这是一份anu澳大利亚国立大学MATH3351/MATH6211的成功案例

$$
\sigma_{y}=\sigma F-(T F) \sigma
$$
where
$$
F=u \sigma+v+w\left[T^{-1}(\sigma)\right]^{-1} \text {. }
$$
One can check the by direct substitution of the operator $\sigma$ and by use of the equation for $\phi$.
Remark 2.33. Theorem $2.32$ is evidently valid for the spectral problem
$$
\lambda \psi=u T \psi+v \psi+w T^{-1} \psi
$$
with the only correction being that the last term for the transform $v[1]$ is absent. The equation goes to the “Riccati equation” analog for the function $\sigma$ :
$$
\mu=u \sigma+v+w\left[T^{-1}(\sigma)\right]^{-1}
$$

MATH3351/MATH6211 COURSE NOTES ：

$$
\xi_{t}=[v / 2+(u+\beta I) T] \xi=Z \xi
$$
which is solved by
$$
\xi=\exp (Z t) \xi_{0} .
$$
Plugging $\Phi$ into (2.135) yields the spectral problem for the difference shift operators:
$$
\mu \Phi(x)=\xi^{-1}[u \xi \Phi(x+1)+v \xi \Phi+w \xi \Phi(x-1)]
$$
Separating variables again, a class of particular solutions is built as
$$
\Phi=\eta \exp (\Sigma x)
$$
hence, we arrive at the matrix spectral problem for $\eta$ :
$$
\mu \eta=\xi^{-1}[u \xi \eta \exp (\Sigma)+v \xi \eta+w \xi \eta \exp (-\Sigma)]
$$
with the operator on the right-hand side and, therefore, spectral parameter $\mu$ parameterized by $t$. Finally, the matrix $\sigma$ is composed as
$$
\sigma=\xi(t) \eta \exp (\Sigma) \eta^{-1} \xi^{-1}(t)
$$