A simple differential equation!

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Main Question or Discussion Point

Just out of pure curiosity, can anyone here give me any advice on the problem of solving the following differential equation

[tex]\left\{\sum_i \vec{\alpha}_i \cdot \nabla_i + \sum_{i neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.[/tex]

where,

[tex][\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}[/tex]

[tex][\alpha_{i\tau} , \alpha_{j\tau}] = \delta_{ij}[/tex]

[tex][\beta_{ij} , \beta_{kl}] = \delta_{ik}\delta_{jl}[/tex]

[tex][\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} =
\left\{\begin{array}{ c c }
0, & \text{ if } i \neq j \\
1, & \text{ if } i=j
\end{array} \right.
[/tex]

Any advice on approaching this problem would be greatly appreciated.

Thank you very much!
 

Answers and Replies

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Just out of pure curiosity, can anyone here give me any advice on the problem of solving the following differential equation

[tex]\left\{\sum_i \vec{\alpha}_i \cdot \nabla_i + \sum_{i neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.[/tex]

where,

[tex][\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}[/tex]

[tex][\alpha_{i\tau} , \alpha_{j\tau}] = \delta_{ij}[/tex]

[tex][\beta_{ij} , \beta_{kl}] = \delta_{ik}\delta_{jl}[/tex]

[tex][\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} =
\left\{\begin{array}{ c c }
0, & \text{ if } i \neq j \\
1, & \text{ if } i=j
\end{array} \right.
[/tex]

Any advice on approaching this problem would be greatly appreciated.

Thank you very much!
It doesn't look simple to me ...lol...
 

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