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Physics and mathematics

of nonlinear processes and models

(University "La Sapienza" , Rome, Italy)











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” electron-positron scattering (in collaboration with Laurie Brown) and electron-electron 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 many-body problem. In this area FC collaborated with Yuri Simonov and others, developing and using the hyperspherical expansion approach to the Schroedinger many-body problem. But the more interesting results of this period are the determination (together with Yuri Simonov) of rigorous conditions that the nuclear two-body 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 many-body problem FC noticed (around 1970) that the one-dimensional quantal many-body problem with a two-body potential proportional to the inverse square of the interparticle distance, and possibly in addition a (one-body, or two-body) harmonic-oscillator 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 three-four decades, in the theory of integrable dynamical systems. It is universally known as the Calogero-Moser 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 Calogero-Moser 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 well-known 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 C-integrable and S-integrable 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 C-integrable 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 many-body problems (mainly classical), introducing new approaches and discovering several novel instances of such systems (including many-body problems in two and three dimensional space, characterized by rotation-invariant 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 many-body 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. Gomez-Ullate, 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