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Francesco Calogero

Bio: Francesco Calogero is an academic researcher from Sapienza University of Rome. The author has contributed to research in topics: Nonlinear system & Dynamical systems theory. The author has an hindex of 40, co-authored 295 publications receiving 10357 citations. Previous affiliations of Francesco Calogero include International Centre for Theoretical Physics & Istituto Nazionale di Fisica Nucleare.


Papers
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TL;DR: In this paper, the quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (harmonical) and/or inverse (centrifugal) potentials is solved.
Abstract: The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl], with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi st...

1,454 citations

Journal ArticleDOI
TL;DR: In this article, the problem of three equal particles interacting pairwise by inversecube forces (centrifugal potential) in addition to linear forces (harmonical potential) is solved in one dimension.
Abstract: The problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is solved in one dimension.

1,015 citations

Journal ArticleDOI
TL;DR: In this article, the problem of N quantum-mechanical equal particles interacting pairwise by inverse cube forces (''centrifugal potential'') in addition to linear forces ( ''harmonical potential''), is considered in a onedimensional space.
Abstract: The problem of N quantum‐mechanical equal particles interacting pairwise by inverse‐cube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is considered in a onedimensional space. An explicit expression for the ground‐state energy and for the corresponding wavefunction is exhibited. A class of excited states is similarly displayed.

488 citations

Book
27 Apr 2012
TL;DR: In this paper, a terse introduction to the spectral transform technique to solve certain classes of nonlinear evolution equations is given. And the properties of the solutions of these nonlinear PDEs are discussed.
Abstract: This is a terse introduction to the spectral transform technique to solve certain classes of nonlinear evolution equations, and to the properties of the solutions of these nonlinear PDEs.

459 citations


Cited by
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Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

Journal ArticleDOI
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.

2,688 citations

Book
01 Jan 1991
TL;DR: The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3) as mentioned in this paper.
Abstract: The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3). In order to reveal level repulsion under conditions of global classical chaos special care may be necessary: (i) subspectra referring to different values of the quantum numbers related to symmetries must be dealt with separately and (ii) for systems with quantum localization only levels whose wavefunctions have overlapping support must be admitted. A “level” may either be an energy eigenvalue E in the case of autonomous systems or, for periodically driven systems, a quasi-energy φ, i.e. an eigenphase of the unitary Floquet operator transporting the wavevector from period to period.

2,495 citations

Journal ArticleDOI
TL;DR: The general properties of the factorized S-matrix in two-dimensional space-time are considered in this article, where the relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed.

1,985 citations

Journal ArticleDOI
TL;DR: A review of the development of random-matrix theory (RMT) during the last fifteen years is given in this paper, with a brief historical survey of the developments of RMT and of localization theory since their inception.

1,750 citations