Let $L=-\partial_{x}^{2}+u(x, t)$ be a Schrödinger operator and let $A$ be another differential operator which does not contain derivatives with respect to $t$ and such that

$$L_{t}=[L, A] .$$

Show that the eigenvalues of $L$ are independent of $t$, and deduce that if $f$ is an eigenfunction of $L$ then so is $f_{t}+A f$. [You may assume that $L$ is self-adjoint.]

Let $f$ be an eigenfunction of $L$ corresponding to an eigenvalue $\lambda$ which is nondegenerate. Show that there exists a function $\hat{f}=\hat{f}(x, t, \lambda)$ such that

$$L \hat{f}=\lambda \hat{f}, \quad \hat{f}_{t}+A \hat{f}=0 .$$

Assume

$$A=\partial_{x}^{3}+a_{1} \partial_{x}+a_{0},$$

where $a_{k}=a_{k}(x, t), k=0,1$ are functions. Show that the system $(*)$ is equivalent to a pair of first order matrix PDEs

$$\partial_{x} F=U F, \quad \partial_{t} F=V F$$

where $F=\left(\hat{f}, \partial_{x} \hat{f}\right)^{T}$ and $U, V$ are $2 \times 2$ matrices which should be determined.