Unitary, it should be!
一般化渦度演算子(複素ヘリカル波を基底にとる)
$ \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \hat W = \AR{cc}{ \DS \alpha \sigma_k k + \frac{1}{\alpha \sigma_k k} & \DS -\frac{1}{\alpha \sigma_k k} \\ \DS \frac{2 \alpha \Omega + 1}{\alpha \sigma_k k} & \DS -\frac{2 \alpha \Omega + 1}{\alpha \sigma_k k} } ；
固有値の計算
$ tr = \alpha \sigma_k k - \frac{2 \Omega }{\sigma_k k} ，$ det = - ( 2 \alpha \Omega + 1 ) ；$ \Lambda^2 - 2 \sigma_k ( \frac{\alpha k}{2} - \frac{\Omega }{k} ) \Lambda - ( 2 \alpha \Omega + 1 ) = 0 ；
$ \Lambda = \sigma_k ( \frac{\alpha k}{2} - \frac{\Omega }{k} ) \pm \sqrt{ ( \frac{\alpha k}{2} - \frac{\Omega }{k} )^2 + ( 2 \alpha \Omega + 1 ) }
$ = \sigma_k ( \frac{\alpha k}{2} - \frac{\Omega }{k} ) \pm \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 }
$ = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) \pm \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 }
$ = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) + \boxed{ \sigma_k s } \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } …$ \LaTeX 原稿のsの定義
$ \Lambda_{\sigma_k}^{+} = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) + \sigma_k \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } $ = \sigma_k \alpha k + \sigma_k \lambda^{-1}
$ \Lambda_{\sigma_k}^{-} = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) - \sigma_k \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } $ = \sigma_k \alpha k - \sigma_k \lambda
$ \Lambda_{\sigma_k}^{s} = \sigma_k \alpha k + \sigma_k s \lambda^{-s} …経験的にはこの括り方の方が渦度演算子の固有ベクトル、ひいては速度場、磁場の展開の計算との「親和性」が高いと思える；
$ \lambda = ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) + \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } ；$ \lambda^{-1} = \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } - ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) ；
$ \Omega\to0 の極限を考えるなら$ \Lambda_{\sigma_k}^{s}=\sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) + \sigma_k s \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } - \sigma_k \frac{2\Omega }{k} = \sigma_k s \lambda^{+s} - \sigma_k \frac{2\Omega }{k}
固有ベクトルの計算
$ \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \hat W - \Lambda_{\sigma_k}^{s_k}(k) I = \AR{cc}{ \DS \alpha \sigma_k k + \frac{1}{\alpha \sigma_k k} - \Lambda_{\sigma_k}^{s_k}(k) & \DS -\frac{1}{\alpha \sigma_k k} \\ \DS \frac{2 \alpha \Omega + 1}{\alpha \sigma_k k} & \DS -\frac{2 \alpha \Omega + 1}{\alpha \sigma_k k} - \Lambda_{\sigma_k}^{s_k}(k) }
$ - ( 2 \alpha \Omega + 1 )= \Lambda_{\sigma_k}^{s_k}(k) \Lambda_{\sigma_k}^{-s_k}(k) なので$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \alpha \sigma_k k + \frac{1}{\alpha \sigma_k k} - \Lambda_{\sigma_k}^{s_k}(k) & \DS -\frac{1}{\alpha \sigma_k k} \\ \DS -\frac{ \Lambda_{\sigma_k}^{s_k}(k) \Lambda_{\sigma_k}^{-s_k}(k) }{\alpha \sigma_k k} & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) \Lambda_{\sigma_k}^{-s_k}(k) }{\alpha \sigma_k k} - \Lambda_{\sigma_k}^{s_k}(k) }
$ \Lambda_{\sigma_k}^{s_k}(k) で括って$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \alpha \sigma_k k + \frac{1}{\alpha \sigma_k k} - \Lambda_{\sigma_k}^{s_k}(k) & \DS -\frac{1}{\alpha \sigma_k k} \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( -\Lambda_{\sigma_k}^{-s_k}(k) \big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( \Lambda_{\sigma_k}^{-s_k}(k) - \alpha \sigma_k k \big) }
$ \Lambda_{\sigma_k}^{s} = \sigma_k \alpha k + \sigma_k s \lambda^{-s} を代入して$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \frac{1}{\alpha \sigma_k k} - \sigma_k s_k \lambda^{-s_k} & \DS -\frac{1}{\alpha \sigma_k k} \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \Big( - \big( \sigma_k \alpha k + \sigma_k (-s_k) \lambda^{+s_k} \big) \Big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( \sigma_k (- s_k) \lambda^{+s_k} \big) }
$ \sigma_k s \lambda^{+s} \sigma_k s \lambda^{-s} = 1 を代入して$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \frac{ \sigma_k s_k \lambda^{s_k} \sigma_k s_k \lambda^{-s_k} }{\alpha \sigma_k k} - \sigma_k s_k \lambda^{-s_k} & \DS -\frac{ \sigma_k s_k \lambda^{s_k} \sigma_k s_k \lambda^{-s_k} }{\alpha \sigma_k k} \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( \sigma_k s_k \lambda^{s_k} - \sigma_k \alpha k \big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( - \sigma_k s_k \lambda^{s_k} \big) }
$ \sigma_k s \lambda^{-s} で括って$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \frac{ \sigma_k s_k \lambda^{-s_k} }{\alpha \sigma_k k} \big( \sigma_k s_k \lambda^{s_k} - \alpha \sigma_k k \big) & \DS \frac{ \sigma_k s_k \lambda^{-s_k} }{\alpha \sigma_k k} \big( - \sigma_k s_k \lambda^{s_k} \big) \phantom{\bigg|}\!\! \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( \sigma_k s_k \lambda^{s_k} - \sigma_k \alpha k \big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( - \sigma_k s_k \lambda^{s_k} \big) \phantom{\bigg|}\!\! } \AR{l}{ s_k \lambda^{s_k} \phantom{\bigg|}\!\! \\ s_k \lambda^{s_k} - \alpha k \phantom{\bigg|}\!\! }
$ \sigma_k を約分して$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{cc}{ \DS \frac{ s_k \lambda^{-s_k} }{\alpha k} \big( s_k \lambda^{s_k} - \alpha k \big) & \DS \frac{ s_k \lambda^{-s_k} }{\alpha k} \big( - s_k \lambda^{s_k} \big) \phantom{\bigg|}\!\! \\ \DS \frac{ \Lambda_{}^{s_k}(k) }{\alpha k} \big( s_k \lambda^{s_k} - \alpha k \big) & \DS \frac{ \Lambda_{}^{s_k}(k) }{\alpha k} \big( - s_k \lambda^{s_k} \big) \phantom{\bigg|}\!\! } \AR{l}{ s_k \lambda^{s_k} \phantom{\bigg|}\!\! \\ s_k \lambda^{s_k} - \alpha k \phantom{\bigg|}\!\! }
固有ベクトルの成分を$ \sigma_k に依存しない形にできるぶんだけ、こちらの固有値の$ s_k の定義がよさげ；
$ F(\vec k,\sigma_k) \bm h(\vec k,\sigma_k) e^{i\vec k\cdot\vec x}
磁場、速度場のペアを固有モードに変換する
$ \footnotesize Z_{\sigma_k}^{s_k}(\vec k) = \int {}^t\vec{\bm{\Phi}}_{\sigma_k}^{s_k}(-\vec k) \, \hat M \, \vec{\bm{V}} \, {\rm{d}}^3\vec x
$ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \int \frac{\bm h_{\sigma_k}^*(\vec k) e^{-i\vec k \cdot \vec x}}{\sqrt{ \lambda_k^{2 s_k} + 1 }} \AR{l}{ s_k \lambda_k^{s_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k} - \alpha k \phantom{\Big|}\!\! }^T \AR{r}{ \DS \bm U + \frac{\bm A}{\alpha} \phantom{\Big|}\!\! \\ \DS - \frac{\bm A}{\alpha} \phantom{\Big|}\!\! } {\rm{d}}^3\vec x
$ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2 s_k} + 1 }} \int e^{-i\vec k\cdot\vec x} \Big( s_k \lambda_k^{s_k} \bm h_{\sigma_k}(-\vec k) \cdot \bm{U} + k \, \bm h_{\sigma_k}(-\vec k) \cdot \bm{A} \Big) {\rm{d}}^3\vec x
$ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2 s_k} + 1 }} \int e^{-i\vec k\cdot\vec x} \Big( s_k \lambda_k^{s_k} \bm h(-\vec k,\sigma_k) \cdot \big( U(\vec k,\sigma_k) \bm h(\vec k,\sigma_k) e^{i\vec k\cdot\vec x} \big) + k \, \bm h_{\sigma_k}(-\vec k) \cdot \big( A(\vec k,\sigma_k) \bm h(\vec k,\sigma_k) e^{i\vec k\cdot\vec x} \big) \Big) {\rm{d}}^3\vec x
$ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2 s_k} + 1 }} \Big( s_k \lambda_k^{s_k} U_{\sigma_k}(\vec k) + k \, A_{\sigma_k}(\vec k) \Big)
$ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2 s_k} + 1 }} \Big( s_k \lambda_k^{s_k} U_{\sigma_k}(\vec k) + \sigma_k B_{\sigma_k}(\vec k) \Big) $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \int \frac{\bm h_{\sigma_k}^*(\vec k) e^{-i\vec k \cdot \vec x}}{\sqrt{ \lambda_k^{2 s_k} + 1 }} \AR{l}{ s_k \lambda_k^{s_k} \\ \sigma_k }^T \AR{r}{ \bm U \\ \bm B } {\rm{d}}^3\vec x
IC:$ (+) = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2} + 1 }} \Big( \lambda_k U_{\sigma_k}(\vec k) + \sigma_k B_{\sigma_k}(\vec k) \Big)
Wh:$ (-) = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{-2} + 1 }} \Big( - \lambda_k^{-1} U_{\sigma_k}(\vec k) + \sigma_k B_{\sigma_k}(\vec k) \Big) $ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \Big( - U_{\sigma_k}(\vec k) + \sigma_k \lambda_k B_{\sigma_k}(\vec k) \Big)
$ \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{c}{ Z_{\sigma_k}^{IC} \\ Z_{\sigma_k}^{Wh} } = \AR{c}{ Z_{\sigma_k}^+ \\ Z_{\sigma_k}^- } = \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \AR{rr}{ \lambda_k & \sigma_k \\ -1 & \sigma_k \lambda_k } \AR{rr}{ U_{\sigma_k} \\ B_{\sigma_k} } $ \cdots det = \sigma_k
$ \footnotesize \boxed{ 理想的には \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \AR{c}{ Z_{\sigma_k}^+ \\ Z_{\sigma_k}^- } = \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \AR{rr}{ \lambda_k & \sigma_k \\ -\sigma_k & \lambda_k } \AR{rr}{ U_{\sigma_k} \\ B_{\sigma_k} } \cdots det = 1 であって欲しい }
…と思っていたが、2x2行列なので、行の交換でdetの符号はいくらでもかわるので、こだわる理由が無い；
$ \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \hat M = \AR{rr}{ \DS 1 + \frac{1}{(\alpha\sigma k)^2} & \DS -\frac{1}{(\alpha\sigma k)^2} \\ \DS -\frac{1}{(\alpha\sigma k)^2} & \DS \frac{1}{(\alpha\sigma k)^2} } $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{rr}{ \DS 1 + \frac{1}{(\alpha k)^2} & \DS -\frac{1}{(\alpha k)^2} \\ \DS -\frac{1}{(\alpha k)^2} & \DS \frac{1}{(\alpha k)^2} }
$ \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{r}{ s_k \lambda_k^{s_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k} - \alpha k \phantom{\Big|}\!\! }^T \AR{rr}{ \DS 1 + \frac{1}{(\alpha k)^2} & \DS -\frac{1}{(\alpha k)^2} \\ \DS -\frac{1}{(\alpha k)^2} & \DS \frac{1}{(\alpha k)^2} } \AR{r}{ s_k \lambda_k^{s_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k} - \alpha k \phantom{\Big|}\!\! } $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{r}{ s_k \lambda_k^{s_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k} - \alpha k \phantom{\Big|}\!\! }^T \AR{r}{ \DS s_k \lambda_k^{s_k} + \frac{1}{\alpha k} \phantom{\Big|}\!\! \\ \DS -\frac{1}{\alpha k} \phantom{\Big|}\!\! } $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize s_k \lambda_k^{s_k} (s_k \lambda_k^{s_k} + \frac{1}{\alpha k}) + (s_k \lambda_k^{s_k} - \alpha k)(-\frac{1}{\alpha k}) $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \lambda_k^{2s_k} + 1