# 数学物理学的实用技能 Practical Skills for Mathematical Physics  PHYS156

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a linear invertible map $T, \mathbf{R} \rightarrow \mathbf{R}$ such that for any $\psi(x, t)$ and $\varphi(x, t) \in \mathbf{R}$. $x \in R^{n}, t \in R$ we have
$$T(\psi \varphi)=T(\psi) T(\varphi), \quad T(1)=1 .$$
The automorphism with the defining allows us to write down a wide class of functional-differential-difference and difference-difference equations starting from
$$\psi_{t}(x, t)=\sum_{m=-M}^{N} U_{m} T^{m} \psi$$
where $M$ and $N$ are integers. For example, the operator $T$ can be chosen as
$$T \psi(x, t)=\psi(q x+\delta, t),$$
where $q \in G L(n, \mathbb{C}), \delta \in R^{n}$. Another choice gives
$$T \psi(x)=W \psi(x) W^{-1}, \quad W \in G L(n, \mathbb{C})$$

## PHYS156COURSE NOTES ：

The formalism for the second DT from may be similarly constructed on the ground of the identity
$$T^{m} \varphi=\prod_{k=0}^{m} T^{k}\left(\sigma^{-}\right) T^{-1} \varphi=B_{m}^{-}\left(\sigma^{-}\right) T^{-1} \varphi$$
The definition of the lattice Bell polynomials of the second type $B_{m}^{-}\left(\sigma^{-}\right)$can be extracted. The evolution equation for $\sigma^{-}$is similar :
$$\sigma_{t}^{-}=\sum_{m=-M}^{N}\left[U_{m} B_{m}^{-}\left(\sigma^{-}\right)-\sigma^{-} T^{-1}\left(U_{m}\right) B_{m-1}^{-}\left(\sigma^{-}\right)\right]$$