Consider the following system of coupled differential equation.
$$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B \end{bmatrix} \times \left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right]$$
I tried diagonalyzng the matrix using the eigenvectors of the coefficient matrix. But, since the matrix is time dependent so are its eigenvectors. So, how can one decouple this system?
Best Answer
You can't. As you say, the eigenvectors of this matrix are time dependent, so it's no use to make linear combinations of $c_1$ and $c_2$ using these eigenvectors, because the time derivative will introduce extra terms by the product rule.
What you can do, is try to write $c_1(t) = \alpha(t) x(t)$ and $c_2(t) = \beta(t) y(t)$, and try to choose $\alpha(t)$ and $\beta(t)$ such that the system simplifies. As it turns out, if you choose $\alpha(t) = e^{-B t}$ and $\beta(t) = e^{B t}$, you obtain \begin{equation} \begin{pmatrix} \frac{\text{d} x}{\text{d} t} \\ \frac{\text{d} y}{\text{d} t} \end{pmatrix} = \begin{pmatrix} 0 & -e^{2 B t} V(t) \\ - e^{-2 B t} V(t) & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \end{equation} This is equivalent to the second order ODE \begin{equation} \frac{\text{d}^2 x}{\text{d} t^2} - \left(2 B + \frac{V'(t)}{V(t)}\right)\frac{\text{d} x}{\text{d} t} - V(t)^2 x = 0, \end{equation} which is of Sturm-Liouville type. For general $V(t)$, there are only general results for the solutions of such equations (see e.g. Titchmarsh, Eigenfunction Expansions). If $V(t)$ is periodic in time though, you can invoke Floquet theory.