Expressions for translation operators (to be checked)
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Marek Nečada
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@ -713,7 +713,7 @@ Abstract.
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\end_layout
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\begin_layout Itemize
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Translation operators: explicit expression, also in sph.
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Translation operators: rewrite in sph.
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harm.
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convention independent form.
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\end_layout
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@ -290,8 +290,8 @@ outgoing
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, respectively, defined as follows:
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\begin_inset Formula
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\begin{align*}
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\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh1lm\left(\uvec r\right),\\
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\vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),
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\end{align*}
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\end_inset
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@ -299,8 +299,8 @@ outgoing
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\begin_inset Formula
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\begin{align*}
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\vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\\
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\vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),\\
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
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\end{align*}
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@ -326,9 +326,9 @@ vector spherical harmonics
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\begin_inset Formula
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\begin{align*}
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\vsh1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\
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\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
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\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
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\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\
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\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
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\end{align*}
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\end_inset
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@ -452,7 +452,7 @@ noprefix "false"
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\end_inset
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inside a ball
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\begin_inset Formula $\openball0{R^{>}}$
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\begin_inset Formula $\openball 0{R^{>}}$
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\end_inset
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with radius
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@ -470,7 +470,7 @@ noprefix "false"
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\end_inset
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to have a complete basis of the solutions in the volume
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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.
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@ -496,7 +496,7 @@ The single-particle scattering problem at frequency
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\end_inset
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and let the whole volume
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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be filled with a homogeneous isotropic medium with wave number
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@ -532,7 +532,7 @@ If there was no scatterer and
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\end_inset
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due to sources outside
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\begin_inset Formula $\openball0R$
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\begin_inset Formula $\openball 0R$
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\end_inset
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would remain.
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@ -793,7 +793,7 @@ literal "true"
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.
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Let the field in
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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have expansion as in
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@ -812,7 +812,7 @@ noprefix "false"
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\end_inset
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to
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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via by electromagnetic radiation is
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@ -1535,10 +1535,37 @@ outside.
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\begin_layout Standard
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In our convention, the regular translation operator can be expressed explicitly
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as
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as (TODO CHECK CAREFULLY FOR POSSIBLE
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\begin_inset Formula $(-1)^{m'}$
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\end_inset
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AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{multline}
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\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
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\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
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\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}.\label{eq:translation operator}
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\end{multline}
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\end_inset
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