Zeldovich regularization explained

Zeldovich regularization refers to a regularization method to calculate divergent integrals and divergent series, that was first introduced by Yakov Zeldovich in 1961.^[1] 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^[2]

\int ^{inf ty} _{0} f(x) dx \equiv \lim _{\alpha\to 0 ^{+}} \int ^{inf ty} _{0} f(x) e ^{- \alpha x ^{2}} dx.

and, for divergent series, by^{[3] [4] [5]}

\sum _{n} c _{n} \equiv \lim _{\alpha\to 0 ^{+}} \sum _{n} c _{n} e ^{- \alpha n ^{2}} .

See also

• Abel's theorem
• Borel summation

Notes and 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.