifications is moderate, and variation in SHAPE
> of calcifications is mild, and IRREGULARITY in shape of calcifications is
> mild, THEN Benign.

$f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases –
$92.86\%$. Radiologist’s comment - I would trust this rule.

## Acknowledgment

This work partially supported by the Russian Science Foundation grant
08-07-00272a and Integration projects of the Siberian Division of the Russian
Academy of science grants 47, 111, 119.

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http://math.nsc.ru/AP/ScientificDiscovery


# Multi-wavelength view of the M87 black hole captured by Event Horizon
Telescope

G. Principe Dipartimento di Fisica, Universitá di Trieste, I-34127 Trieste,
Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127
Trieste, Italy  INAF Istituto di Radioastronomia, Via P. Gobetti, 101, I-40129
Bologna, Italy  M. Lucchini MIT Kavli Institute for Astrophysics and Space
Research, MIT, 70 Vassar Street, Cambridge, MA 02139, USA  API–Anton Pannekoek
Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  on behalf of Event Horizon Telescope (EHT) Multi-
Wavelength Science Working Group  EHT Collaboration  Fermi-LAT Collaboration
H.E.S.S. Collaboration  MAGIC Collaboration  VERITAS Collaboration  EAVN
Collaboration

galaxies: individual: M 87 – black hole physics – galaxies: jets

##  1 Abstract

In 2017, the first image of the center of the M87 galaxy was captured by the
Event Horizon Telescope (EHT). It revealed a ring morphology and a size
consistent with theoretical expectations for the light pattern around a weakly
accreting supermassive black hole of $\sim$6.5 billion solar masses. In
parallel to the EHT measurements, an extensive multi-wavelength campaign with
ground- and space-based facilities from radio all the way up to the TeV energy
range was organized. In this note we will give an overview of the results from
this campaign. M87 was found to be in a historically low state. At X-ray
energies the emission from the core dominates over HST-1. We present the most
complete simultaneous, multi-wavelength spectrum of the active nucleus to
date.

##  2 Introduction

M87 is the most prominent elliptical galaxy within the Virgo Cluster, located
just $16.8\pm 0.8$ Mpc away (Blakeslee et al., 2009; Bird et al., 2010). In
addition of being among the closest active galactic nuclei (AGN), M87 also
hosts the first example of an extragalactic jet to have been detected by
astronomers (Curtis, 1918). This famous one-sided jet has been well-studied in
almost every waveband from radio (down to sub-parsec scales) to gamma rays
(e.g., Reid et al., 1989; Biretta et al., 1999; Marshall et al., 2002; Abdo et
al., 2009). Despite the many studies that have been performed, the conditions
under which such jets are launched, as well as the origin of gamma-ray
emission in radio galaxies are two of the most enduring questions in
astrophysics today (see e.g., Blandford & Znajek, 1977; Blandford & Payne,
1982).

In 2017, the first direct image of an super-massive black hole (SMBH) “shadow”
in the center of M87 galaxy was captured by the Event Horizon Telescope (EHT)
(Event Horizon Telescope Collaboration et al., 2019). It revealed a ring
morphology and a size consistent with theoretical expectations for the light
pattern around a weakly accreting supermassive black hole of $\sim$6.5 billion
solar masses. The mass of the SMBH was derived by comparing the size of blurry
ring that was measured to that predicted by models (Event Horizon Telescope
Collaboration et al., 2019).

To test the newest generation of models, as well as to investigate the origin
of the high-energy emission, it is important to have extensive, quasi-
simultaneous multi-wavelength (MWL) monitoring of AGNs, providing both
spectral and imaging data over a wide range of physical scales. We present
here an overview of the results from the 2017 EHT campaign on M87, combining
very long baseline interferometry (VLBI) imaging and spectral index maps at
longer wavelengths, with spectral data from submillimeter (submm) to TeV gamma
rays (covering more than 17 decades in frequency).

##  3 Observational campaign and results

In parallel to the EHT measurements, an extensive multi-wavelength campaign
with ground- and space-based facilities from radio to TeV gamma rays was
organized. This provided the most extensive, quasi-simultaneous, broadband
spectrum of M87 yet taken, together with the highest ever resolution mm-VLBI
images using the EHT from its 2017 April campaign.

M87 was found to be in a historically low state, both for its core and for the
nearest knot HST-1. In the X-ray range, the emission from the core dominates
over HST-1. This provides the ideal observing conditions for a multiwavelength
campaign combining data over a large range of spatial resolution, since M87’s
core was dominating the total flux in the radio through X-ray bands. This made
it possible to combine core flux constraints with the more spatially precise
VLBI data. Figure 1(a) shows the broadband SED of M87 over a the range between
a frequency of $\simeq\\!1$ GHz and photon energy of $\simeq\\!1$ TeV. The
emission measured by the instruments contributing to this broadband SED spans
a factor of $\gtrsim 10^{8}$ in angular scale: from the effective spatial
resolution of the EHT, $\simeq\\!20$ $\mu$as (Event Horizon Telescope
Collaboration et al., 2019), to the resolving power of Fermi-LAT in low-energy
$\gamma$-rays, $\simeq\\!2^{\circ}$ (Principe et al., 2018). In Figure 1 we
model the SED focusing on the EHT data. At $\gamma$-ray energy bands, one can
see the SSC emission components although they underestimate the observed
gamma-ray flux density. In Figure 1 (c) the SED is modelled focusing on the
higher energy data. Without a strong prior constraint on the size, as in the
previous case, we found that most model parameters are highly degenerate and
cannot be constrained satisfactorily by our fit.

Figure 1: Top: Observed broadband SED of M87 quasi-simultaneous with the EHT
campaign in April 2017 with fluxes measured by various instruments highlighted
with different colors and markers. For the mm-radio VLBI, the upper limits on
emission size for several representative frequencies are labelled to clarify
the size-scale. An illustration of the resolved flux differences depending on
spatial resolution is shown by the comparison of the differing EHT and ALMA-
only 230 GHz fluxes and size limits. (b): SED fit focusing on the EHT data.
Blue and green lines display the resulting SEDs for models including radiative
cooling on the initial single power-law eDF and a broken power-law,
respectively. (c) SED with model 2 fit focusing on the higher energy data. The
figures are taken from EHT MWL Science Working Group et al. (2021).

##  4 Conclusions

The goal of this extensive, quasi-simultaneous observational campaign was to
provide this important legacy data set, which will serve as a resource for the
entire community, to enable the best possible modelling outcomes and a
reference point for theory. Based on simple single-zone models, it appears
that M87’s complex, broadband spectral energy distribution cannot be modeled
by a single zone. The stratified nature of the radio through millimeter bands
is clearly indicated by the high dynamic-range images that place significant
size constraints on the emitting geometry per frequency band. Furthermore it
is not yet clear where the gamma rays originate, but it is possible to rule
out that they are originated within the EHT region by leptonic processes.
Although a $\sim$100 times larger region can provide the measured gamma-ray
emission, it would, however, require an uncomfortably high particle domination
compared to the energy stored in the magnetic fields, which may be
inconsistent with the observed jet velocity and parabolic geometrical profile.
Further information on the results and a description of the analysis performed
for this campaign, as well as the multi-wavelength data as a legacy data
repository, can be found in EHT MWL Science Working Group et al. (2021).

The support given by the LAT AGN Working Group coordinators to review this
note, is gratefully acknowledged. We refer to the related publication (EHT MWL
Science Working Group et al., 2021) for the complete acknowledgments.

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