eigenfunction of W in HMHD

eigenvalue problem of generalized vorticity operator

in (U,V)-repr. ( where $ \footnotesize C := C_e/C_i and the eigenvalue of $ \footnotesize \nabla\times is$ \footnotesize \sigma \lambda , $ \footnotesize \sigma = \pm 1 )

$ \newcommand{\f}{\displaystyle} \footnotesize \hat W_D \left[\begin{array}{rr} U \\ V \end{array}\right] = \left[\begin{array}{rr} \f C_i\alpha\sigma\lambda + \frac{\f C_i}{\f\alpha\sigma\lambda} & - \frac{\f C_i}{\f\alpha\sigma\lambda} \\ - \frac{\f C_e}{\f\alpha\sigma\lambda} & \phantom{\Big|} \frac{\f C_e}{\f\alpha\sigma\lambda} \end{array}\right] \left[\begin{array}{rr} \phantom{\Big|\!\!} U \\ \phantom{\Big|\!\!} V \end{array}\right] = C_i\left[\begin{array}{rr} \f \alpha\sigma\lambda + \frac{\f 1}{\f\alpha\sigma\lambda} & - \frac{\f 1}{\f\alpha\sigma\lambda} \\ - \frac{\f C}{\f\alpha\sigma\lambda} & \phantom{\Big|\!\!} \frac{\f C}{\f\alpha\sigma\lambda} \end{array}\right] \left[\begin{array}{rr} \phantom{\Big|\!\!} U \\ \phantom{\Big|\!\!} V \end{array}\right] = \Lambda \left[\begin{array}{rr} U \\ V \end{array}\right]

modified eigenvalue problem

$ \newcommand{\f}{\displaystyle} \footnotesize \left[\begin{array}{rr} \f \alpha\sigma\lambda + \frac{1}{\alpha\sigma\lambda} & \f - \frac{1}{\alpha\sigma\lambda} \\ \f - \frac{C}{\alpha\sigma\lambda} & \f \frac{C}{\alpha\sigma\lambda} \end{array}\right] \left[\begin{array}{rr} U \\ V \end{array}\right] = \frac{\Lambda}{C_i} \left[\begin{array}{rr} U \\ V \end{array}\right]

$ \Longleftrightarrow $ \newcommand{\f}{\displaystyle}\newcommand{\U}{\underline} \footnotesize \left[\begin{array}{rr} \f \alpha\sigma\lambda + \frac{1}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i} & \f - \frac{1}{\alpha\sigma\lambda} \\ \f - \frac{C}{\alpha\sigma\lambda} & \f \frac{C}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i} \end{array}\right] \left[\begin{array}{c} \phantom{\Big|\!\!} U \\ \phantom{\Big|\!\!} V \end{array}\right] = \left[\begin{array}{rr} \f \frac{\U\Lambda}{C_i} -\frac{C}{\alpha\sigma\lambda} & \f - \frac{1}{\alpha\sigma\lambda} \\ \f - \frac{C}{\alpha\sigma\lambda} & \f \frac{C}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i} \end{array}\right] \left[\begin{array}{rr} U \phantom{\Big|\!\!} \\ V \phantom{\Big|\!\!} \end{array}\right] = \left[\begin{array}{rr} 0 \\ 0 \end{array}\right]

$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]}\newcommand{\D}{\displaystyle} \footnotesize \Longrightarrow \A{c}{U\\V} \propto \A{l}{\D \frac{C}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i} \phantom{\bigg|} \\ \D \frac{C}{\alpha\sigma\lambda} \phantom{\bigg|} } = \frac{\Lambda/C_i}{\alpha\sigma\lambda} \A{l}{\D \frac{\underline \Lambda}{C_i} - \alpha\sigma\lambda \phantom{\bigg|} \\ \D \frac{\underline \Lambda}{C_i} \phantom{\bigg|} } \propto \A{r}{\D \alpha\sigma\lambda - \frac{\underline \Lambda}{C_i} \phantom{\bigg|} \\ \D - \frac{\underline \Lambda}{C_i} \phantom{\bigg|} } = \A{l}{ s \gamma^{\sigma s} \phantom{\bigg|} \\ s \gamma^{\sigma s} - \alpha\sigma\lambda \phantom{\bigg|} }

the last vector components imply that, when$ \bm U is defined by$ \bm U:= s\gamma^{\sigma s} \bm\phi_\sigma , then$ \bm V = \bm U - \alpha \bm J = (s\gamma^{\sigma s} - \alpha\sigma\lambda) \bm\phi_\sigma = -\frac{\Lambda_\sigma^{-s}}{C_i}\bm\phi_\sigma then$ \bm J = \sigma\lambda \bm\phi_\sigma then$ \bm B = \bm\phi_\sigma then$ \alpha\bm\Omega + \bm B = (\alpha\sigma\lambda s \gamma^{\sigma s} + 1) \bm\phi_\sigma= (\underbrace{ \alpha\sigma\lambda + s \gamma^{-\sigma s}}_{\Lambda_\sigma^s/C_i} ) s \gamma^{\sigma s} \bm\phi_\sigma = \frac{\Lambda_\sigma^s}{C_i} \bm U then$ \frac{\Lambda_\sigma^s}{C_i} \bm V = -\frac{\Lambda_\sigma^{-s}}{C_i} \frac{\Lambda_\sigma^s}{C_i} \bm\phi_\sigma = - C \bm B

eigenvalue

$ \footnotesize \frac{\Lambda}{C_i} = \frac{\Lambda_{\sigma}^s}{C_i} := \frac12\left(\alpha\sigma\lambda + \frac{1+C}{\alpha\sigma\lambda}\right) + \frac{s}{2} \sqrt{ \left( \alpha\sigma\lambda + \frac{1+C}{\alpha\sigma\lambda} \right)^2 - 4 C }

$ \footnotesize = \sigma \left( \frac{\alpha\lambda}{2} + \frac{1+C}{2\alpha\lambda} \right) + s \sqrt{ \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right)^2 + 1 }

（cf. graph of $ x+\frac ax+s\sqrt{(x-\frac ax)^2+1} ：）

$ \footnotesize = \sigma \alpha \lambda + s \bigg[ \sqrt{ \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right)^2 + 1 } - \sigma s \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right) \bigg]

$ \footnotesize = \sigma \alpha \lambda + s \gamma^{ - \sigma s } where$ \footnotesize \gamma := \sqrt{ \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right)^2 + 1 } + \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right)

$ \footnotesize = \frac{1+C}{\alpha\sigma\lambda} + \sigma \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right) + s \sqrt{ \left( \frac{\alpha\lambda}{2} - \frac{1+C}{2\alpha\lambda} \right)^2 + 1 }

$ \footnotesize = \frac{1+C}{\alpha\sigma\lambda} + s \gamma(\lambda;\alpha,C)^{\sigma s} \ \overset{C=-1}{\longrightarrow} \ = s \gamma(\lambda;\alpha,0)^{\sigma s}

if $ \footnotesize C = - 1 ,$ \footnotesize \frac{\Lambda}{C_i} = \sigma \alpha \lambda + s \bigg[ \sqrt{ \left( \frac{\alpha\lambda}{2} \right)^2 + 1 } - \sigma s \frac{\alpha\lambda}{2} \bigg] = s \bigg[ \sqrt{ \left( \frac{\alpha\lambda}{2} \right)^2 + 1 } + \sigma s \frac{\alpha\lambda}{2} \bigg] = s \gamma^{\sigma s}

"conjugate" eigenvalues of the operator

underline notation is defined by$ \footnotesize \underline\Lambda := \Lambda_\sigma^{-s} when$ \footnotesize \Lambda := \Lambda_\sigma^s .

$ \footnotesize \newcommand{\U}{\underline} \frac{\Lambda}{C_i} + \frac{\U\Lambda}{C_i} = \alpha\sigma\lambda + \frac{1+C}{\alpha\sigma\lambda} ,$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline} \f \frac{\Lambda}{C_i}\frac{\U\Lambda}{C_i} = C ,

check the eigenfunction coefficient

$ \newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \footnotesize ( \frac{\hat W}{C_i} - \frac{\Lambda}{C_i}\hat I ) \vec\Psi = \left[\begin{array}{rr} \d \frac{\U\Lambda}{C_i} -\frac{C}{\alpha\sigma\lambda} & \d - \frac{1}{\alpha\sigma\lambda} \\ \d - \frac{C}{\alpha\sigma\lambda} & \d \frac{C}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i} \end{array}\right] \left[\begin{array}{rr} \d -\frac{\U\Lambda}{C_i} + {\alpha\sigma\lambda} \\ \d -\frac{\U\Lambda}{C_i}\end{array}\right] $ \newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \footnotesize = \left[\begin{array}{rr} \d (\frac{\U\Lambda}{C_i} -\frac{C}{\alpha\sigma\lambda})(-\frac{\U\Lambda}{C_i} + {\alpha\sigma\lambda}) + (- \frac{1}{\alpha\sigma\lambda}) (-\frac{\U\Lambda}{C_i}) \\ \d (- \frac{C}{\alpha\sigma\lambda})(-\frac{\U\Lambda}{C_i} + {\alpha\sigma\lambda}) + (\frac{C}{\alpha\sigma\lambda} - \frac{\Lambda}{C_i}) (-\frac{\U\Lambda}{C_i}) \end{array}\right]_{\phantom{\Large q}}

$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f = \left[\begin{array}{rr} \d -(\frac{\U\Lambda}{C_i})^2 + (\frac{C+1}{\f\alpha\sigma\lambda} + {\f\alpha\sigma\lambda})(\frac{\U\Lambda}{C_i} ) - C \\ \d - C + \frac{\Lambda}{C_i}\frac{\U\Lambda}{C_i} \end{array}\right] = \left[\begin{array}{rr} 0 \phantom{\Big|\!\!} \\ 0 \phantom{\Big|\!\!} \end{array}\right]

obtained eigenvector of generalized velocity

$ \newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \footnotesize \left[\begin{array}{c} \bm U_\sigma^s \phantom{\Big|\!\!} \\ \bm V_\sigma^s \phantom{\Big|\!\!} \end{array}\right] = Z_\sigma^s(\lambda; t) \left[\begin{array}{r} \d \alpha\sigma\lambda - \frac{\U\Lambda}{C_i} \\ \d -\frac{\U\Lambda}{C_i} \end{array}\right] \bm\phi_\sigma(\lambda; x) ,

obtained eigenvector of generalized momentum

$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f \left[\begin{array}{r} U + A/\alpha \\ -A/\alpha \end{array}\right]^* = \overline{Z_\sigma^s(\lambda; t)} \left[\begin{array}{r} \d -\frac{\U\Lambda}{C_i} + {\f\alpha\sigma\lambda} + \frac{1}{\f\alpha\sigma\lambda} \\ \d -\frac{1}{\f\alpha\sigma\lambda} \end{array}\right] \overline{\phi_\sigma(\lambda; x)} $ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f = \overline{Z_\sigma^s(\lambda; t)} \left[\begin{array}{r} \d \frac{\Lambda}{C_i} - \frac{C}{\f\alpha\sigma\lambda} \\ \d -\frac{1}{\f\alpha\sigma\lambda} \end{array}\right] \overline{\phi_\sigma(\lambda; x)}

check eigenvalue relation (by operating generalized curl to generalized momentum vector)

$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f \left[\begin{array}{cc} \d C_i \alpha\sigma\lambda & 0 \\ 0 & \d C_e \alpha\sigma\lambda \end{array}\right] \left[\begin{array}{r} \d \frac{\Lambda}{C_i} - \frac{C}{\f\alpha\sigma\lambda} \\ \d -\frac{1}{\f\alpha\sigma\lambda} \end{array}\right] = C_i \left[\begin{array}{cc} \d \alpha\sigma\lambda & 0 \phantom{\Big|} \\ 0 \phantom{\Big|} & \d C \alpha\sigma\lambda \end{array}\right] \left[\begin{array}{r} \d \frac{\Lambda}{C_i} - \frac{C}{\f\alpha\sigma\lambda} \\ \d -\frac{1}{\f\alpha\sigma\lambda} \end{array}\right] = C_i \left[\begin{array}{r} \d {\f\alpha\sigma\lambda}\frac{\Lambda}{C_i} - {C} \\ \d -{C} \end{array}\right] = C_i \frac{\Lambda}{C_i} \left[\begin{array}{r} \d {\f\alpha\sigma\lambda} - \frac{\U\Lambda}{C_i} \\ \d -\frac{\U\Lambda}{C_i} \end{array}\right]

固有関数を(U,B)で書くときれいな見た目

$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \A{c}{ \bm U(\vec k) \\ \bm B(\vec k) } = \sum_{\sigma=\pm1,s=\pm1} \frac{ Z(\vec k,\sigma,s) }{ \sqrt{1+\gamma^{2\sigma}} } \A{r}{ s \gamma^{s\sigma} \bm\phi_\sigma(\vec k) \\ \bm\phi_\sigma(\vec k) }

$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} = \frac{ Z(\vec k,+,+) }{ \sqrt{1+\gamma^{2}} } \A{r}{ \gamma \bm\phi_+(\vec k) \\ \bm\phi_+(\vec k) } + \frac{ Z(\vec k,+,-) }{ \sqrt{\gamma^{2}+1} } \A{r}{ - \bm\phi_+(\vec k) \\ \gamma \bm\phi_+(\vec k) } + \frac{ Z(\vec k,-,+) }{ \sqrt{\gamma^{2}+1} } \A{r}{ \bm\phi_-(\vec k) \\ \gamma \bm\phi_-(\vec k) } + \frac{ Z(\vec k,-,-) }{ \sqrt{1+\gamma^{2}} } \A{r}{ \gamma \bm\phi_-(\vec k) \\ - \bm\phi_-(\vec k) }

↑これはColiorisが入っていてもOK

$ \newcommand{\A}[2]{\left[\begin{array}{#1}#2\end{array}\right]} \A{r}{ s' \gamma^{s'\sigma} \bm\phi_\sigma(\vec k') & \bm\phi_\sigma(\vec k') } \A{r}{ s \gamma^{s\sigma} \bm\phi_\sigma(\vec k) \\ \bm\phi_\sigma(\vec k) } = ( s' \gamma^{s'\sigma} s \gamma^{s\sigma} + 1 ) \bm\phi_\sigma(\vec k') \cdot \bm\phi_\sigma(\vec k)

if $ s' = s then$ s' \gamma^{s'\sigma} s \gamma^{s\sigma} = \gamma^{2\sigma}

if $ s' = - s then$ s' \gamma^{s'\sigma} s \gamma^{s\sigma} = - \gamma^{0} = - 1

integrand of natural pair of general velocities and momenta

$ U(U+A/\alpha)+V(-A/\alpha)=U^2+A(U-V)/\alpha=U^2+AJ

substituting the eigenvector

$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f (-\frac{\U\Lambda}{C_i} +{\f\alpha\sigma\lambda})( -\frac{\U\Lambda}{C_i} + {\f\alpha\sigma\lambda} + \frac{1}{\f\alpha\sigma\lambda}) + (-\frac{\U\Lambda}{C_i})(-\frac{1}{\f\alpha\sigma\lambda}) = (-\frac{\U\Lambda}{C_i} +{\f\alpha\sigma\lambda})^2 + (-\frac{\U\Lambda}{C_i} +{\f\alpha\sigma\lambda})(\frac{1}{\f\alpha\sigma\lambda}) + (-\frac{\U\Lambda}{C_i})(-\frac{1}{\f\alpha\sigma\lambda})

$ \newcommand{\f}{\footnotesize}\newcommand{\U}{\underline}\newcommand{\d}{\displaystyle} \f = \boxed{(-\frac{\U\Lambda}{C_i} +{\f\alpha\sigma\lambda})^2 + 1} as is expected