Nonlocal potentials and complex angular momentum theory. (English) Zbl 1218.81099

When two composite particles collide, the fermionic character of the components emerges and the antisymmetrization of the system generates repulsive exchange-forces followed by echoes due to Pauli’s exclusion principle. The notion of echo was introduced in parallel with the notion of resonance by K. W. McVoy [Ann. Phys. 43, 91–125 (1965)]. Despite of this parallel, the status of the echoes has remained obscure in terms of the analytic properties in energy of the scattering functions while the pole-resonance conceptual correspondence was successfully achieved by various authors, e.g., V. de Alfaro and T. Regge [Potential scattering. Amsterdam: North-Holland Publishing Comp. (1965; Zbl 0141.23202)].
In this paper, the authors study singularities of the partial scattering amplitude \(T\) for appropriate classes of nonlocal potentials, both in the complex momentum \(k\)-plane and in the complex angular momentum \(\lambda\)-plane. Their results extend Regge’s formalism to the case of nonlocal potentials and provide a framework for analyzing both resonances and echoes in terms of the polar singularities of \(T\) in the upper or lower half plane of \(\lambda\).


81U05 \(2\)-body potential quantum scattering theory
35B34 Resonance in context of PDEs


Zbl 0141.23202
Full Text: DOI arXiv


[1] Nussenzveig, H. M., Causality and Dispersion Relations (1972), New York: Academic Press, New York · Zbl 0093.44202
[2] Mcvoy, K. W., Giant Resonances and Neutron-Nucleus Total Cross Sections, Ann. Phys., 43, 91-125 (1967)
[3] Cindro, N., Počanič, D.: Resonances in heavy-ion reactions—structural vs. diffractional models. In: Resonances Models and Phenomena. Lecture Notes in Physics, vol. 211, pp. 158-181. Springer, Berlin (1993)
[4] De Alfaro, V.; Regge, T., Potential Scattering (1965), Amsterdam: North-Holland, Amsterdam · Zbl 0141.23202
[5] Newton, R.G.: The Complex j-plane. W. A. Benjamin, New York · Zbl 0119.44004
[6] Bethe, H. A., Nuclear many-body problem, Phys. Rev., 103, 1353-1390 (1956) · Zbl 0075.22202
[7] Wildermuth, K.; Tang, Y. C., A Unified Theory of the Nucleus (1977), Braunschweig: Vieweg, Braunschweig
[8] Tang, Y. C.; Lemere, M.; Thompson, D. R., Resonating-group method for nuclear many-body problems, Phys. Rep., 47, 167-223 (1978)
[9] Lemere, M.; Tang, Y. C.; Thompson, D. R., Study of the α +^16O system with the resonating-group method, Phys. Rev. C, 14, 23-27 (1976)
[10] Wildermuth, K., McClure, W.: Cluster representation of nuclei. Springer Tracts in Modern Physics, vol. 41. Springer, Berlin (1966)
[11] De Micheli, E.; Viano, G. A., Unified scheme for describing time delay and time advance in the interpolation of rotational bands of resonances, Phys. Rev. C, 68, 064606 (2003)
[12] De Micheli, E.; Viano, G. A., Time delay and time advance in resonance theory, Nucl. Phys. A, 735, 515-539 (2004)
[13] Bros, J.: On the notions of scattering state, potential and wave-function in quantum field theory: an analytic-viewpoint. In: Kashiwara, M., Kawai, T. (eds.) Prospect of Algebraic Analysis, vol. 1, pp. 49-74. Academic Press, New York (1988)
[14] Bros, J.; Viano, G. A., Complex angular momentum in general quantum field theory, Ann. Henri Poincaré, 1, 101-172 (2000) · Zbl 1136.81392
[15] Bros, J.; Viano, G. A., Complex angular momentum diagonalization of the Bethe-Salpeter structure in general quantum field theory, Ann. Henri Poincaré, 4, 85-126 (2003) · Zbl 1090.81514
[16] Reed, M., Simon, B.: Methods of Modern Mathematical Physics—Scattering Theory, vol. 3. Academic Press, New York (1979) · Zbl 0405.47007
[17] Bertero, M.; Talenti, G.; Viano, G. A., Scattering and bound states solutions for a class of non-local potentials (s-wave), Commun. Math. Phys., 6, 128-150 (1967)
[18] Bertero, M.; Talenti, G.; Viano, G. A., Bound states and Levinson’s theorem for a class of non-local potentials (s-wave), Nucl. Phys. A, 113, 625-640 (1968)
[19] Bertero, M.; Talenti, G.; Viano, G. A., A note on non-local potentials, Nucl. Phys. A, 115, 395-404 (1968)
[20] Bertero, M.; Talenti, G.; Viano, G. A., Eigenfunction expansions associated with Schrödinger two-particle operators, Nuovo Cimento, 62A, 10, 27-87 (1969)
[21] Smithies, F., The Fredholm theory of integral equations, Duke Math. J., 8, 107-130 (1941) · JFM 67.0376.02
[22] Yosida, K., Functional Analysis (1965), Berlin: Springer, Berlin · Zbl 0126.11504
[23] Ikebe, T., Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory, Arch. Ration. Mech. Anal., 5, 1-34 (1960) · Zbl 0145.36902
[24] Ikebe, T., On the phase-shift formula for the scattering operator, Pac. J. Math., 15, 511-523 (1965) · Zbl 0138.45004
[25] Simon, B.: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton Univ. Press, Princeton (1971) (see also the references quoted therein) · Zbl 0232.47053
[26] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), New York: Dover, New York
[27] Newton, R. G., Analytic properties of radial wave functions, J. Math. Phys., 1, 319-347 (1960) · Zbl 0090.19303
[28] Tichonov, A. N.; Samarskij, A. A., Uravnenija Matematičeskoi Fiziki (1977), Moscow: Mir, Moscow
[29] Von Neumann, J.; Wigner, E. P., Über Merkwurdige Diskrete Eigenwerte, Z. Physik, 30, 465-467 (1929) · JFM 55.0520.04
[30] Fonda, L., Bound states embedded in the continuum and the formal theory of scattering, Ann. Phys., 22, 123-132 (1963) · Zbl 0122.45802
[31] Goursat, E., Cours d’Analyse Mathématique, Tome III (1956), Paris: Gauthier-Villars, Paris · JFM 46.0375.13
[32] Bros, J.; Pesenti, D., Fredholm resolvents of meromorphic kernels with complex parameters, J. Math. Pures et Appl., 62, 215-252 (1983) · Zbl 0531.35067
[33] Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, p. 110. Springer, New York (1977) · Zbl 0363.47006
[34] Martin, A., On the validity of Levinson’s theorem for non-local interactions, Nuovo Cimento, 7, 607-627 (1958) · Zbl 0080.22601
[35] Watson, G.N.: Theory of Bessel Functions, p. 389. Cambridge University Press, Cambridge (1952)
[36] Erdelyi, A., Bateman Manuscript Project—Higher Trascendental Functions (1953), New York: McGraw-Hill, New York
[37] Boas, R. P., Entire Functions (1954), New York: Academic Press, New York
[38] Widder, D. V., The Laplace Transform (1972), Princeton: Princeton University Press, Princeton · JFM 67.0384.01
[39] Hoffman, K.: Banach spaces of analytic functions, p. 125, Prentice-Hall, Englewood Cliffs (1962) · Zbl 0117.34001
[40] Bros, J.; Viano, G. A., Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid-III, Forum Math., 9, 165-191 (1997) · Zbl 0878.43016
[41] Erdelyi, A., Bateman Manuscript Project—Tables of Integral Transforms (1954), New York: McGraw-Hill, New York
[42] Bros, J.; Viano, G. A., Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid-II, Forum Math., 8, 659-722 (1996) · Zbl 0878.43015
[43] Newton, R. G., Scattering Theory of Waves and Particles (1966), New York: McGraw-Hill, New York
[44] Martin, A., Some simple inequalities in scattering by complex potentials, Nuovo Cimento, 23, 641-654 (1962) · Zbl 0112.45002
[45] Sommerfeld, A.: Partial Differential Equations in Physics, vol. 6. Academic Press, New York (1964)
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