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Zeldovich regularization

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Zeldovich regularization refers to a regularization method to calculate divergent integrals and divergent series, that was first introduced by Yakov Zeldovich in 1961. Zeldovich was originally interested in calculating the norm of the Gamow wave function which is divergent since there is an outgoing spherical wave. Zeldovich regularization uses a Gaussian type-regularization and is defined, for divergent integrals, by

∫ 0 ∞ f ( x ) d x ≡ lim α → 0 + ∫ 0 ∞ f ( x ) e − α x 2 d x . {\displaystyle \int _{0}^{\infty }f(x)dx\equiv \lim _{\alpha \to 0^{+}}\int _{0}^{\infty }f(x)e^{-\alpha x^{2}}dx.}

and, for divergent series, by

∑ n c n ≡ lim α → 0 + ∑ n c n e − α n 2 . {\displaystyle \sum _{n}c_{n}\equiv \lim _{\alpha \to 0^{+}}\sum _{n}c_{n}e^{-\alpha n^{2}}.}

See also

References

  1. Zel’Dovich, Y. B. (1961). On the theory of unstable states. Sov. Phys. JETP, 12, 542.
  2. Garrido, E., Fedorov, D. V., Jensen, A. S., & Fynbo, H. O. U. (2006). Anatomy of three-body decay III: Energy distributions. Nuclear Physics A, 766, 74-96.
  3. Mur, V. D., Pozdnyakov, S. G., Popruzhenko, S. V., & Popov, V. S. (2005). Summation of divergent series and Zeldovich’s regularization method. Physics of Atomic Nuclei, 68, 677-685.
  4. Mur, V. D., Pozdnyakov, S. G., Popov, V. S., & Popruzhenko, S. V. E. (2002). On the Zel’dovich regularization method in the theory of quasistationary states. Journal of Experimental and Theoretical Physics Letters, 75, 249-252.
  5. Orlov, Y. V., & Irgaziev, B. F. (2008). On the normalization of the Gamov resonant wave function in the configuration space. Bulletin of the Russian Academy of Sciences: Physics, 72, 1539-1543.
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