# Casimir Energies for Isorefractive or Diaphanous Balls

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## Abstract

**:**

## 1. Introduction

## 2. The Diaphanous Ball

## 3. The Dual Electromagnetic $\mathit{\delta}$ Sphere

#### 3.1. Uniform Asymptotic Expansion

#### 3.2. First Approximation

#### 3.3. $O\left({\lambda}^{2}\right)$ Contribution

#### 3.4. General Analysis

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The ratio of the remaining contributions to the energy relative to the lowest order approximant, ${R}_{l}/{E}^{\left(2\right)}$, for isorefractive dielectric-diamagnetic balls. Plotted are the negatives of this ratio for $l=1$ (solid, black), $l=2$ (dotted, blue), and $l=3$ (dashed, red).

**Figure 2.**The energy for isorefractive dielectric-diamagnetic balls. Plotted are the first approximation (dotted, blue), the second approximation (dashed, red), and the total (solid, black).

**Figure 3.**Energy estimate $\tilde{E}$ for a diaphanous ball (in units of $1/a$) based on the leading terms in the UAE as a function of the coupling $\lambda $. Plotted is the exact approximant (solid curve), the contribution of the leading order ${E}^{\left(2\right)}$ (with the $1/\Delta $ divergence removed) (dotted curve), and the sum of the first two leading orders ${E}^{\left(2\right)}+{E}^{\left(4\right)}$ (dashed curve). Although the exact approximant is finite at $\lambda =3$, it possesses an infinite slope there, having changed sign for a slightly smaller value of $\lambda $.

**Figure 4.**Zeroes of ${\Delta}^{E}$ (right) and ${\Delta}^{H}$ (left), and of their product, shown by the plots of their magnitudes, for $l=1$ and $\lambda =10$. As $\lambda $ increases, the zeroes approach $\lambda /2$ for all l.

**Figure 5.**The integrand of the energy ${w}_{l}(x,\lambda )$ given by Equation (26) for $\lambda =1$ and $l=1,10,100$, shown by the dotted lines. The solid lines show the unsubtracted $\mathrm{ln}{\Delta}^{E}{\Delta}^{H}$ integrand. The removal of the leading UAE contributions greatly improves the behavior of high x values, but leaves large and growing contributions for moderate values of x.

**Table 1.**${E}^{\left(2\right)}$, ${E}^{\left(4\right)}$, ${R}_{1}$, ${R}_{2}$, ${R}_{3}$, and the sum E for various values of $\xi $.

$\mathit{\xi}$ | ${\mathit{E}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathit{a}$ | ${\mathit{E}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathit{a}$ | ${\mathit{R}}_{\mathbf{1}}\mathit{a}$ | ${\mathit{R}}_{\mathbf{2}}\mathit{a}$ | ${\mathit{R}}_{\mathbf{3}}\mathit{a}$ | $\mathit{Ea}$ |
---|---|---|---|---|---|---|

1 | $0.046875$ | $-0.0005135$ | $-0.0001517$ | $-0.0000223$ | $-6\times {10}^{-6}$ | $0.04618$ |

$0.5$ | $0.011719$ | $0.0005456$ | $-0.0000384$ | $-6\times {10}^{-6}$ | $-1.6\times {10}^{-6}$ | $0.0122184$ |

$0.01$ | $4.687\times {10}^{-6}$ | $3.081\times {10}^{-7}$ | $-1.793\times {10}^{-8}$ | $-2.74\times {10}^{-9}$ | $-7.53\times {10}^{-10}$ | $4.974\times {10}^{-6}$ |

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Milton, K.A.; Brevik, I.
Casimir Energies for Isorefractive or Diaphanous Balls. *Symmetry* **2018**, *10*, 68.
https://doi.org/10.3390/sym10030068

**AMA Style**

Milton KA, Brevik I.
Casimir Energies for Isorefractive or Diaphanous Balls. *Symmetry*. 2018; 10(3):68.
https://doi.org/10.3390/sym10030068

**Chicago/Turabian Style**

Milton, Kimball A., and Iver Brevik.
2018. "Casimir Energies for Isorefractive or Diaphanous Balls" *Symmetry* 10, no. 3: 68.
https://doi.org/10.3390/sym10030068