Bounded solutions of a system of copled liniar Schrodinger equations

[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.

 

[jvicens]

Hello,

Thank you for your post. The system of coupled linear Schrodinger equations you are studying is very interesting and has many applications in physics, particularly in the study of quantum mechanics and condensed matter systems.

To find the bound states and their eigenvalues, you can use various mathematical techniques such as perturbation theory, variational methods, or numerical methods. One approach you can try is to use the variational method, where you approximate the solution of the equations by a trial function and minimize the energy functional to find the best approximation. This method has been successfully applied to similar systems of coupled equations in the past.

Another approach is to use perturbation theory, where you expand the solution in a power series and solve for the coefficients iteratively. This method can be more challenging for systems with multiple coupled equations, but it has been used in the past for similar problems in quantum mechanics.

As for references, I recommend looking into textbooks on quantum mechanics or mathematical methods in physics, as well as research articles on similar systems. Some possible keywords to search for include "coupled Schrodinger equations," "bound states," and "eigenvalues."

I hope this helps and good luck with your research!