# Bounded solutions of a system of copled liniar Schrodinger equations

1. Sep 20, 2008

### soarce

bounded solutions of a system of coupled liniar Schrodinger equations

Hi.

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

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

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

$\beta_{p,c}>0$, $m$ is integer.

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

$\lim_{r\rightarrow\infty} a_j(r)=0$

I am interested in finding the bound states $(f,h,g,q)$ 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