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Asymptotic Summation of Slow Converging and Rapidly Oscillating Series

Alexander M. Chebotarev

TL;DR

The paper addresses the problem of summing slowly converging, rapidly oscillating Poisson-weighted series with an extremely large mean $N$ by deriving a Gaussian-approximation-based asymptotic expansion. It restricts the sum to a near-peak interval, replaces the Poisson distribution with its Gaussian correction via the Stirling expansion, and converts the discrete sum into a Gaussian integral with explicit $O(N^{-1})$ error, obtaining a closed-form expression for the main term. The approach is extended to related sums and multi-parameter variants, with additional theorems providing explicit asymptotics for $Z_F$, $Z_s$, a tilde variant, and a two-variable sum. The Appendix compiles robust technical tools (trapezoidal errors, derivative estimates of $P_N$, Gaussian-correction terms, and Komatsu-type bounds) that underpin the rigor of the asymptotics, enabling precise analytic approximations of quantum observables in interferometric settings with large photon numbers.

Abstract

Mean values of some observables describing quantum interaction between the Bose field in a cavity and a movable mirror can be represented as expectations of rapidly oscillating functions w.r.t. the Poisson measure with a large mean value ($N\approx 10^{23}$) corresponding to the average number of photons in laser beam. Straightforward summation of the series is impossible because over $2\sqrt N$ summands make a significant contribution. We derive an analytical expression approximating this sum with the error $O(N^{-1})$.

Asymptotic Summation of Slow Converging and Rapidly Oscillating Series

TL;DR

The paper addresses the problem of summing slowly converging, rapidly oscillating Poisson-weighted series with an extremely large mean by deriving a Gaussian-approximation-based asymptotic expansion. It restricts the sum to a near-peak interval, replaces the Poisson distribution with its Gaussian correction via the Stirling expansion, and converts the discrete sum into a Gaussian integral with explicit error, obtaining a closed-form expression for the main term. The approach is extended to related sums and multi-parameter variants, with additional theorems providing explicit asymptotics for , , a tilde variant, and a two-variable sum. The Appendix compiles robust technical tools (trapezoidal errors, derivative estimates of , Gaussian-correction terms, and Komatsu-type bounds) that underpin the rigor of the asymptotics, enabling precise analytic approximations of quantum observables in interferometric settings with large photon numbers.

Abstract

Mean values of some observables describing quantum interaction between the Bose field in a cavity and a movable mirror can be represented as expectations of rapidly oscillating functions w.r.t. the Poisson measure with a large mean value () corresponding to the average number of photons in laser beam. Straightforward summation of the series is impossible because over summands make a significant contribution. We derive an analytical expression approximating this sum with the error .

Paper Structure

This paper contains 9 sections, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Estimate of the weight of tails
  3. Estimates of the oscillating sum
  4. Approximation of related series
  5. Appendix: Simple a priori estimates
  6. Trapezoidal approximation
  7. Derivatives of the Poisson distribution
  8. Corrections to the Gaussian distribution
  9. Komatsu inequalities

Figures (3)

  • Figure 1: The picture shows the Poisson distribution $P_N(n)$ for $N=10^4$, which practically coincides with the Gaussian distribution $N(10^4,10^4)$, and the approximation error (double-well curve scaled by $10^5$), representing the difference between the Poisson and normal distributions with the Stirling corrections.
  • Figure 2: The derivatives of the Poisson distribution become more flatten as $N$ increases. Each derivation add a new critical point. The picture shows $P^{(1)}_N(n)$ with two critical points, and $100*P^{(2)}_N(n)$ with three critical points, for $N=10^4$.
  • Figure 3: The picture shows the absolute approximation error $E$ of the sum $Z$ by integral (3.2), as a decreasing function of $A$ and $B$ for $N=1000$.