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Bounded solutions of a system of copled liniar Schrodinger equations

  1. Sep 20, 2008 #1
    bounded solutions of a system of coupled liniar Schrodinger equations

    Hi.

    I study the following system of four coupled liniar Schrodinger equations:

    [itex]
    i\delta \left(\begin{array}{c}f&h&g&q \end{array}\right) =
    \left(\begin{array}{cccc}
    -L_p&-a_1&-a_2&-a_2\\
    a_1&L_p&a_2&a_2\\
    -a_3&-a_3&-L_c&-a_4\\
    a_3&a_3&a_4&L_c
    \end{array}\right)
    \left(\begin{array}{c}f&h&g&q \end{array}\right)
    [/itex]

    where [itex]L_{p,c}=\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr}-\frac{m^2}{r^2}-\beta_{p,c}[/itex]

    [itex]\beta_{p,c}>0[/itex], [itex]m[/itex] is integer.

    The coeficients [itex]a_j(r)[/itex] are real functions given numerically, they have no singularity on [itex][0,\infty)][/itex], and they fulfill

    [itex]\lim_{r\rightarrow\infty} a_j(r)=0[/itex]

    I am interested in finding the bound states [itex](f,h,g,q)[/itex] and their eigenvalues, or at least finding some condition for their existence. I will appreciate any idea or any reference (book or article) on how one might solve this problem. I must say that I am physicist.

    Thank you.
     
    Last edited: Sep 20, 2008
  2. jcsd
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