这是一份liverpool利物浦大学MATH326的成功案例

$$
\langle x \mid p\rangle=h^{-1 / 2} e^{i x p / \hbar}
$$
are improper eigenstates ${ }^{2}$ of $\widehat{p}$ and $\widehat{H}{0}$, normalized according to Dirac’s delta function, $$ \left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right) . $$ Closure relations (or resolutions of the unit operator $\hat{1}$ ) may therefore be written in momentum or coordinate representation as $$ \widehat{1}=\int{-\infty}^{\infty} d x|x\rangle\left\langle x\left|=\int_{-\infty}^{\infty} d p\right| p\right\rangle\langle p|
$$

MATH325 COURSE NOTES ：

The properties of $T(p)$ as a function of the complex momentum $p$ are of importance for many application. Let the potential function $V(x)$ be such that
$$
\int_{-\infty}^{\infty} d x|V(x)|\left(1+x^{2}\right)<\infty . $$ Then $T(p)$ is meromorphic in $\operatorname{Im} p>0$ with a finite number $n_{b}$ of simple poles $i \beta_{1}, i \beta_{2}, \ldots,, i \beta_{n}, \beta_{j}>0$ on the imaginary axis. The numbers $-\beta_{j}^{2} /(2 m)$ are the eigenvalues of $H$. Moreover,
$$
T(p)=1+O(1 / p) \text { as }|p| \rightarrow \infty, \quad \operatorname{Im} p \geq 0,
$$
and there can only be a zero at the real axis, at $p=0$,
$$
|T(p)|>0 \quad \operatorname{Im} p \geq 0, p \neq 0 .
$$
In the generic case $T(0)=0$, and
$$
T(p)=\gamma p+o(p), \gamma \neq 0, \text { as } p \rightarrow 0, \quad \operatorname{Im} p \geq 0 .
$$