Contour and surface integrals in potential scattering

Abstract

When the Schrödinger equation for stationary states is studied for a system described by a central potential in n-dimensional Euclidean space, the radial part of stationary states is an even function of a parameter \(\lambda \) which is a linear combination of angular momentum quantum number l and dimension n, i.e., \(\lambda =l+{(n-2)\over 2}\). Thus, without setting a priori \(n=3\), complex values of \(\lambda \) can be achieved, in particular, by keeping l real and complexifying n. For suitable values of such an auxiliary complexified dimension, it is therefore possible to obtain results on scattering amplitude and phase shift that are completely equivalent to the results obtained in the sixties for Yukawian potentials in \(\mathbf{R}^{3}\). Moreover, if both l and n are complexified, the possibility arises of recovering the partial wave amplitude from residues of a function of two complex variables. Thus, the complex angular momentum formalism can be imbedded into a broader framework, where a correspondence exists between the scattering amplitude and a skew curve in \(\mathbf{R}^{3}\).

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    Recall from Ref. [3] that the potential V(z) is Yukawian if one can express it in the form

    $$\begin{aligned} V(z)=\int _{m}^{\infty }C(\mu )e^{-\mu z}\mathrm {d}\mu , \end{aligned}$$

    where \(C(\mu )\) is a distribution. Other relevant cases occur when \(C(\mu )\) is a function of bounded variation, for which the potential reads as

    $$\begin{aligned} V(z)=\int _{m}^{\infty }{e^{-\mu z}\over z}\mathrm {d}C_{\mu }, \end{aligned}$$

    or when C is an absolutely continuous function with first derivative \(C'(\mu )=\sigma (\mu )\), for which V(z) takes the form

    $$\begin{aligned} V(z)=\int _{m}^{\infty }{e^{-\mu z}\over z}\sigma (\mu )\mathrm {d}\mu . \end{aligned}$$

    The Yukawa potential is recovered from the first formula when \(C(\mu )\) is just a constant.

  2. 2.

    The assumption of Poincaré is not however too restrictive, since a theorem holds, according to which a function that is meromorphic over the whole of \(\mathbf{C}^{2}\), including the point at infinity, is a rational function [14].

References

  1. 1.

    T. Regge, Introduction to complex orbital momenta. Nuovo Cim. 14, 951–976 (1959)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    A. Bottino, A.M. Longoni, T. Regge, Potential scattering for complex energy and angular momentum. Nuovo Cim. 23, 954–1004 (1962)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    V. de Alfaro, T. Regge, Potential Scattering (North-Holland, Amsterdam, 1965)

    Google Scholar 

  4. 4.

    G.N. Watson, The diffraction of electric waves by the earth. Proc. R. Soc. Lond. 95, 83–99 (1918)

    ADS  MATH  Google Scholar 

  5. 5.

    A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949)

    Google Scholar 

  6. 6.

    A. Bottino, A retrospective look at Regge poles, arXiv:1807.02456 [physics.hist-ph]

  7. 7.

    A. Chatterjee, Large-N expansions in quantum mechanics, atomic physics ands some \(O(n)\) invariant systems. Phys. Rep. 186, 249–372 (1990)

    ADS  Article  Google Scholar 

  8. 8.

    G. Esposito, Complex parameters in quantum mechanics. Found. Phys. Lett. 11, 535–547 (1998)

    MathSciNet  Article  Google Scholar 

  9. 9.

    E.T. Whittaker, G.N. Watson, Modern Analysis (Cambridge University Press, Cambridge, 1927)

    Google Scholar 

  10. 10.

    T. Daudé, F. Nicoleau, Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles. Ann. H. Poincaré 17, 2849–2904 (2016)

    Article  Google Scholar 

  11. 11.

    H. Poincaré, Sur les résidus des intégrales doubles. Acta Math. 9, 321–380 (1887)

    MathSciNet  Article  Google Scholar 

  12. 12.

    J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe. Bull. Soc. Math. France 87, 81–180 (1959)

    MathSciNet  Article  Google Scholar 

  13. 13.

    L.A. Aizenberg and A.P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Transl. Math. Mon. 58 (AMS, Providence, 1983)

  14. 14.

    R. Caccioppoli, Theory of Functions of Several Complex Variables (Federico II University, Naples, 1947)

    Google Scholar 

  15. 15.

    V. Del Duca and L. Magnea, The long road from Regge poles to the LHC, arXiv:1812.05829 [hep-ph]

  16. 16.

    A. Folacci, M. Ould El Hadji, Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole. Phys. Rev. D 99, 104079 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    A. Folacci, M. Ould El Hadji, Regge pole description of scattering of gravitational waves by a Schwarzschild black hole. Phys. Rev. D 100, 064009 (2019)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author is grateful to the Dipartimento di Fisica “Ettore Pancini” for hospitality and support. The preprint version of this paper was arXiv:2001.11217 [hep-th].

Author information

Affiliations

Authors

Corresponding author

Correspondence to Giampiero Esposito.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Esposito, G. Contour and surface integrals in potential scattering. Eur. Phys. J. Plus 135, 501 (2020). https://doi.org/10.1140/epjp/s13360-020-00514-5

Download citation