Correction to: Algebraic Approach to Casimir Force Between Two $\delta$-like Potentials

Correction to: Ann. Henri Poincaré 22 (2021) 1751–1781 https://doi.org/10.1007/s00023-020-00994-2

In the original article, equations (92a) and (92b) contain a sign mistake, and they should have the form

$$\begin{aligned} \big ( \varphi , ( h_{ \hspace{0.05em}\textbf{a}} - h ) \psi \big )&= \frac{ 2 }{ \pi } \int \limits _{ 0 }^{ +\infty } dr \, r^{ 2 } \big ( \varphi , [ G( -r^{ 2 } ) - G_{ 0 }( -r^{ 2 } ) ] \psi \big ), \end{aligned}$$

(92a)

$$\begin{aligned} \big ( \nabla \varphi , ( h_{ \hspace{0.05em}\textbf{a}}^{ -1 } - h^{ -1 } ) \nabla \psi \big )&= -\frac{ 2 }{ \pi } \int \limits _{ 0 }^{ +\infty } dr \, \big ( \nabla \varphi , [ G( -r^{ 2 } ) - G_{ 0 }( -r^{ 2 } ) ] \nabla \psi \big ), \end{aligned}$$

(92b)

This implies that all terms on the right hand sides of the formulas (97), (98), (99), (101), (103), (104) and (108) should have opposite signs.

In addition, a 1/2 factor is missing in equation (106). It should read

$$\begin{aligned} \mathcal {E}_{ \hspace{0.1em}\text {single} }( \textbf{x}, 0 ) = \frac{ 1 }{ 4 } \int \limits _{ 0 }^{ +\infty } dr \, \frac{ 1 }{ \alpha + 2 \pi ^{ 2 } r } \left( 1 + 2 r | \hspace{0.05em} \textbf{x} \hspace{0.05em} | \right) \frac{ e^{ -2 r | \hspace{0.05em} \textbf{x} \hspace{0.05em} | } }{ | \hspace{0.05em} \textbf{x} \hspace{0.05em} | ^{ 4 } }. \end{aligned}$$

(106)

Both mistakes have been pointed out in the article by D. Fermi and L. Pizzocchero, Annales Henri Poincaré 24 (2023) 2363. It is also true, as noticed by these authors, that the corrected form of our formula (106), which refers to the simple special case of local energy outside one strictly point-like impurity, is in agreement with a formula obtained earlier for this case by these authors (contrary to the claim expressed in our original article). This fact does not have substantial consequences for the rest of our discussion in the article.

We also want to point out a few typographical errors. In the title of “Appendix D” the symbol $Q_{ \uplambda }^{ ( i ) }( k )$ should be replaced by $Q_{ \uplambda }^{ ( i ) }( a, l )$. In equation (140) the function $M_{ q / a }$ needs to be replaced by $M_{ \uplambda q / a }$. In equation (143) $n_{ \uplambda }( \uplambda l )$ should be replaced by $n_{ \uplambda }( l )$. In the same equation in the expression for $Q_{ \uplambda }^{ ( 2 ) }( a, l )$ instead of $r_{ \uplambda }( l )$ in the denominator it should be $r_{ \uplambda }( l )^{ 2 }$. None of these errors have any consequences for the rest of the article.