# 数学物理学的高级课题 Advanced Topics in Mathematical Physics MATH3351/MATH6211

$$\sigma_{y}=\sigma F-(T F) \sigma$$
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
$$\sigma=\xi(t) \eta \exp (\Sigma) \eta^{-1} \xi^{-1}(t)$$