Unitary, it should be! (copy)
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} } ；
$ 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 ；
固有値の計算
$ \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{ s } \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } …$ \LaTeX 原稿のsの定義から換えてみる
$ \Lambda_{\sigma_k}^{+} = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) + \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } $ = \sigma_k \alpha k + \lambda^{-\sigma_k}
$ \Lambda_{\sigma_k}^{-} = \sigma_k \alpha k - \sigma_k ( \frac{\alpha k}{2} + \frac{\Omega }{k} ) - \sqrt{ ( \frac{\alpha k}{2} + \frac{\Omega }{k} )^2 + 1 } $ = \sigma_k \alpha k - \lambda^{+\sigma_k}
$ \Lambda_{\sigma_k}^{s} = \sigma_k \alpha k + s \lambda^{-s\sigma_k} …ここでIC/Whの区別が$ s から$ s\sigma に変わる
$ \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} ) ；
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 + s \lambda^{-s\sigma_k} 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} - s_k \lambda^{-s_k\sigma_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 + (-s_k) \lambda^{+s_k\sigma_k} \big) \Big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( (- s_k) \lambda^{+s_k\sigma_k} \big) }
$ s_k \lambda^{s_k\sigma_k} s_k \lambda^{-s_k\sigma_k} = 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{ s_k \lambda^{s_k\sigma_k} s_k \lambda^{-s_k\sigma_k} }{\alpha \sigma_k k} - s_k \lambda^{-s_k\sigma_k} & \DS -\frac{ s_k \lambda^{s_k\sigma_k} s_k \lambda^{-s_k\sigma_k} }{\alpha \sigma_k k} \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( s_k \lambda^{s_k\sigma_k} - \sigma_k \alpha k \big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( -s_k \lambda^{s_k\sigma_k } \big) }
$ = \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( s_k \lambda^{s_k\sigma_k} - \alpha \sigma_k k \big) & \DS \frac{ \sigma_k s_k \lambda^{-s_k} }{\alpha \sigma_k k} \big( - s_k \lambda^{s_k\sigma_k} \big) \phantom{\bigg|}\!\! \\ \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( s_k \lambda^{s_k\sigma_k} - \alpha \sigma_k k \big) & \DS \frac{ \Lambda_{\sigma_k}^{s_k}(k) }{\alpha \sigma_k k} \big( - s_k \lambda^{s_k\sigma_k} \big) \phantom{\bigg|}\!\! } \AR{l}{ s_k \lambda^{s_k\sigma_k} \phantom{\bigg|}\!\! \\ s_k \lambda^{s_k\sigma_k} - \alpha \sigma_k k \phantom{\bigg|}\!\! }
$ 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^{2s_k\sigma_k} + 1 }} \AR{l}{ s_k \lambda_k^{s_k\sigma_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k\sigma_k} - \alpha \sigma_k 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 \sigma_k} + 1 }} \int e^{-i\vec k\cdot\vec x} \Big( s_k \lambda_k^{s_k\sigma_k} \bm h_{\sigma_k}(-\vec k) \cdot \bm{U} + \sigma_k 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 \sigma_k} + 1 }} \int e^{-i\vec k\cdot\vec x} \Big( s_k \lambda_k^{s_k\sigma_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) + \sigma_kk \, \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 \sigma_k} + 1 }} \Big( s_k \lambda_k^{s_k\sigma_k} U_{\sigma_k}(\vec k) + \sigma_kk \, A_{\sigma_k}(\vec k) \Big)
$ = \newcommand{\DS}{\displaystyle} \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2 s_k \sigma_k} + 1 }} \Big( s_k \lambda_k^{s_k\sigma_k} U_{\sigma_k}(\vec k) + B_{\sigma_k}(\vec k) \Big)
$ \sigma_k = +1
IC:$ (+,+) = \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2} + 1 }} \Big( \lambda_k U_{\sigma_k}(\vec k) + B_{\sigma_k}(\vec k) \Big)
Wh:$ (+,-) = \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{-2} + 1 }} \Big( - \lambda_k^{-1} U_{\sigma_k}(\vec k) + B_{\sigma_k}(\vec k) \Big) $ = \footnotesize \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \Big( - U_{\sigma_k}(\vec 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_+^{IC} \\ Z_+^{Wh} } = \AR{c}{ Z_+^+ \\ Z_+^- } = \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \AR{rr}{ \lambda_k & 1 \\ -1 & \lambda_k } \AR{rr}{ U_+ \\ B_+ }
$ \sigma_k = -1
Wh:$ (-,+) = \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{-2} + 1 }} \Big( \lambda_k^{-1} U_{\sigma_k}(\vec k) + B_{\sigma_k}(\vec k) \Big) $ = \footnotesize \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \Big( U_{\sigma_k}(\vec k) + \lambda_k B_{\sigma_k}(\vec k) \Big)
IC:$ (-,-) = \footnotesize \frac{ 1 }{\sqrt{ \lambda_k^{2} + 1 }} \Big( - \lambda_k U_{\sigma_k}(\vec 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_-^{Wh} \\ Z_-^{IC} } = \AR{c}{ Z_-^+ \\ Z_-^- } = \frac{ 1 }{\sqrt{ 1 + \lambda_k^{2} }} \AR{rr}{ 1 & \lambda_k \\ - \lambda_k & 1 } \AR{rr}{ U_- \\ B_- }
det=1は作れたけど変換に大きな違いはない。detは1行目と２行目の入れ替えるとマイナスつくので、こだわる理由がない；$ \sigma_k = -1 の展開係数の符号がかわるだけ；$ \sigma_k と$ s_k の分離が出来てない；
$ \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{l}{ s_k \lambda_k^{s_k\sigma_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k\sigma_k} - \alpha \sigma_k 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{l}{ s_k \lambda_k^{s_k\sigma_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k\sigma_k} - \alpha \sigma_k k \phantom{\Big|}\!\! } $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \AR{l}{ s_k \lambda_k^{s_k\sigma_k} \phantom{\Big|}\!\! \\ s_k \lambda_k^{s_k\sigma_k} - \alpha \sigma_k k \phantom{\Big|}\!\! }^T \AR{r}{ \DS s_k \lambda_k^{s_k\sigma_k} + \frac{1}{\alpha \sigma_k k} \phantom{\Big|}\!\! \\ \DS -\frac{1}{\alpha \sigma_k k} \phantom{\Big|}\!\! } $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize s_k \lambda_k^{s_k\sigma_k} (s_k \lambda_k^{s_k\sigma_k} + \frac{1}{\alpha \sigma_k k}) + (s_k \lambda_k^{s_k\sigma_k} - \alpha \sigma_k k)(-\frac{1}{\alpha \sigma_k k}) $ = \newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\DS}{\displaystyle} \footnotesize \lambda_k^{2s_k\sigma_k} + 1