<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%è¡¨é¡Œ  SPMODEL ãƒ¬ãƒ•ã‚¡ãƒ¬ãƒ³ã‚¹ãƒžãƒ‹ãƒ¥ã‚¢ãƒ«
%
%      3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã®æ‹¡æ•£æ–¹ç¨‹å¼ã®è§£æžè§£
%      ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã‚’åˆ©ç”¨ã—ãŸè¨ˆç®—ãƒã‚§ãƒƒã‚¯ç”¨
%
%å±¥æ­´  2006/02/16 ä½œæˆ  ç«¹åºƒçœŸä¸€
%      2006/02/25 æ”¹è¨‚ 
%
\documentclass[a4j,12pt]{jarticle}
\usepackage{Dennou6}

\Dtitle[çƒæ®»é&nbsp;˜åŸŸã§ã®æ‹¡æ•£æ–¹ç¨‹å¼è§£]
       {3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã§ã®\\æ‹¡æ•£åž‹æ–¹ç¨‹å¼ã®è§£æžè§£}
\Dauthor{ç«¹åºƒçœŸä¸€}
\Ddate{å¹³æˆ 18 å¹´ 2 æœˆ 25 æ—¥}

\Dparskip

\begin{document}

\maketitle

ã“ã®æ–‡ç«&nbsp;ã§ã¯, çƒé¢èª¿å’Œå‡½æ•°--ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã‚’åˆ©ç”¨ã—ãŸ
3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã§ã®æµä½“è¨ˆç®—ã®ãŸã‚ã® SPMODEL ãƒ©ã‚¤ãƒ–ãƒ©ãƒª(spml)ã®
ãƒ¢ã‚¸ãƒ¥ãƒ¼ãƒ« wt\_module ã®ãƒ†ã‚¹ãƒˆã®ãŸã‚ã«, 
3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã§ã®æ‹¡æ•£åž‹æ–¹ç¨‹å¼ã®ã•ã¾ã–ã¾ãªå¢ƒç•Œæ¡ä»¶ã«å¯¾ã™ã‚‹
è§£ã‚’(ã§ãã‚‹ã&nbsp;ã‘)è§£æžçš„ã«æ±‚ã‚ã‚‹ã“ã¨ã‚’è¡Œã†. 

  \section{æ‹¡æ•£æ–¹ç¨‹å¼}

    åŠå¾„ $r_i,r_o$ ã®åŒå¿ƒçƒé¢ã®é–“ã® 3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã§ã®æ‹¡æ•£å•é¡Œã‚’è€ƒãˆã‚‹. 
    æ”¯é…æ–¹ç¨‹å¼ã¯ 3 æ¬¡å…ƒæ‹¡æ•£æ–¹ç¨‹å¼
    %
    \begin{equation}
      \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼}
      \DP{\phi}{t} = \kappa \Dlapla \phi
    \end{equation}
    %
    $\phi$ ã¯æ‹¡æ•£ã™ã‚‹ç‰©ç†é‡ã§ã‚ã‚Š, ä¾‹ãˆã°æ¸©åº¦, çµ„æˆæ¿ƒåº¦, ç£å&nbsp;´ãªã©ã§ã‚ã‚‹.
    $x$, $y$ ã¯ãã‚Œãžã‚Œå£ã«å¹³è¡ŒãŠã‚ˆã³åž‚ç›´æ–¹å‘ã®åº§æ¨™, 
    $\kappa$ ã¯æ‹¡æ•£çŽ‡(æ­£æ•°)ã§ã‚ã‚‹. 
    $\Dlapla$ ã¯ 3 æ¬¡å…ƒãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã§ã‚ã‚Š, 
    ç·¯åº¦çµŒåº¦å‹•å¾„åº§æ¨™ç³»$ (\lambda,\varphi,r)$ ã§ã®è¡¨ç¾ã¯
    %
    \begin{equation}
      \Dlapla = \frac{1}{r^2}\DP{}{r}\left(r^2\DP{}{r}\right)
             + \frac{1}{r^2\cos\varphi}\DP{}{\varphi}
                 \left(\cos\varphi\DP{}{\varphi}\right)
             + \frac{1}{r^2\cos^2\varphi}\DP[2]{}{\lambda}.
    \end{equation}
    %

    ä¸€èˆ¬çš„ãªå¢ƒç•Œæ¡ä»¶ã¯, å†…å¤–å¢ƒç•Œçƒé¢ã§ã®ç‰©ç†é‡ã¨ãã® $r$ å¾®åˆ†ã®
    ç·šå½¢çµåˆã§ä¸Žãˆã‚‰ã‚Œã‚‹(æ··åˆåž‹å¢ƒç•Œæ¡ä»¶). 
    %
    \begin{eqnarray}
      \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸‹}
      &amp; &amp; \alpha_{1,i}\DP{\phi}{r} + \alpha_{0,i}\phi =0, 
          \quad {\rm at} \quad r=r_i,\\
      \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸Š}
      &amp; &amp; \alpha_{1,o}\DP{\phi}{r} + \alpha_{0,o}\phi =0, 
          \quad {\rm at} \quad r=r_o,
    \end{eqnarray}
    %
    ã“ã“ã§ $\alpha_{i,j}$ ã¯å®šæ•°ä¿‚æ•°ã§ã‚ã‚Š, ä»¥ä¸‹ã®æ¡ä»¶ã‚’æº€ãŸã—ã¦ã„ã‚‹
    ã‚‚ã®ã¨ã™ã‚‹. 
    %
    \begin{equation}
      \alpha_{1,o}\cdot\alpha_{0,o} \ge 0, \quad
      \alpha_{1,i}\cdot\alpha_{0,i} \le 0, \quad
    \end{equation}
    %
    ãŸã&nbsp;ã— $\alpha_{1,i}$ ã¨ $\alpha_{0,i}$, 
    ãŠã‚ˆã³ $\alpha_{1,o}$ ã¨ $\alpha_{0,o}$ ã¯åŒæ™‚ã«ã¯
    0 ã«ã¯ãªã‚‰ãªã„ã‚‚ã®ã¨ã™ã‚‹. 

    \subsection{çƒé¢èª¿å’Œé–¢æ•°å±•é–‹}

      ç·¯åº¦ãƒ»çµŒåº¦æ–¹å‘ã«ã¯çƒé¢èª¿å’Œé–¢æ•°ã§å±•é–‹ã—ã¦è§£ãã“ã¨ãŒã§ãã‚‹. 
      ä»¥ä¸‹, ãã® 1 æ³¢æ•°æˆåˆ†ã®ã¿å–ã‚Šã&nbsp;ã—ã¦æ‰±ã†. 
      æ™‚é–“ã«é–¢ã—ã¦ã‚‚æŒ‡æ•°åž‹ã®è§£ã‚’ä»®å®šã—, 
      %
      \begin{equation}
        \Deqlab{çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \phi(x,y,t)=\tilde{\phi}_{lm}(r)Y_{lm}(\lambda,\varphi)e^{\sigma t}
      \end{equation}
      %
      ã“ã‚Œã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, 
      %
      \begin{equation}
        \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
          \sigma \tilde{\phi}_{lm} = 
              \kappa \left[
                \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\phi}_{lm}}{r}\right)
                  -\frac{l(l+1)}{r^2}\tilde{\phi}_{lm}\right],
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ã¯, 
      %
      \begin{eqnarray}
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸Šçƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        &amp; &amp; \alpha_{1,i}\DD{\tilde{\phi}_{lm}}{y} 
             + \alpha_{0,i}\tilde{\phi}_{lm} =0 
               \quad {\rm at} \quad y=r_i,\\
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸‹çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        &amp; &amp; \alpha_{1,o}\DD{\phi}{y} 
              + \alpha_{0,o}\tilde{\phi}_{lm} =0
               \quad {\rm at} \quad y=r_o.
      \end{eqnarray}
      %

    \subsection{æœ€å¤§æˆé•·çŽ‡ã®è¦‹ç©ã‚‚ã‚Š}

      æˆé•·çŽ‡ã®æœ€å¤§å€¤ã‚’é&nbsp;˜åŸŸç©åˆ†ã‹ã‚‰è¦‹ç©ã‚‚ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹} $r^2\tilde{\phi}_{lm}^*$ ã‚’ã‹ã‘ã¦
      $r$ æ–¹å‘ã«å…¨é&nbsp;˜åŸŸç©åˆ†ã‚’è¡Œã„, éƒ¨åˆ†ç©åˆ†ã‚’ç¹°ã‚Šè¿”ã™ã¨, 
      %
      \begin{eqnarray*}
        \sigma \int_{r_i}^{r_o} r^2|\tilde{\phi}_{lm}|^2 dr &amp;=&amp;
               \int_{r_i}^{r_o} dr r^2 \tilde{\phi}_{lm}^*  
                \kappa \left[
                    \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\phi}_{lm}}{r}\right)
                   -\frac{l(l+1)}{r^2}\tilde{\phi}_{lm}\right]\\
        &amp;=&amp; \kappa \int_{r_i}^{r_o} dr \tilde{\phi}_{lm}^*
                     \DD{}{r}\left(r^2\DD{\tilde{\phi}_{lm}}{r}\right)
             -\kappa l(l+1) \int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr\\
        &amp;=&amp;  \kappa\left[
                  r^2\tilde{\phi}_{lm}^*\DD{\tilde{\phi}_{lm}}{r}\right]_{r_i}^{r_o} 
             -\kappa \int_{r_i}^{r_o} 
                  r^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2 dr
             -\kappa l(l+1) \int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr\\
      \end{eqnarray*}
      %
      ã“ã“ã§, å¢ƒç•Œæ¡ä»¶ã‚’ç”¨ã„ã¦å³è¾ºç¬¬ 1 é&nbsp;…ç›®ã‚’è¦‹ç©ã‚‚ã‚‹ã¨
      %
      \begin{eqnarray*}
        &amp; &amp; \left[r^2\tilde{\phi}_{lm}^*\DD{\tilde{\phi}_{lm}}{y}\right]_{r_i}^{r_o}\\
        &amp;=&amp; \left\{
        \begin{array}{cc}
          \Ddsty -\frac{\alpha_{1,o}}{\alpha_{0,o}}
               r_o^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2_{r=r_o}
          &amp; (\alpha_{0,o}\neq0)\\
          \Ddsty -\frac{\alpha_{0,o}}{\alpha_{1,o}}
               r_o^2|\tilde{\phi}_{lm}|^2_{r=r_o}
          &amp; (\alpha_{1,o}\neq0)
        \end{array}
        \right\}
        - \left\{
        \begin{array}{cc}
          \Ddsty -\frac{\alpha_{1,i}}{\alpha_{0,i}}
               r_i^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2_{r=r_i}
          &amp; (\alpha_{0,i}\neq0)\\
          \Ddsty -\frac{\alpha_{0,i}}{\alpha_{1,i}}
               r_i^2|\tilde{\phi}_{lm}|^2_{r=r_i}
          &amp; (\alpha_{1,i}\neq0)
        \end{array}
        \right\}\\
        &amp;=&amp; -\left\{
        \begin{array}{cc}
          \Ddsty \frac{\alpha_{1,o}}{\alpha_{0,o}}
               r_o^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2_{r=r_o}
          &amp; (\alpha_{0,o}\neq0)\\
          \Ddsty \frac{\alpha_{0,o}}{\alpha_{1,o}}
               r_o^2|\tilde{\phi}_{lm}|^2_{r=r_o}
          &amp; (\alpha_{1,o}\neq0)
        \end{array}
        \right\}
        + \left\{
        \begin{array}{cc}
          \Ddsty  \frac{\alpha_{1,i}}{\alpha_{0,i}}
               r_i^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2_{r=r_i}
          &amp; (\alpha_{0,i}\neq0)\\
          \Ddsty \frac{\alpha_{0,i}}{\alpha_{1,i}}
               r_i^2 |\tilde{\phi}_{lm}|^2_{r=r_i}
          &amp; (\alpha_{1,i}\neq0)
        \end{array}
        \right\}\\
        &amp; \le &amp; 0.
      \end{eqnarray*}
      %
      ã—ãŸãŒã£ã¦, 
      %
      \begin{eqnarray*}
        \sigma 
        &amp;\le&amp; 
             -\kappa \int_{r_i}^{r_o} 
                  r^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2 dr
             -\kappa l(l+1) \int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr
         &lt; -\kappa l(l+1) \int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr,\\
        \sigma &amp;&lt;&amp; 
            \frac{\Ddsty -\kappa l(l+1)\int_{r_i}^{r_o} r^2|\tilde{\phi}_{lm}|^2 dr}
                 {\Ddsty \int_{r_i}^{r_o} r^2|\tilde{\phi}_{lm}|^2 dr}
         &lt;
            \frac{\Ddsty -\kappa l(l+1)\int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr}
                 {\Ddsty r_i^2\int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 dr}
          = -\kappa \frac{l(l+1)}{r_i^2}.
      \end{eqnarray*}
      %
      ã‚ˆã£ã¦æˆé•·çŽ‡ã®æœ€å¤§å€¤ãŒ
      %
      \begin{equation}
        \sigma  &lt; -\kappa \frac{l(l+1)}{r_i^2}
      \end{equation}
      %
      ã¨è¦‹ç©ã‚‚ã‚‰ã‚Œã‚‹.

    \subsection{ä¸€èˆ¬è§£}

      æ”¯é…æ–¹ç¨‹å¼ã‚’å¤‰å½¢ã™ã‚‹ã¨
      %
      \begin{displaymath}
        \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\phi}_{lm}}{r}\right)
          +\left[\frac{-\sigma}{\kappa} -\frac{l(l+1)}{r^2}\right]\tilde{\phi}_{lm},
      \end{displaymath}
      %
      ã•ã‚‰ã«å‹•å¾„åº§æ¨™ã‚’
      \begin{equation}
        \tilde{r} = \sqrt{\frac{-\sigma}{\kappa}}r, \quad
        r  = \sqrt{\frac{\kappa}{-\sigma}}\tilde{r},
      \end{equation}
      %
      ã¨å¤‰å½¢ã™ã‚‹ã¨,
      %
      \begin{displaymath}
        \frac{1}{\tilde{r}^2}\DD{}{\tilde{r}}
             \left(\tilde{r}^2\DD{\tilde{\phi}_{lm}}{\tilde{r}}\right)
          +\left[1 -\frac{l(l+1)}{\tilde{r}^2}\right]\tilde{\phi}_{lm},
      \end{displaymath}
      %
      ã“ã®å¾®åˆ†æ–¹ç¨‹å¼ã¯çƒãƒ™ãƒƒã‚»ãƒ«ã®å¾®åˆ†æ–¹ç¨‹å¼ã§ã‚ã‚Š, ãã®ä¸€èˆ¬è§£ã¯
      $l$ æ¬¡ã®çƒãƒ™ãƒƒã‚»ãƒ«å‡½æ•° ${\cal J}_l(\tilde{r}),{\cal N}_l(\tilde{r})$ ã®
      ç·šå½¢çµåˆã¨ã—ã¦è¡¨ã•ã‚Œã‚‹.
      %
      \begin{equation}
        \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}
        \tilde{\phi}_{lm} = A_1 {\cal J}_l(\tilde{r}) + A_2 {\cal N}_l(\tilde{r})
               = A_1 {\cal J}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right) 
               + A_2 {\cal N}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right)
      \end{equation}


    \subsection{ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      ä¸¡ç«¯ã§ã®å¢ƒç•Œæ¡ä»¶ãŒãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹ã®
      %
      \begin{equation}
        \Deqlab{ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \tilde{\phi}_{lm} =0 \quad {\rm at} \quad y=r_i,r_o
      \end{equation}
      %
      å&nbsp;´åˆã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           {\cal J}_l\left(r_i \alpha\right) 
           &amp; {\cal N}_l\left(r_i \alpha\right)\\
           {\cal J}_l\left(r_o \alpha\right) 
           &amp; {\cal N}_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      \begin{equation}
               {\cal J}_l\left(r_i \alpha\right) 
          \cdot{\cal N}_l\left(r_o \alpha\right)
        -      {\cal N}_l\left(r_i \alpha\right)
          \cdot{\cal J}_l\left(r_o \alpha\right) 
         =0.
      \end{equation}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 
      \footnote{
        çƒãƒ™ãƒƒã‚»ãƒ«é–¢æ•°ã®å½¢ã¯
        %
        \begin{equation}
          {\cal J}_l(x) = a_l(x) \sin x + b_l(x) \cos x, \quad
          {\cal J}_l(x) = b_l(x) \sin x - a_l(x) \cos x,
        \end{equation}
        %
        ã¨è¡¨ã•ã‚Œã‚‹. ã“ã“ã§ $a_l(x), b_l(x)$ ã¯ $x$ ã®æœ‰ç†å¤šé&nbsp;…å¼ã§ã‚ã‚‹. 
        ã“ã®å½¢ã‚’ä»£å…¥ã™ã‚‹ã¨
        %
        \begin{eqnarray*}
         &amp; &amp;
            [a_l(r_i\alpha)\sin(r_i\alpha) + b_l(r_i\alpha)\cos(r_i\alpha)]
            [b_l(r_o\alpha)\sin(r_o\alpha) - a_l(r_o\alpha)\cos(r_o\alpha)]\\
         &amp;&amp;-[b_l(r_i \alpha)\sin(r_i\alpha) - a_l(r_i\alpha)\cos(r_i\alpha)]
            [a_l(r_o \alpha)\sin(r_o\alpha) + b_l(r_o\alpha)\cos (r_o\alpha)]\\
         &amp;=&amp;
             a_l(r_i\alpha)b_l(r_o\alpha)\sin(r_i\alpha)\sin(r_o\alpha)
            -a_l(r_i\alpha)a_l(r_o\alpha)\sin(r_i\alpha)\cos(r_o\alpha)\\
         &amp;&amp; +b_l(r_i\alpha)b_l(r_o\alpha)\cos(r_i\alpha)\sin(r_o\alpha)
            -b_l(r_i\alpha)a_l(r_o\alpha)\cos(r_i\alpha)\cos(r_o\alpha)\\
         &amp;&amp; -b_l(r_i \alpha)a_l(r_o \alpha)\sin(r_i\alpha)\sin (r_o\alpha)
            -b_l(r_i \alpha)b_l(r_o\alpha)\sin(r_i\alpha)\cos (r_o\alpha)\\
         &amp;&amp; +a_l(r_i\alpha)a_l(r_o \alpha)\cos(r_i\alpha)\sin (r_o\alpha)
            +a_l(r_i\alpha)b_l(r_o\alpha)\cos(r_i\alpha)\cos (r_o\alpha)\\
         &amp;=&amp;
             a_l(r_i\alpha)b_l(r_o\alpha)
             [\sin(r_i\alpha)\sin(r_o\alpha)+\cos(r_i\alpha)\cos(r_o\alpha)]\\
         &amp;&amp; +a_l(r_i\alpha)a_l(r_o\alpha)
             [-\sin(r_i\alpha)\cos(r_o\alpha)+\cos(r_i\alpha)\sin (r_o\alpha)]\\
         &amp;&amp; +b_l(r_i\alpha)b_l(r_o\alpha)
             [\cos(r_i\alpha)\sin(r_o\alpha)-\sin(r_i\alpha)\cos (r_o\alpha)]\\
         &amp;&amp; +b_l(r_i\alpha)a_l(r_o\alpha)
             [-\cos(r_i\alpha)\cos(r_o\alpha)-\sin(r_i\alpha)\sin (r_o\alpha)]\\
         &amp;=&amp;
             a_l(r_i\alpha)b_l(r_o\alpha)\cos[(r_o-r_i\alpha)]
            +a_l(r_i\alpha)a_l(r_o\alpha)\sin[(r_o-r_i)\alpha)]\\
         &amp;&amp; +b_l(r_i\alpha)b_l(r_o\alpha)\sin[(r_o-r_i)\alpha)]
            -b_l(r_i\alpha)a_l(r_o\alpha)\cos[(r_o-r_i\alpha)]\\
         &amp;=&amp;
             [a_l(r_i\alpha)b_l(r_o\alpha)-b_l(r_i\alpha)a_l(r_o\alpha)]
             \cos[(r_o-r_i)\alpha)]\\
         &amp;&amp; +[a_l(r_i\alpha)a_l(r_o\alpha)+b_l(r_i\alpha)b_l(r_o\alpha)]
             \sin[(r_o-r_i)\alpha]\\
         &amp;=&amp; \sin[(r_o-r_i)\alpha+\theta_l]=0, 
        \end{eqnarray*}
        %
        ãŸã&nbsp;ã—
        \begin{eqnarray*}
          \sin\theta_l  &amp;=&amp;
          \frac{a_l(r_i\alpha)b_l(r_o\alpha)-b_l(r_i\alpha)a_l(r_o\alpha)}
               {\sqrt{
                 [a_l(r_i\alpha)b_l(r_o\alpha)-b_l(r_i\alpha)a_l(r_o\alpha)]^2
                +[a_l(r_i\alpha)a_l(r_o\alpha)+b_l(r_i\alpha)b_l(r_o\alpha)]^2}},\\        \cos\theta  &amp;=&amp;
          \frac{a_l(r_i\alpha)a_l(r_o\alpha)+b_l(r_i\alpha)b_l(r_o\alpha)}
               {\sqrt{
                 [a_l(r_i\alpha)b_l(r_o\alpha)-b_l(r_i\alpha)a_l(r_o\alpha)]^2
                +[a_l(r_i\alpha)a_l(r_o\alpha)+b_l(r_i\alpha)b_l(r_o\alpha)]^2}},
        \end{eqnarray*}
        %
        ã—ãŸãŒã£ã¦ $\theta$ ãŒ $\alpha$ ã«å¯¾ã—ã¦ãã‚Œã»ã©æ€¥æ¿€ã«
        å¤‰åŒ–ã—ãªã„ãªã‚‰ã°, ã“ã®å¼ã®å½¢ã¯ $\sin[(r_o-r_i)\alpha$ ã«ã‚ˆã£ã¦
        ã&nbsp;ã„ãŸã„ã•ã&nbsp;ã¾ã‚‹ã&nbsp;ã‚ã†. 
        ã—ãŸãŒã£ã¦
        %
        \begin{eqnarray}
          &amp; &amp;(r_o-r_i)\alpha+\theta_l = n\pi, \quad n=1,2,\ldots, \nonumber\\
          &amp; &amp; {\rm i.e.}\quad
          \alpha =\sqrt{\frac{-\sigma}{\kappa}}= \frac{n\pi-\theta_l}{r_o-r_i}, 
          \quad
          \sigma = -\kappa \left(\frac{n\pi-\theta_l}{r_o-r_i}\right)^2
              \quad n=1,2,\ldots.
        \end{eqnarray}
        %
        ç‰¹ã« $l=0$ ã®å&nbsp;´åˆ, $a_0(x) =1/x , b_0=0$ ã§ã‚ã‚‹ã‹ã‚‰
        %
        \begin{eqnarray*}
          \sin\theta_0 = 0, \quad \cos\theta_0 = 1 \quad 
          {\rm i.e.} \quad \theta_0=0,
        \end{eqnarray*}
        %
        ã¨ãªã‚‹ã®ã§
        \begin{eqnarray}
          \alpha =\sqrt{\frac{-\sigma}{\kappa}}= \frac{n\pi}{r_o-r_i}, 
          \quad
          \sigma = -\kappa \left(\frac{n\pi}{r_o-r_i}\right)^2
              \quad n=1,2,\ldots
        \end{eqnarray}
      }
      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
           -{\cal N}_l\left(r_i \alpha\right)\\
           {\cal J}_l\left(r_i \alpha\right) 
        \end{array}
        \right).
      \end{equation}

    \subsection{ä¸¡ç«¯ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      ä¸¡ç«¯ã§ã®å¢ƒç•Œæ¡ä»¶ãŒãƒŽã‚¤ãƒžãƒ³åž‹ã®
      %
      \begin{equation}
        \Deqlab{ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \DD{\tilde{\phi}_{lm}}{r} =0 \quad {\rm at} \quad y=r_i,r_o
      \end{equation}
      %
      å&nbsp;´åˆã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           \alpha{\cal J}'_l\left(r_i \alpha\right) 
           &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)\\
           \alpha{\cal J}'_l\left(r_o \alpha\right) 
           &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      \begin{equation}
        \alpha^2[
               {\cal J}'_l\left(r_i \alpha\right) 
          \cdot{\cal N}'_l\left(r_o \alpha\right)
        -      {\cal N}'_l\left(r_i \alpha\right)
          \cdot{\cal J}'_l\left(r_o \alpha\right) ]
         =0.
      \end{equation}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
           -{\cal N}'_l\left(r_i \alpha\right)\\
           {\cal J}'_l\left(r_i \alpha\right) 
        \end{array}
        \right).
      \end{equation}

    \subsection{å¤–å´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å†…å´ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      å¤–å´ã§ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹, å†…å´ãŒãƒŽã‚¤ãƒžãƒ³åž‹ã®å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ,
      %
      \begin{equation}
        \Deqlab{å¤–å´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å†…å´ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \tilde{\phi}_{lm} =0 \quad {\rm at} \quad y=r_o, \quad
        \DD{\tilde{\phi}_{lm}}{r} =0 \quad {\rm at} \quad y=r_i,
      \end{equation}
      %
      ã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
            \alpha{\cal J}'_l\left(r_i \alpha\right) 
           &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)\\
             {\cal J}_l\left(r_o \alpha\right) 
           &amp; {\cal N}_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      \begin{equation}
        \alpha[
               {\cal J}'_l\left(r_i \alpha\right) 
          \cdot{\cal N}_l\left(r_o \alpha\right)
        -      {\cal N}'_l\left(r_i \alpha\right)
          \cdot{\cal J}_l\left(r_o \alpha\right) ]
         =0.
      \end{equation}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
           -{\cal N}'_l\left(r_i \alpha\right)\\
           {\cal J}'_l\left(r_i \alpha\right) 
        \end{array}
        \right).
      \end{equation}

    \subsection{å¤–å´ãƒŽã‚¤ãƒžãƒ³å†…å´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      å¤–å´ã§ãƒŽã‚¤ãƒžãƒ³åž‹, å†…å´ãŒãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹ã®å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ,
      %
      \begin{equation}
        \Deqlab{å¤–å´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å†…å´ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \DD{\tilde{\phi}_{lm}}{r} =0 \quad {\rm at} \quad y=r_o,\quad
        \tilde{\phi}_{lm} =0 \quad {\rm at} \quad y=r_i
      \end{equation}
      %
      ã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
            {\cal J}_l\left(r_i \alpha\right) 
           &amp; {\cal N}_l\left(r_i \alpha\right)\\
           \alpha{\cal J}'_l\left(r_o \alpha\right) 
           &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      \begin{equation}
        \alpha[
               {\cal J}_l\left(r_i \alpha\right) 
          \cdot{\cal N}'_l\left(r_o \alpha\right)
        -      {\cal N}_l\left(r_i \alpha\right)
          \cdot{\cal J}'_l\left(r_o \alpha\right) ]
         =0.
      \end{equation}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
           -{\cal N}_l\left(r_i \alpha\right)\\
           {\cal J}_l\left(r_i \alpha\right) 
        \end{array}
        \right).
      \end{equation}

    \subsection{é&nbsp;˜åŸŸå¤–éƒ¨ã®å¢ƒç•Œé&nbsp;˜åŸŸã§ãƒ©ãƒ—ãƒ©ã‚¹æ–¹ç¨‹å¼ã«å¾“ã†å&nbsp;´åˆ}

      æœ‰é™ã®ç†±ä¼å°ŽçŽ‡ã‚’æŒã¤ç‰©è³ªã§çƒæ®»å†…å¤–é&nbsp;˜åŸŸãŒæº€ãŸã•ã‚Œã¦ã„ã‚‹å&nbsp;´åˆã®
      ç†±æ‹¡æ•£å•é¡Œ, ã‚ã‚‹ã„ã¯ãƒãƒ­ã‚¤ãƒ€ãƒ«ç£å&nbsp;´ã®æ‹¡æ•£ã®å•é¡Œã®å&nbsp;´åˆã«ã¯
      çƒæ®»é&nbsp;˜åŸŸã®æ‹¡æ•£æ–¹ç¨‹å¼ã‚’æ¬¡ã®ã‚ˆã†ãªæ··åˆåž‹ã®å¢ƒç•Œæ¡ä»¶ã®ä¸‹ã§
      è§£ãã“ã¨ã«ãªã‚‹
      \footnote{
        é&nbsp;˜åŸŸå¤–éƒ¨ã®æ”¯é…æ–¹ç¨‹å¼ã¯
        %
        \begin{displaymath}
          \Dlapla \phi =0, 
        \end{displaymath}
        %
        ç·¯åº¦çµŒåº¦ã«é–¢ã—ã¦çƒé¢èª¿å’Œé–¢æ•°å¤‰æ›ã™ã‚‹ã¨
        %
        \begin{displaymath}
          \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
          \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\phi}_{lm}}{r}\right)
                    -\frac{l(l+1)}{r^2}\tilde{\phi}_{lm}=0
        \end{displaymath}
        %
        ã“ã®æ–¹ç¨‹å¼ã®ç‹¬ç«‹ãª 2 ã¤ã®è§£ã¯ $r^{-(l+1)}, r^l$ ã§ã‚ã‚‹. 
        ãã‚Œãžã‚Œ $r=0,\infty$ ã§ç™ºæ•£ã—ãªã„è§£ã‚’æ±‚ã‚ã‚‹ã¨
        \begin{displaymath}
          \tilde{\phi}_{lm} = C_1 r^{-(l+1)}
              \quad \left(r \ge r_o \right),\qquad
          \tilde{\phi}_{lm} = C_2 r^l
              \quad \left(r \le r_i \right),
        \end{displaymath}
        %
        å¢ƒç•Œ $r=r_o$ ã§å€¤ã¨ 1 éšŽå¾®åˆ†ãŒé€£ç¶šã§ã‚ã‚‹ã“ã¨ã‹ã‚‰
        \begin{eqnarray*}
          \tilde{\phi}_{lm}|_{r=r_o} =  C_1 r^{-(l+1)r_o}, \quad
          \left.\DD{\tilde{\phi}_{lm}}{r}\right|_{r_o} =  -(l+1) C_1 r^{-(l+2)r_o}
        \end{eqnarray*}
        %
        ã“ã‚Œã‚‰ã‚ˆã‚Š $C_1$ ã‚’æ¶ˆåŽ»ã™ã‚‹ã¨ $r=r_o$ ã§ã®å¢ƒç•Œæ¡ä»¶
        %
        \begin{displaymath}
          \left.\DD{\tilde{\phi}_{lm}}{r}\right|_{r_o}
              +\frac{(l+1)}{r_o}\tilde{\phi}_{lm}|_{r_o}=0,
        \end{displaymath}
        %
        ãŒå¾—ã‚‰ã‚Œã‚‹. $r=r_i$ ã®å&nbsp;´åˆã‚‚åŒæ§˜ã«è¨ˆç®—ã™ã‚‹ã¨, 
        %
        \begin{displaymath}
          \left.\DD{\tilde{\phi}_{lm}}{r}\right|_{r_o}
              -\frac{l}{r_i}\tilde{\phi}_{lm}|_{r_o}=0.
        \end{displaymath}
      }.
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯æ··åˆåž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå±•é–‹}
         \left.\DD{\tilde{\phi}_{lm}}{r}\right|_{r_o}
              +\frac{(l+1)}{r_o}\tilde{\phi}_{lm}|_{r_o}=0,\quad
         \left.\DD{\tilde{\phi}_{lm}}{r}\right|_{r_o}
              -\frac{l}{r_i}\tilde{\phi}_{lm}|_{r_o}=0.
      \end{equation}
      %
      ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨, 
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           {\cal J}'_l\left(r_i \alpha\right)
               -l/r_i{\cal J}_l\left(r_i \alpha\right) 
           &amp; {\cal N}'_l\left(r_i \alpha\right)
               -l/r_i{\cal N}_l\left(r_i \alpha\right)\\
           {\cal J}'_l\left(r_o \alpha\right) 
             +(l+1)/r_o{\cal J}_l\left(r_o \alpha\right) 
           &amp; {\cal N}'_l\left(r_o \alpha\right)
             +(l+1)/r_o{\cal N}_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      ã“ã“ã§çƒãƒ™ãƒƒã‚»ãƒ«é–¢æ•°ã®æ€§è³ª
      \begin{eqnarray*}
        &amp; &amp; {\cal J}'_l(x) 
          = \frac{l}{x}{\cal J}_l(x) - {\cal J}_{l+1}(x) 
          = {\cal J}_{l-1}(x) - \frac{l+1}{x}{\cal J}_l(x), \\
        &amp; &amp; {\cal N}'_l(x) 
          = \frac{l}{x}{\cal N}_l(x) - {\cal N}_{l+1}(x) 
          = {\cal N}_{l-1}(x) - \frac{l+1}{x}{\cal N}_l(x)
      \end{eqnarray*}
      %
      ã‚’é©ç”¨ã™ã‚‹ã¨, 
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
             -{\cal J}_{l+1}\left(r_i \alpha\right)
           &amp; -{\cal N}_{l+1}\left(r_i \alpha\right)\\
              {\cal J}_{l-1}\left(r_o \alpha\right) 
           &amp;  {\cal N}_{l-1}\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      \begin{equation}
               {\cal J}_{l+1}\left(r_i \alpha\right) 
          \cdot{\cal N}_{l-1}\left(r_o \alpha\right)
        -      {\cal N}_{l+1}\left(r_i \alpha\right)
          \cdot{\cal J}_{l-1}\left(r_o \alpha\right) 
         =0.
      \end{equation}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

  \section{éžåœ§ç¸®æµä½“ã®ç²˜æ€§æ‹¡æ•£è§£}

    åŠå¾„ $r_i,r_o$ ã®åŒå¿ƒçƒé¢ã®é–“ã® 3 æ¬¡å…ƒçƒæ®»é&nbsp;˜åŸŸã§ã®
    éžåœ§ç¸®æµä½“ã®é‹å‹•ã‚’è€ƒãˆã‚‹. 
    éžåœ§ç¸®é€Ÿåº¦å&nbsp;´ã¯ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’ç”¨ã„ã¦
    æ¬¡ã®ã‚ˆã†ã«è¡¨ã™ã“ã¨ãŒã§ãã‚‹. 
    %
    \begin{eqnarray}
      \Deqlab{é€Ÿåº¦åˆ†è§£}
      \Dvect{u} &amp;=&amp; \Drot(\psi \Dvect{r}) + \Drot\Drot(\phi\Dvect{r}).
    \end{eqnarray}
    %
    ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹è¿‘ä¼¼ã®ä¸‹ã§ã“ã‚Œã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å¾“ã†æ–¹ç¨‹å¼ã¯
    \footnote{
      æ–¹ç¨‹å¼ã®å°Žå‡ºã«é–¢ã—ã¦ã¯
      `3 æ¬¡å…ƒå›žè»¢çƒæ®»å†…ã§ã®ç²˜æ€§æµä½“(éžåœ§ç¸®ãƒŠãƒ“ã‚¨ãƒ¼ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹æ–¹ç¨‹å¼)ã®å®šå¼åŒ–'
      ã‚’å‚ç…§ã•ã‚ŒãŸã„. 
    }
    % 
    \begin{eqnarray}
      \Deqlab{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
          \DP{}{t} (L_2 \psi)
       &amp;=&amp; \nu L_2\Dlapla \psi, \\
      \Deqlab{ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
         \DP{}{t} (L_2 \Dlapla \phi)
       &amp;=&amp; \nu L_2\Dlapla\Dlapla \phi,
    \end{eqnarray}
    ã“ã“ã§å¼ä¸­ã®æ¼”ç®—å­ $L_2$ ã¯
    åŠå¾„ 1 ã®çƒé¢ä¸Šã® 2 æ¬¡å…ƒãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã®é€†ç¬¦å·ã®ã‚‚ã®ã§ã‚ã‚Š,
    % 
    \begin{eqnarray}
       L_{2} &amp;\equiv&amp;  -r^{2}\Dlapla +\DP{}{r}r^{2}\DP{}{r}  \nonumber\\
             &amp;=&amp; -\left[
                   \frac{1}{\cos^{2}\varphi}\DP[2]{}{\lambda}
                 +
                   \frac{1}{\cos\varphi}\DP{}{\varphi}
                   \left(\cos\varphi\right)\DP{}{\varphi} \right].
    \end{eqnarray}
 
    é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å†…å¤–çƒé¢ã«ã¦æ³•ç·šæ–¹å‘ã®
    é€Ÿåº¦æˆåˆ†ãŒ 0 ã¨ãªã‚‹ã“ã¨ã§ã‚ã‚‹. 
    ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°,
    \begin{equation}
      \phi = 0, 
        \quad \mbox{at} \quad r=r_i, r_o
    \end{equation}
    %
    åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒè‡ªç”±ã™ã¹ã‚Šã®å&nbsp;´åˆã¯
    \begin{equation}
      \Deqlab{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
      \DP[2]{\phi}{r} = \DP{}{r}\left(\frac{\psi}{r}\right) = 0,
      \quad \mbox{at} \quad r=r_i,r_o
    \end{equation}
    %
    ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
    \begin{eqnarray*}
      \Deqlab{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
         \DP{\phi}{r} = 0, \quad
         \psi = 0
         \quad \mbox{at} \quad r=r_i, r_o
    \end{eqnarray*}

    \subsection{çƒé¢èª¿å’Œé–¢æ•°å±•é–‹}

      ç·¯åº¦ãƒ»çµŒåº¦æ–¹å‘ã«ã¯çƒé¢èª¿å’Œé–¢æ•°ã§å±•é–‹ã—ã¦è§£ãã“ã¨ãŒã§ãã‚‹. 
      ä»¥ä¸‹, ãã® 1 æ³¢æ•°æˆåˆ†ã®ã¿å–ã‚Šã&nbsp;ã—ã¦æ‰±ã†. 
      æ™‚é–“ã«é–¢ã—ã¦ã‚‚æŒ‡æ•°åž‹ã®è§£ã‚’ä»®å®šã—, 
      %
      \begin{equation}
        \Deqlab{ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \psi(x,y,t)=\tilde{\psi}_{lm}(r)Y_{lm}(\lambda,\varphi)e^{\sigma t},
        \quad
        \phi(x,y,t)=\tilde{\phi}_{lm}(r)Y_{lm}(\lambda,\varphi)e^{\sigma t}
      \end{equation}
      %
      ã“ã‚Œã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, 
      %
      \begin{eqnarray*}
          &amp; &amp;
          \sigma l(l+1)\tilde{\psi}_{lm} =
              \nu l(l+1) \left[
                \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\psi}_{lm}}{r}\right)
                  -\frac{l(l+1)}{r^2}\tilde{\psi}_{lm}\right],\\
          &amp; &amp;
          \sigma l(l+1)
                 \left[\frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}\right]\tilde{\phi}_{lm}
            =
              \nu l(l+1) 
                 \left[\frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}\right]^2 \tilde{\phi}_{lm}
      \end{eqnarray*}
      %
      ã—ãŸãŒã£ã¦ $l\neq 0$ ã®ãƒ¢ãƒ¼ãƒ‰ã«å¯¾ã—ã¦
      \begin{eqnarray}
        \Deqlab{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
          &amp; &amp;
          \sigma \tilde{\psi}_{lm} =
              \nu \left[
                \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\psi}_{lm}}{r}\right)
                  -\frac{l(l+1)}{r^2}\tilde{\psi}_{lm}\right],\\
          &amp; &amp;
        \Deqlab{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
          \sigma 
                 \left[\frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}\right]\tilde{\phi}_{lm}
             = \nu 
                 \left[\frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}\right]^2 \tilde{\phi}_{lm}.
      \end{eqnarray}
      
      å¢ƒç•Œæ¡ä»¶ã¯è‡ªç”±ã™ã¹ã‚Šã®å&nbsp;´åˆ, 
      %
      åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      \begin{equation}
        \Deqlab{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
        \tilde{\phi}_{lm} = \DD[2]{\tilde{\phi}_{lm}}{r} = 0, \quad
        \DD{}{r}\left(\frac{\tilde{\psi}_{lm}}{r}\right) = 0,
        \quad \mbox{at} \quad r=r_i,r_o
      \end{equation}
      %
      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
      \begin{eqnarray*}
        \Deqlab{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
           \tilde{\phi}_{lm} = \DD{\tilde{\phi}_{lm}}{r} = 0, \quad
           \tilde{\psi}_{lm} = 0
           \quad \mbox{at} \quad r=r_i, r_o
      \end{eqnarray*}

    \subsection{æœ€å¤§æˆé•·çŽ‡ã®è¦‹ç©ã‚‚ã‚Š}

      å¢ƒç•Œæ¡ä»¶ãŒç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, æœ€å¤§æˆé•·çŽ‡ãŒè²&nbsp;ã¨ãªã‚‹ã“ã¨ã‚’
      æ”¯é…æ–¹ç¨‹å¼ã‚’å…¨ç©åˆ†ã™ã‚‹ã“ã¨ã§æ±‚ã‚ã‚‰ã‚Œã‚‹. 
      ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«é–¢ã—ã¦ã¯, æ‹¡æ•£æ–¹ç¨‹å¼ã®ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬
      å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆã¨åŒã˜ã§ã‚ã‚‹ã‹ã‚‰æœ€å¤§æˆé•·çŽ‡ãŒ
      %
      \begin{equation}
        \sigma  &lt; -\nu \frac{l(l+1)}{r_i^2}
      \end{equation}
      %
      ã¨è©•ä¾¡ã•ã‚Œã‚‹. 

      ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«ã¤ã„ã¦ã¯, æ¼”ç®—å­ ${\cal D}_l$ ã‚’
      %
      \begin{displaymath}
        {\cal D}_l \equiv \frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}
      \end{displaymath}
      %
      ã¨å®šç¾©ã™ã‚‹ã¨, ä»»æ„ã® 2 ã¤ã®é–¢æ•° $f(r),g(r)$ ã«å¯¾ã—ã¦
      %
      \begin{eqnarray*}
        \int_{r_i}^{r_o} f(r) {\cal D}_l g(r) r^2 dr
        &amp;=&amp; \int_{r_i}^{r_o} f
             \left[\frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)
                        -\frac{l(l+1)}{r^2}\right]g r^2 dr\\
        &amp;=&amp; \int_{r_i}^{r_o} f \DD{}{r}\left(r^2\DD{g}{r}\right)
             - \int_{r_i}^{r_o} l(l+1)fg dr\\
        &amp;=&amp; \left[ f r^2\DD{g}{r} \right]_{r_i}^{r_o}
             - \int_{r_i}^{r_o} \DD{f}{r}\left(r^2\DD{g}{r}\right) dr
             - \int_{r_i}^{r_o} l(l+1)fg dr\\
        &amp;=&amp; \left[ f r^2\DD{g}{r} \right]_{r_i}^{r_o}
             - \left[ r^2\DD{f}{r} g \right]_{r_i}^{r_o}
             + \int_{r_i}^{r_o} g \DD{}{r}\left(r^2\DD{f}{r}\right)
             - \int_{r_i}^{r_o} l(l+1)fg dr\\
        &amp;=&amp; \left[ f r^2\DD{g}{r} \right]_{r_i}^{r_o}
             - \left[ r^2\DD{f}{r} g \right]_{r_i}^{r_o}
             + \int_{r_i}^{r_o} g(r) {\cal D}_l f(r) r^2 dr
      \end{eqnarray*}
      %
      $f=\psi^*_{lm}(r), g={\cal D}_l\psi_{lm}(r)$ ã¨é¸ã¶ã¨
      å¢ƒç•Œæ¡ä»¶\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}ã‚’ç”¨ã„ã¦
      \begin{eqnarray*}
        \int_{r_i}^{r_o} \psi^*_{lm}(r)
                 {\cal D}_l{\cal D}_l\psi_{lm}(r) r^2 dr        
        &amp;=&amp; \left[ \psi^*_{lm} r^2\DD{{\cal D}_l\psi_{lm}}{r} 
            \right]_{r_i}^{r_o}
             - \left[ r^2\DD{\psi^*_{lm}}{r}
                     {\cal D}_l\psi_{lm} \right]_{r_i}^{r_o}
             + \int_{r_i}^{r_o} {\cal D}_l\psi^*_{lm}{\cal D}_l\psi_{lm} r^2 dr\\
        &amp;=&amp; \int_{r_i}^{r_o} |{\cal D}_l\psi_{lm}|^2 r^2 dr
      \end{eqnarray*}
      
      ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å¼
      \Deqref{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«æ–¹ç¨‹å¼çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}ã«
      $\phi^*_{lm}r^2$ ã‚’ã‹ã‘ã¦ç©åˆ†ã™ã‚‹ã¨
      \begin{eqnarray*}
        \sigma \int_{r_i}^{r_o}
             \tilde{\phi}^*_{lm}{\cal D}_l\tilde{\phi}_{lm}r^2 dr 
        &amp;=&amp; \nu \int_{r_i}^{r_o}
             \tilde{\phi}^*_{lm}{\cal D}_l{\cal D}_l\tilde{\phi}_{lm}r^2 dr,\\
      \end{eqnarray*}
      %
      å·¦è¾ºã¯æ‹¡æ•£æ–¹ç¨‹å¼ã®å³è¾ºã®å¼å¤‰å½¢ã¨åŒæ§˜ã«è¨ˆç®—ã§ãã¦, 
      \begin{displaymath}
        \sigma \int_{r_i}^{r_o}
             \tilde{\phi}^*_{lm}{\cal D}_l\tilde{\phi}_{lm}r^2 dr 
        =-\sigma \int_{r_i}^{r_o} \left[
              r^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2 
             + l(l+1) \int_{r_i}^{r_o} |\tilde{\phi}_{lm}|^2 \right]dr
      \end{displaymath}
      %
      ã§ã‚ã‚‹ã‹ã‚‰, 
      \begin{displaymath}
        -\sigma \int_{r_i}^{r_o} \left[
              r^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2 
                       + l(l+1) |\tilde{\phi}_{lm}|^2 \right]dr
        = \nu\int_{r_i}^{r_o} |{\cal D}_l\psi_{lm}|^2 r^2 dr, 
      \end{displaymath}
      %
      ã‚ˆã£ã¦
      \begin{displaymath}
        \sigma = - \nu
           \frac{\Ddsty \int_{r_i}^{r_o} |{\cal D}_l\psi_{lm}|^2 r^2 dr}
                {\Ddsty \int_{r_i}^{r_o} \left[
                        r^2\left|\DD{\tilde{\phi}_{lm}}{r}\right|^2 
                        + l(l+1) |\tilde{\phi}_{lm}|^2 \right]dr}.
      \end{displaymath}


    \subsection{ä¸€èˆ¬è§£}

      æ‹¡æ•£æ–¹ç¨‹å¼ã®å&nbsp;´åˆã¨åŒæ§˜ã«æ”¯é…æ–¹ç¨‹å¼ã‚’å¤‰å½¢ã™ã‚‹ã¨
      %
      \begin{eqnarray*}
        &amp; &amp; 
        \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\psi}_{lm}}{r}\right)
          +\left[\frac{-\sigma}{\kappa} -\frac{l(l+1)}{r^2}\right]
             \tilde{\psi}_{lm},\\
        &amp; &amp;
        \left[
          \frac{1}{r^2}\DD{}{r}\left(r^2\DD{}{r}\right)-\frac{l(l+1)}{r^2}
        \right]
        \left\{
          \frac{1}{r^2}\DD{}{r}\left(r^2\DD{\tilde{\psi}_{lm}}{r}\right)
            +\left[\frac{-\sigma}{\kappa} -\frac{l(l+1)}{r^2}\right]
        \right\}
             \tilde{\psi}_{lm},
      \end{eqnarray*}
      %
      æˆé•·çŽ‡ $\sigma &lt;0$ ã«å¯¾ã—ã¦ã®ã“ã‚Œã‚‰ã®ä¸€èˆ¬è§£ã¯
      %
      \begin{eqnarray}
        \Deqlab{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}
        &amp; &amp;
        \tilde{\psi}_{lm} 
               = A_1 {\cal J}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right) 
               + A_2 {\cal N}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right),\\
        &amp; &amp;
        \Deqlab{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}
        \tilde{\phi}_{lm} 
               = B_1 r^l + B_2 r^{-(l+1)}
               + B_3 {\cal J}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right) 
               + B_4 {\cal N}_l\left(\sqrt{\frac{-\sigma}{\kappa}}r\right).
      \end{eqnarray}


    \subsection{ä¸¡ç«¯ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã®ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã¯æ‹¡æ•£æ–¹ç¨‹å¼ã§ã®ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å•é¡Œã¨åŒã˜ã§ã‚ã‚‹. 

    \subsection{ä¸¡ç«¯è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆã®ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚’æ›¸ãæ›ãˆã‚‹ã¨
      %
      \begin{displaymath}
        \DD{\tilde{\psi}_{lm}}{r} - \frac{\tilde{\psi}_{lm}}{r}=0, 
        \quad {\rm at} \quad r=r_i,r_o.
      \end{displaymath}
      %
      ã“ã‚Œã«\Deqref{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           \alpha{\cal J}'_l\left(r_i \alpha\right) 
            -{\cal J}_l\left(r_i \alpha\right)/r_i
           &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)
            -{\cal N}_l\left(r_i \alpha\right)/r_i\\
           \alpha{\cal J}'_l\left(r_o \alpha\right) 
            -{\cal J}_l\left(r_o \alpha\right)/r_o
           &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
            -{\cal N}_l\left(r_o \alpha\right)/r_o
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      %
      \begin{eqnarray*}
       &amp; &amp;  [\alpha{\cal J}'_l\left(r_i \alpha\right) 
            -{\cal J}_l\left(r_i \alpha\right)/r_i]
          [\alpha{\cal N}'_l\left(r_o \alpha\right)
            -{\cal N}_l\left(r_o \alpha\right)/r_o]\\
       &amp; &amp;
         - [\alpha{\cal N}'_l\left(r_i \alpha\right)
            -{\cal N}_l\left(r_i \alpha\right)/r_i]
         [\alpha{\cal J}'_l\left(r_o \alpha\right) 
            -{\cal J}_l\left(r_o \alpha\right)/r_o]
       = 0.
      \end{eqnarray*}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
          -[\alpha{\cal N}'_l\left(r_i \alpha\right)
            -{\cal N}_l\left(r_i \alpha\right)/r_i]\\
           \alpha{\cal J}'_l\left(r_i \alpha\right) 
            -{\cal J}_l\left(r_i \alpha\right)/r_i
        \end{array}
        \right).
      \end{equation}

    \subsection{å¤–å´è‡ªç”±ã™ã¹ã‚Šå†…å´ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã®ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚’æ›¸ãæ›ãˆã‚‹ã¨
      %
      \begin{displaymath}
        \DD{\tilde{\psi}_{lm}}{r} - \frac{\tilde{\psi}_{lm}}{r}=0, 
        \quad {\rm at} \quad r=r_o, \quad
        \tilde{\psi}_{lm}=0, 
        \quad {\rm at} \quad r=r_i.
      \end{displaymath}
      %
      ã“ã‚Œã«\Deqref{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
            {\cal J}_l\left(r_i \alpha\right)
           &amp; {\cal N}_l\left(r_i \alpha\right)\\
           \alpha{\cal J}'_l\left(r_o \alpha\right) 
            -{\cal J}_l\left(r_o \alpha\right)/r_o
           &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
            -{\cal N}_l\left(r_o \alpha\right)/r_o
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      %
      \begin{displaymath}
          {\cal J}_l\left(r_i \alpha\right)
          [\alpha{\cal N}'_l\left(r_o \alpha\right)
            -{\cal N}_l\left(r_o \alpha\right)/r_o]
         - {\cal N}_l\left(r_i \alpha\right)
         [\alpha{\cal J}'_l\left(r_o \alpha\right) 
            -{\cal J}_l\left(r_o \alpha\right)/r_o]
       = 0.
      \end{displaymath}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
            -{\cal N}_l\left(r_i \alpha\right)\\
             {\cal J}_l\left(r_i \alpha\right)
        \end{array}
        \right).
      \end{equation}

    \subsection{å¤–å´ç²˜ç€å†…å´è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆã®ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚’æ›¸ãæ›ãˆã‚‹ã¨
      %
      \begin{displaymath}
        \tilde{\psi}_{lm}=0, 
        \quad {\rm at} \quad r=r_o, \quad
        \DD{\tilde{\psi}_{lm}}{r} - \frac{\tilde{\psi}_{lm}}{r}=0, 
        \quad {\rm at} \quad r=r_i.
      \end{displaymath}
      %
      ã“ã‚Œã«\Deqref{ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           \alpha{\cal J}'_l\left(r_i \alpha\right) 
            -{\cal J}_l\left(r_i \alpha\right)/r_i
           &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)
            -{\cal N}_l\left(r_i \alpha\right)/r_i\\
            {\cal J}_l\left(r_o \alpha\right)
           &amp; {\cal N}_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      %
      \begin{displaymath}
           [\alpha{\cal J}'_l\left(r_i \alpha\right) 
            -{\cal J}_l\left(r_i \alpha\right)/r_o]
           {\cal N}_l\left(r_o \alpha\right)
         - [\alpha{\cal N}'_l\left(r_i \alpha\right)
            -{\cal N}_l\left(r_i \alpha\right)/r_o]
          {\cal J}_l\left(r_o \alpha\right)
        = 0.
      \end{displaymath}
      %
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          A_1 \\ A_2
        \end{array}
        \right)
        ={\rm const}\cdot
        \left(
        \begin{array}{c}
            -{\cal N}_l\left(r_o \alpha\right)\\
             {\cal J}_l\left(r_o \alpha\right)
        \end{array}
        \right).
      \end{equation}
      
    \subsection{ä¸¡ç«¯ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã®ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \tilde{\phi}_{lm}= \DD{\tilde{\phi}_{lm}}{r}=0, 
        \quad {\rm at} \quad r=r_o,r_i.
      \end{displaymath}
      %
      ä¸€èˆ¬è§£\Deqref{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ã“ã‚Œã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      \begin{equation}
        \left(
        \begin{array}{cccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)
             &amp; {\cal N}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)
             &amp; {\cal N}_l\left(r_o \alpha\right)\\
           lr_i^{l-1} &amp; -(l+1)r_i^{-(l+2)} 
             &amp; \alpha{\cal J}'_l\left(r_i \alpha\right)
             &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)\\
           lr_o^{l-1} &amp; -(l+1)r_o^{-(l+2)} 
             &amp; \alpha{\cal J}'_l\left(r_o \alpha\right)
             &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 \\ B_3 \\ B_4
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      %
      \begin{equation}
        \left|
        \begin{array}{cccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)
             &amp; {\cal N}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)
             &amp; {\cal N}_l\left(r_o \alpha\right)\\
           lr_i^{l-1} &amp; -(l+1)r_i^{-(l+2)} 
             &amp; \alpha{\cal J}'_l\left(r_i \alpha\right)
             &amp; \alpha{\cal N}'_l\left(r_i \alpha\right)\\
           lr_o^{l-1} &amp; -(l+1)r_o^{-(l+2)} 
             &amp; \alpha{\cal J}'_l\left(r_o \alpha\right)
             &amp; \alpha{\cal N}'_l\left(r_o \alpha\right)
        \end{array}
        \right| = 0.
      \end{equation}
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $B_1, B_2, B_3, B_4$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      å…ˆã®é€£ç«‹æ–¹ç¨‹å¼ã«ãŸã¨ãˆã° $B_4=-1$ ã‚’ä»£å…¥ã—ã¦æ•´ç†ã—ãŸ
      %
      \begin{equation}
        \left(
        \begin{array}{ccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)\\
           lr_i^{l-1} &amp; -(l+1)r_i^{-(l+2)} 
             &amp; \alpha{\cal J}'_l\left(r_i \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 \\ B_3
        \end{array}
        \right)
        = 
        \left(
        \begin{array}{c}
          {\cal N}_l\left(r_i \alpha\right)\\
          {\cal N}_l\left(r_o \alpha\right)\\
          \alpha{\cal N}'_l\left(r_i \alpha\right)
        \end{array}
        \right)
      \end{equation}
      %
      ã‚’è§£ãã“ã¨ã§å®šã‚ã‚‰ã‚Œã‚‹. 

    \subsection{ä¸¡ç«¯è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆã®ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯\Deqref{é€Ÿåº¦ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \tilde{\phi}_{lm}= \DD[2]{\tilde{\phi}_{lm}}{r}=0, 
        \quad {\rm at} \quad r=r_o,r_i.
      \end{displaymath}
      %
      ã“ã‚Œã«ä¸€èˆ¬è§£\Deqref{ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«çƒé¢èª¿å’Œå‡½æ•°å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      \begin{equation}
        \left(
        \begin{array}{cccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)
             &amp; {\cal N}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)
             &amp; {\cal N}_l\left(r_o \alpha\right)\\
           l(l-1)r_i^{l-2} &amp; (l+1)(l+2)r_i^{-(l+3)} 
             &amp; \alpha^2{\cal J}''_l\left(r_i \alpha\right)
             &amp; \alpha^2{\cal N}''_l\left(r_i \alpha\right)\\
           l(l-1)r_o^{l-2} &amp; -(l+1)(l+2)r_o^{-(l+3)} 
             &amp; \alpha^2{\cal J}''_l\left(r_o \alpha\right)
             &amp; \alpha^2{\cal N}''_l\left(r_o \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 \\ B_3 \\ B_4
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      \footnote{
        çƒãƒ™ãƒƒã‚»ãƒ«å‡½æ•°ã® 2 éšŽå¾®åˆ†ã¯çƒãƒ™ãƒƒã‚»ãƒ«ã®æº€ãŸã™å¾®åˆ†æ–¹ç¨‹å¼ã‹ã‚‰
        \begin{displaymath}
          {\cal J}''= -\frac{2}{r}{\cal J}'
                      -\left[1-\frac{l(l+1)}{r^2}\right]{\cal J}, \quad
          {\cal N}''= -\frac{2}{r}{\cal N}'
                      -\left[1-\frac{l(l+1)}{r^2}\right]{\cal N}
        \end{displaymath}
        ã¨è¨ˆç®—ã•ã‚Œã‚‹. 
      }
      %
      \begin{equation}
        \left|
        \begin{array}{cccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)
             &amp; {\cal N}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)
             &amp; {\cal N}_l\left(r_o \alpha\right)\\
           l(l-1)r_i^{l-2} &amp; (l+1)(l+2)r_i^{-(l+3)} 
             &amp; \alpha^2{\cal J}''_l\left(r_i \alpha\right)
             &amp; \alpha^2{\cal N}''_l\left(r_i \alpha\right)\\
           l(l-1)r_o^{l-2} &amp; -(l+1)(l+2)r_o^{-(l+3)} 
             &amp; \alpha^2{\cal J}''_l\left(r_o \alpha\right)
             &amp; \alpha^2{\cal N}''_l\left(r_o \alpha\right)
        \end{array}
        \right| = 0.
      \end{equation}
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $B_1, B_2, B_3, B_4$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      å…ˆã®é€£ç«‹æ–¹ç¨‹å¼ã«ãŸã¨ãˆã° $B_4=-1$ ã‚’ä»£å…¥ã—ã¦æ•´ç†ã—ãŸ
      %
      \begin{equation}
        \left(
        \begin{array}{ccc}
           r_i^l &amp; r_i^{-(l+1)}
             &amp; {\cal J}_l\left(r_i \alpha\right)\\
           r_o^l &amp; r_o^{-(l+1)}
             &amp; {\cal J}_l\left(r_o \alpha\right)\\
           l(l-1)r_i^{l-2} &amp; (l+1)(l+2)r_i^{-(l+3)} 
             &amp; \alpha^2{\cal J}''_l\left(r_i \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 \\ B_3
        \end{array}
        \right)
        = 
        \left(
        \begin{array}{c}
          {\cal N}_l\left(r_i \alpha\right)\\
          {\cal N}_l\left(r_o \alpha\right)\\
          \alpha^2{\cal N}''_l\left(r_i \alpha\right)\\
        \end{array}
        \right)
      \end{equation}
      %
      ã‚’è§£ãã“ã¨ã§å®šã‚ã‚‰ã‚Œã‚‹. 

      

\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 
</pre></body></html>