
Brief overview of the main scientific results obtained by
Francesco
Calogero
(FC)
In 1960, at the very
beginning of his scientific career, FC obtained some results
connected with “high energy” electronpositron scattering (in
collaboration with Laurie Brown) and electronelectron scattering
(in collaboration with Charles Zemach). The first kind of results
were soon afterwards overtaken by a much more complete analysis
performed by Nicola Cabibbo and Raoul Gatto; the second kind
provided instead, for decades, the standard first reference for
experimentalists working in this field.
Then (in 1962) FC
performed, in collaboration with Asim Barut, the first numerical
computation, in the context of nonrelativistic scattering theory, of
Regge pole trajectories.
The first completely
autonomous result of FC was the introduction of the variable phase
approach to potential scattering (in 1963). Using this approach FC
proved that the scattering amplitude produced by potentials
representable as superpositions of Yukawa functions, when
analytically continued in angular momentum, vanishes as the angular
momentum diverges to positive and negative infinity (this technical
result was of crucial importance in the theory of Regge poles). Via
the variable phase approach – and variations of it – FC also
obtained several results in nonrelativistic scattering theory,
including novel (rigorous) upper and lower limits on the number of
bound states in a given central potential. Much later novel results
of this kind were obtained by FC together with Fabian Brau.
In 1967 FC published the
monograph Variable phase approach to potential scattering
(Academic Press, New York, 244 pages; translated into Russian in
1972), which has been, and still is, widely used as a teaching and
research tool.
In these years FC also
worked in quantum field theory and elementary particles theory, but
became more and more dissatisfied with this kind of research (as
generally performed at the time) and eventually moved his main
research activity to the nuclear manybody problem. In this area FC
collaborated with Yuri Simonov and others, developing and using the
hyperspherical expansion approach to the Schroedinger manybody
problem. But the more interesting results of this period are the
determination (together with Yuri Simonov) of rigorous conditions
that the nuclear twobody potential must satisfy, in the context of
nonrelativistic quantum mechanics, to be consistent with the
property of saturation of nuclear forces (demonstrated by the fact
that the binding energy of each nucleon, and the nucleon density, is
roughly the same in all heavy nuclei).
Motivated by his interest
in the nuclear manybody problem FC noticed (around 1970) that the
onedimensional quantal manybody problem with a twobody potential
proportional to the inverse square of the interparticle distance,
and possibly in addition a (onebody, or twobody)
harmonicoscillator potential (to confine the particles), is
exactly solvable. This problem, both in its quantal version, and
especially in its classical version (subsequently shown to be
completely integrable by Jurgen Moser), has played a significant
role, over the last threefour decades, in the theory of integrable
dynamical systems. It is universally known as the CalogeroMoser
model; it has been investigated, together with many variations of
it, by many (perhaps hundreds) of scholars, both theoretical and
mathematical physicists as well as applied and pure mathematicians,
acquiring the status of basic lore. There are indeed scores of
papers, published in main scientific journals, which explicitly
mention (often in their title) the CalogeroMoser system and give no
reference to the papers in which it was originally introduced (nor,
quite often, to any paper at all by either FC or Jurgen Moser!); in
addition of course to hundreds of papers that do quote the original
paper by FC in which these findings were first presented.
Another remarkable result
by FC (in 1975) has been the extension, in the classical context, of
this solvable problem to include potentials described by elliptic
functions. To arrive at this result new functional equations were
also introduced and solved, opening a research line that has
subsequently been investigated by FC and by other wellknown
mathematicians.
In 1976 FC also introduced
a new approach to the identification and investigation of
integrable nonlinear PDEs; a research area then in full bloom,
after the breakthrough discovery by Gardner, Green, Kruskal and
Miura, in 1967, of the integrability of the Korteweg – de Vries
nonlinear PDE and of the associated solitonic phenomenology. In this
field FC (alone, and in collaboration with others, mainly with his
former student, and subsequently colleague, Antonio Degasperis)
obtained many results: the first clarification of the spectral
significance of Baecklund transformations; the identification of
several new integrable PDEs; the discovery of a new soliton
phenomenology (“boomerons” and “trappons”, namely solitons that do
not move with constant velocity but boomerang back in the remote
future to where they came from in the remote past, or are trapped to
oscillate around some fixed position determined by the initial
data); necessary conditions for the integrability of nonlinear PDEs;
the notion of Cintegrable and Sintegrable nonlinear PDEs, namely
integrable PDEs solvable by appropriate Changes of variables or by
the Spectral transform technique; the study of initial/boundary
value problems, in the context of certain Cintegrable equations;
the introduction of the notion of “universal” nonlinear PDEs (as the
limit of large classes of PDEs), and the related explanation of the
remarkable fact that certain nonlinear PDEs are both integrable
and widely applicable. Many results in this field were
presented in the monograph jointly authored by FC and Antonio
Degasperis, Spectral transform and solitons (North Holland,
Amsterdam, 1982, 516 pages; translated into Russian in 1985).
In the meantime, and up to
now, FC also continued to work, mainly alone but also with several
collaborators, on integrable and solvable manybody problems (mainly
classical), introducing new approaches and discovering several novel
instances of such systems (including manybody problems in two and
three dimensional space, characterized by rotationinvariant
Newtonian equations of motion), and also several purely mathematical
results related to these findings (including properties of the zeros
of classical polynomials, the identification of remarkable matrices,
the discovery of certain mathematical identities that eventually
made their way into classical compilations of mathematical results,
such as the versions edited by Alan Jeffrey of the Tables of
Integrals, Series and Products by Gradshteyn and Ryzhik
(Academic Press, fifth edition, 1994, see section 15.823). In 2001
FC published a monograph in this field largely based on results by
himself (in some cases, with collaborators): Classical manybody
problems amenable to exact treatments (Lecture Notes in Physics
Monograph m66, Springer, 2001, 750 pages).
Since the late 1970s, and
especially in the 1980s and early 1990s, the scientific activity on
integrable systems of FC and his collaborators, and of other
colleagues in Italy influenced by FC, contributed to put Italy on
the world map as one of the leading centers of research in this
area. FC was also instrumental, through many international workshops
of which he was the main scientific organizer, to foster contacts in
this area among scientists from Western Europe, the United States
and Japan, and scientists from East Europe, the Soviet Union and
China, overcoming the political difficulties due to the cold war
climate. Let us also note in this respect that the discovery of many
integrable dynamical systems (with a finite or infinite number of
degrees of freedom), and the investigation of their properties with
all the consequent ramifications, has been a very important, perhaps
the main, “scientific revolution” in mathematical physics of the
last half century.
FC is still engaged, mainly
in collaboration with Antonio Degasperis, in the investigation of
integrable nonlinear PDEs: for instance relatively recently new
solutions have been obtained of the system of coupled PDEs
describing the resonant interaction of three waves (of applicative
interest in many contexts), and new integrable (systems of)
nonlinear PDEs have been identified. The main current research
interest of FC is focused on isochronous systems, namely
dynamical systems possessing an open domain of initial data,
having full dimensionality in their phase space (and possibly
including the entire phase space), out of which emerge solutions
all of which are completely periodic with the same fixed
period. FC has shown that “such systems are not rare”: in fact any
dynamical system can be modified so that the modified system is
isochronous. The main current research activity of FC is
connected with various related developments.
In collaboration with D.
GomezUllate, Paolo Santini and Matteo Sommacal, the investigation
of certain dynamical systems which are isochronous in certain
regions of their phase space but behave in a complicated manner in
other regions has opened a new understanding (in terms of travel
over Riemann surfaces) of interesting mechanisms subtending the
transition from simple to complicated motions (the latter possibly
providing instances of “deterministic chaos”).
Mainly in collaboration
with Mario Bruschi and Riccardo Droghei, the investigation of the
behavior of isochronous dynamical systems in the neighborhood of
their equilibria has led to interesting mathematical findings (in
particular, the identification of Diophantine properties of certain
matrices and certain polynomials, including essentially all those
belonging to the Askey scheme of orthogonal polynomials).
And, mainly in
collaboration with François Leyvraz, many interesting examples of
isochronous systems have been obtained and investigated, including
some models having remarkable implications for basic concepts of
physics (such as the issue of the arrow of time in statistical
mechanics and in cosmology, and the distinction among integrable and
chaotic systems), and other simple dynamical systems (obtained in a
trilateral collaboration involving also Matteo Sommacal) potentially
relevant as models of (isochronous) chemical reactions.
Many of these results are
reviewed in a recently published monograph (F. Calogero,
Isochronous Systems, Oxford University Press, 2008, 250 pages),
several others have been obtained and published after the completion
of that book.
February 2010

