Kleinert · 제9장 통계역학

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int_{t_{a}}^{\infty} d t_{b} \exp \left\{i E\left(t_{b}-t_{a}\right) / \hbar\right\}\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{b} t_{a}\right) $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \exp \left[\frac{i}{\hbar} \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)-\frac{\mathbf{p}^{2}}{2 M \hbar}\left(\tau_{b}-\tau_{a}\right)\right] $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-i \int_{\tau_{a}}^{\infty} d \tau_{b} \exp \left\{E\left(\tau_{b}-\tau_{a}\right) / \hbar\right\}\left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{b} \tau_{a}\right) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)} \frac{i \hbar}{E-\hbar^{2} \mathbf{k}^{2} / 2 M+i \eta} $$

$$ e^{i \mathbf{k R}}=\sum_{l=0} a_{l}(i k R) \sum_{\mathbf{m}} Y_{l \mathbf{m}}(\hat{\mathbf{k}}) Y_{l \mathbf{m}}^{*}(\hat{\mathbf{R}}) $$

$$ \int d \hat{\mathbf{k}} Y_{l \mathbf{m}}(\hat{\mathbf{k}})=\delta_{l 0} \delta_{\mathbf{m} 0} \sqrt{S_{D}} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 M i}{(2 \pi)^{D}} \int_{0}^{\infty} d k k^{D-1} \frac{1}{k^{2}+\kappa^{2}} a_{0}(i k R) $$

$$ \kappa \equiv \sqrt{-2 M E / \hbar^{2}} $$

$$ a_{0}(i k R)=(2 \pi)^{D / 2} J_{D / 2-1}(k R) /(k R)^{D / 2-1}, $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 M i}{(2 \pi)^{D / 2}} R^{1-D / 2} \int_{0}^{\infty} d k k^{D / 2} \frac{1}{k^{2}+\kappa^{2}} J_{D / 2-1}(k R) $$

$$ \int_{0}^{\infty} d k \frac{k^{\nu+1}}{\left(k^{2}+a^{2}\right)^{\mu+1}} J_{\nu}(k b)=\frac{a^{\nu-\mu} b^{\mu}}{2^{\mu} \Gamma(\mu+1)} K_{\nu-\mu}(a b) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{i M}{\pi \hbar} K_{0}\left(\kappa\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right|\right) $$

$$ \left|\mathbf{x}_{b}-\mathbf{x}_{a}\right|=\sqrt{r_{b}^{2}+r_{a}^{2}-2 r_{a} r_{b} \cos \left(\varphi_{b}-\varphi_{a}\right)} $$

$$ K_{0}\left(\kappa\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right|\right)=\sum_{m=-\infty}^{\infty} I_{m}\left(\kappa r_{<}\right) K_{m}\left(\kappa r_{>}\right) e^{i m\left(\varphi_{b}-\varphi_{a}\right)} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-\frac{2 i M}{\hbar} \sum_{m} I_{m}\left(\kappa r_{<}\right) K_{m}\left(\kappa r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} . $$

$$ \begin{align*} I_{\mu}\left(e^{-i \pi / 2} z\right) & =e^{-i \pi \mu / 2} J_{\mu}(z) \\ K_{\mu}\left(e^{-i \pi / 2} z\right) & =i \frac{\pi}{2} e^{i \pi \mu / 2} H_{\mu}^{(1)}(z) \end{align*} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E+i \eta}=\frac{\pi M}{\hbar} \sum_{m} J_{m}\left(k r_{<}\right) H_{m}^{(1)}\left(k r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} . $$

$$ \begin{align*} H_{\mu}^{(1)}\left(e^{i \pi} z\right) & =-H_{-\mu}^{(2)}(z) \equiv-e^{-i \pi \mu} H_{\mu}^{(2)}(z) \\ J_{\mu}\left(e^{i \pi} z\right) & =e^{i \pi \mu} J_{\mu}(z) \end{align*} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E-i \eta}=-\frac{\pi M}{\hbar} \sum_{m} J_{m}\left(k r_{<}\right) H_{m}^{(2)}\left(k r_{>}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} . $$

$$ J_{\mu}(z)=\frac{1}{2}\left[H_{\mu}^{(1)}(z)+H_{\mu}^{(2)}(z)\right] $$

$$ \operatorname{disc}\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\frac{2 \pi M}{\hbar} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{b}\right) J_{m}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{b}\right)} $$

$$ \begin{align*} \int_{-\infty}^{\infty} & \frac{d E}{2 \pi \hbar} \operatorname{disc}\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} \\ & =\int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} \frac{2 \pi M}{\hbar} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{b}\right) J_{m}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{b}\right)}=\delta\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) \end{align*} $$

$$ \int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} \frac{2 \pi M}{\hbar}=\int_{0}^{\infty} d k k $$

$$ \begin{align*} \int_{-\infty}^{\infty} \frac{d E}{2 \pi \hbar} \operatorname{disc}\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =\int \frac{d^{2} k}{(2 \pi)^{2}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)}=\frac{1}{2 \pi} \int_{0}^{\infty} d k k J_{0}\left(k\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right|\right) \\ & =\sum_{m=-\infty}^{\infty} \int_{0}^{\infty} d k k J_{m}\left(k r_{b}\right) J_{m}\left(k r_{a}\right) \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \\ & =\frac{1}{\sqrt{r_{b} r_{a}}} \delta\left(r_{b}-r_{a}\right) \delta\left(\varphi_{b}-\varphi_{a}\right) \end{align*} $$

$$ \begin{align*} \left(r_{b} \mid r_{a}\right)_{E, l}= & -i \sqrt{M \omega / \hbar} \sqrt{M \omega r_{b} r_{a} / \hbar} \int_{\tau_{a}}^{\infty} d \tau_{b} e^{E\left(\tau_{b}-\tau_{a}\right) / \hbar} \frac{1}{\sin \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]} \\ & \times e^{-\frac{1}{\hbar} \frac{M \omega}{2} \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right)\right]\left(r_{b}^{2}+r_{a}^{2}\right)} I_{\mu}\left(\frac{M \omega r_{b} r_{a}}{\hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\right) \end{align*} $$

$$ \begin{align*} \int_{0}^{\infty} & d x[\operatorname{coth}(x / 2)]^{2 \nu} e^{-\beta \cosh x} J_{\mu}(\alpha \sinh x) \\ & =\frac{\Gamma((1+\mu) / 2-\nu)}{\alpha \Gamma(\mu+1)} W_{\nu, \mu / 2}\left(\sqrt{\alpha^{2}+\beta^{2}}+\beta\right) M_{-\nu, \mu / 2}\left(\sqrt{\alpha^{2}+\beta^{2}}-\beta\right) \end{align*} $$

$$ \begin{gather*} \int_{0}^{\infty} \frac{d y}{\sinh y} e^{2 \nu y} \exp \left[-\frac{1}{2} t\left(\alpha_{b}-\alpha_{a}\right) \operatorname{coth} y\right] J_{\mu}\left(t \frac{\sqrt{\alpha_{b} \alpha_{a}}}{\sinh y}\right) \\ =\frac{\Gamma((1+\mu) / 2-\nu)}{t \sqrt{\alpha_{b} \alpha_{a} \Gamma(\mu+1)}} W_{\nu, \mu / 2}\left(t \alpha_{b}\right) M_{-\nu, \mu / 2}\left(-t \alpha_{a}\right) \end{gather*} $$

$$ M_{-\nu, \mu / 2}(z) \equiv e^{-i(\mu+1) \pi / 2} M_{\nu, \mu / 2}(-z) $$

$$ \begin{align*} \int_{0}^{\infty} \frac{d y}{\sinh y} e^{2 \nu y} & \exp \left[-\frac{1}{2} t\left(\alpha_{b}+\alpha_{a}\right) \operatorname{coth} y\right] I_{\mu}\left(\frac{t \sqrt{\alpha_{b} \alpha_{a}}}{\sinh y}\right) \\ & =\frac{\Gamma((1+\mu) / 2-\nu)}{t \sqrt{\alpha_{b} \alpha_{a} \Gamma(\mu+1)}} W_{\nu, \mu / 2}\left(t \alpha_{b}\right) M_{\nu, \mu / 2}\left(t \alpha_{a}\right) \end{align*} $$

$$ \begin{align*} & \alpha_{b}>\alpha_{a}>0, \quad \operatorname{Re}[(1+\mu) / 2-\nu]>0, \\ & \operatorname{Re} t>0, \quad|\arg t|<\pi . \end{align*} $$

$$ y=\omega\left(t_{b}-t_{a}\right), \quad \alpha_{b}=\frac{M}{\hbar} \omega r_{b}^{2}, \quad \alpha_{a}=\frac{M \omega}{\hbar} r_{a}^{2}, \quad \nu=E / 2 \omega \hbar $$

$$ \left(r_{b} \mid r_{a}\right)_{E, l}=-i \frac{1}{\omega} \frac{1}{\sqrt{r_{b} r_{a}}} \frac{\Gamma((1+\mu) / 2-\nu)}{\Gamma(\mu+1)} W_{\nu, \mu / 2}\left(\frac{M \omega}{\hbar} r_{b}^{2}\right) M_{\nu, \mu / 2}\left(\frac{M \omega}{\hbar} r_{a}^{2}\right) $$

$$ \nu=\nu_{r} \equiv(1+\mu) / 2+n_{r} $$

$$ \Gamma((1+\mu) / 2-\nu) \stackrel{\nu \sim \nu_{r}}{\sim}-\frac{(-1)^{n_{r}}}{n_{r}!} \frac{1}{\nu-\nu_{r}} . $$

$$ E=\hbar \omega\left(2 n_{r}+l+D / 2\right) $$

$$ n \equiv 2 n_{r}+l $$

$$ E_{n}=\hbar \omega(n+D / 2) $$

$$ \begin{align*} & R_{n_{r} l}\left(r_{b}\right) R_{n_{r} l}\left(r_{a}\right)=\frac{1}{\sqrt{r_{b} r_{a}}} \frac{2(-1)^{n_{r}}}{\Gamma(\mu+1) n_{r}!} \\ & \quad \times W_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}\left(M \omega r_{b}^{2} / \hbar\right) M_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}\left(M \omega r_{a}^{2} / \hbar\right) \end{align*} $$

$$ \begin{align*} W_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}(z) & =e^{-z / 2} z^{(1+\mu) / 2} U\left(-n_{r}, 1+\mu, z\right) \\ M_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}(z) & =e^{-z / 2} z^{(1+\mu) / 2} M\left(-n_{r}, 1+\mu, z\right) \end{align*} $$

$$ M_{-(1+\mu) / 2-n_{r}, \frac{\mu}{2}}(z)=e^{-z / 2} z^{(1+\mu) / 2} M\left(1+\mu+n_{r}, 1+\mu, z\right) $$

$$ M(a, b, z)=e^{z} M(b-a, b,-z) $$

$$ M\left(1+\mu+n_{r}, 1+\mu, z\right)=e^{z} M\left(-n_{r}, 1+\mu,-z\right) $$

$$ M_{-(1+\mu) / 2-n_{r}, \frac{\mu}{2}}(z)=e^{z / 2} z^{(1+\mu) / 2} M\left(-n_{r}, 1+\mu,-z\right) $$

$$ M(a, b, z) \equiv{ }_{1} F_{1}(a ; b ; z)=1+\frac{a}{b} z+\frac{a(a+1)}{b(b+1)} \frac{z}{z!}+\ldots $$

$$ U(a, b, z)=\frac{\pi}{\sin \pi b}\left[\frac{M(a, b, z)}{\Gamma(1+a-b) \Gamma(b)}-z^{1-b} \frac{M(1+a-b, 2-b, z)}{\Gamma(a) \Gamma(2-b)}\right] $$

$$ U\left(-n_{r}, 1+\mu, z\right)=\frac{\Gamma(-\mu)}{\Gamma\left(-n_{r}-\mu\right)} M\left(-n_{r}, 1+\mu, z\right) $$

$$ \begin{align*} & W_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}\left(z_{b}\right) M_{(1+\mu) / 2+n_{r}, \frac{\mu}{2}}\left(z_{a}\right)=\frac{\Gamma(-\mu)}{\Gamma\left(-n_{r}-\mu\right)} e^{-\left(z_{b}+z_{a}\right) / 2} \\ & \times\left(z_{b} z_{a}\right)^{(1+\mu) / 2} M\left(-n_{r}, 1+\mu, z_{b}\right) M\left(-n_{r}, 1+\mu, z_{a}\right) \end{align*} $$

$$ \begin{align*} R_{n_{r} l}\left(r_{b}\right) & R_{n_{r} l}\left(r_{a}\right)=\sqrt{\frac{M \omega}{\hbar}} \frac{2(-)^{n_{r}} \Gamma(-\mu)}{\Gamma\left(-n_{r}-\mu\right) \Gamma(1+\mu) n_{r}!} \\ & \times e^{-M \omega\left(r_{b}^{2}+r_{a}^{2}\right) / 2 \hbar}\left(M \omega r_{b} r_{a} / \hbar\right)^{1 / 2+\mu} \\ & \times M\left(-n_{r}, 1+\mu, M \omega r_{b}^{2} / \hbar\right) M\left(-n_{r}, 1+\mu, M \omega r_{a}^{2} / \hbar\right) \end{align*} $$

$$ \frac{(-)^{n_{r}} \Gamma(-\mu)}{\Gamma\left(-n_{r}-\mu\right)}=\frac{\Gamma\left(n_{r}+1+\mu\right)}{\Gamma(1+\mu)} $$

$$ \begin{align*} R_{n_{r} l}(r)=C_{n_{r} l} & (M \omega / \hbar)^{1 / 4}\left(M \omega r^{2} / \hbar\right)^{l / 2+(D-1) / 4} \\ & \times e^{-M \omega r^{2} / 2 \hbar} M\left(-n_{r}, l+D / 2, M \omega r^{2} / \hbar\right) \end{align*} $$

$$ C_{n_{r} l}=\frac{\sqrt{2}}{\Gamma(1+\mu)} \sqrt{\frac{\left(n_{r}+\mu\right)!}{n_{r}!}} $$

$$ L_{n}^{\mu}(z) \equiv \frac{(n+\mu)!}{n!\mu!} M(-n, \mu+1, z) $$

$$ \int_{0}^{\infty} d z e^{-z} z^{\mu} L_{n}^{\mu}(z) L_{n^{\prime}}^{\mu}(z)=\delta_{n n^{\prime}} \frac{(n+\mu)!}{n!} $$

$$ \int_{0}^{\infty} d r R_{n_{r} l}(r) R_{n_{r}^{\prime} l}(r)=\delta_{n_{r} n_{r}^{\prime}} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\sum_{n_{r}=0}^{\infty} R_{n_{r} l}\left(r_{b}\right) R_{n_{r} l}\left(r_{a}\right) e^{-E_{n}\left(\tau_{b}-\tau_{a}\right) / \hbar} $$

$$ E_{n}=\hbar \omega(n+D / 2)=\hbar \omega\left(2 n_{r}+l+D / 2\right) $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{1}{\left(r_{b} r_{a}\right)^{(D-1) / 2}} \sum_{l=0}^{\infty}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l} \sum_{m} Y_{l m}\left(\hat{\mathbf{x}}_{b}\right) Y_{l m}^{*}\left(\hat{\mathbf{x}}_{a}\right) $$

$$ \psi_{n_{r} l m}(\mathbf{x})=\frac{1}{r^{(D-1) / 2}} R_{n_{r} l}(r) Y_{l m}(\hat{\mathbf{x}}) $$

$$ Y_{\mathrm{e}, \varnothing}(\hat{x})=\frac{1}{\sqrt{2}}(\Theta(x) \pm \Theta(-x)) $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)=\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mathrm{e}} Y_{\mathrm{e}}\left(\hat{x}_{b}\right) Y_{\mathrm{e}}\left(\hat{x}_{a}\right)+\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\varnothing} Y_{\varnothing}\left(\hat{x}_{b}\right) Y_{\varnothing}\left(\hat{x}_{a}\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mathrm{e}, \varnothing}=\left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) \pm\left(-x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mathrm{e}, \emptyset} & =\frac{1}{\sqrt{2 \pi}} \sqrt{\frac{M \omega / \hbar}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \\ & \times \exp \left[-\frac{M \omega}{2 \hbar}\left(r_{b}^{2}+r_{a}^{2}\right) \cot \omega\left(\tau_{b}-\tau_{a}\right)\right] 2\left\{\begin{array}{c} \cosh \\ \sinh \end{array}\right\}\left(M \omega r_{b} r_{a} / \hbar\right) \end{align*} $$

$$ \sqrt{z} I_{\mp \frac{1}{2}}(z)=\frac{2}{2 \pi}\left\{\begin{array}{l} \cosh z \\ \sinh z \end{array}\right. $$

$$ E= \begin{cases}\hbar \omega\left(2 n_{r}+\frac{1}{2}\right) & \text { even } \\ \hbar \omega\left(2 n_{r}+\frac{3}{2}\right) & \text { odd }\end{cases} $$

$$ E=\hbar \omega\left(n+\frac{1}{2}\right) $$

$$ \begin{align*} & R_{n_{r}, \mathrm{e}}(r)=(M \omega / \hbar)^{1 / 4} \sqrt{\frac{2 \Gamma\left(n_{r}+\frac{1}{2}\right)}{\pi n_{r}!}} M\left(-n_{r}, \frac{1}{2}, M \omega r^{2} / \hbar\right) \\ & R_{n_{r}, \emptyset}(r)=(M \omega / \hbar)^{1 / 4} \sqrt{\frac{2 \Gamma\left(n_{r}+\frac{3}{2}\right)}{(\pi / 4) n_{r}!}} \sqrt{\frac{M \omega r^{2}}{\hbar}} M\left(-n_{r}, \frac{3}{2}, M \omega r^{2} / \hbar\right) \end{align*} $$

$$ \begin{align*} M\left(-n, \frac{1}{2}, x^{2}\right) & =\frac{n!}{(2 n)!}(-)^{n} H_{2 n}(x) \\ M\left(-n, \frac{3}{2}, x^{2}\right) & =\frac{n!}{(2 n+1)!}(-)^{n} H_{2 n+1}(x) / 2 \sqrt{x} \end{align*} $$

$$ \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)=(2 \pi)^{1 / 2} 2^{-2 z+1 / 2} \Gamma(2 z) $$

$$ R_{n}(r)=N_{n} \sqrt{2} \lambda_{\omega}^{-1 / 2} e^{-r^{2} / 2 \lambda_{\omega}^{2}} H_{n}\left(r / \lambda_{\omega}\right), \quad n=0,1,2, \ldots $$

$$ \lambda_{\omega} \equiv \sqrt{\frac{\hbar}{M \omega}}, \quad N_{n}=\frac{1}{\sqrt{2^{n} n!\sqrt{\pi}}} $$

$$ \begin{align*} & \lim _{n_{r} \rightarrow \infty}\left\{\Gamma\left(1-n_{r}-b\right) U\left(-a, b, \mp z / n_{r}\right)\right\} \\ & =z^{-\frac{1}{2}(b-1)}\left\{\begin{array}{l} 2 K_{b-1}(2 \sqrt{z}) \\ -i \pi e^{i \pi b} H_{b-1}^{(1)}(2 \sqrt{z}) \quad(\operatorname{Im} z>0), \end{array}\right. \\ & \lim _{n_{r} \rightarrow \infty} M\left(-a, b, \mp z / n_{r}\right) / \Gamma(b)=z^{-\frac{1}{2}(b-1)}\left\{\begin{array}{l} I_{b-1}(2 \sqrt{z}) \\ J_{b-1}(2 \sqrt{z}), \end{array}\right. \end{align*} $$

$$ R_{n_{r} l}(r) \xrightarrow{n_{r} \longrightarrow \infty} C_{n_{r} l}\left(M \omega r^{2} / \hbar\right)^{(\mu / 2+1 / 2)}\left(k^{2} r^{2} / 2\right)^{-\mu / 2} \Gamma(1+\mu) J_{\mu}(k r), $$

$$ C_{n_{r} l} \xrightarrow{n_{r} \longrightarrow \infty} \sqrt{2}\left(\frac{E}{2 \hbar \omega}\right)^{\mu / 2} \frac{1}{\Gamma(1+\mu)} $$

$$ R_{n_{r} l}(r) \xrightarrow{n_{r} \longrightarrow \infty} r^{1 / 2} \sqrt{M \omega / \hbar} \sqrt{2} J_{\mu}(k r) . $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\sum_{n_{r}} R_{n_{r} l}\left(r_{b}\right) R_{n_{r} l}\left(r_{b}\right) e^{-E_{n}\left(\tau_{b}-\tau_{a}\right) / \hbar} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mu}=\sqrt{r_{b} r_{a}} \int_{0}^{\infty} d k k J_{\mu}\left(k r_{b}\right) J_{\mu}\left(k r_{a}\right) e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} \exp \left[i k\left(x_{b}-x_{a}\right)-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)\right] $$

$$ \begin{align*} \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{\mathrm{e}, \emptyset} & =\int_{0}^{\infty} \frac{d k}{\pi}\left[\cos k\left(r_{b}-r_{a}\right) \pm \cos k\left(r_{b}+r_{a}\right)\right] e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} \\ & =2 \int_{0}^{\infty} \frac{d k}{\pi}\left\{\begin{array}{cc} \cos k r_{b} & \cos k r_{a} \\ \sin k r_{b} & \sin k r_{a} \end{array}\right\} e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} \end{align*} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) & =\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right)} e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} \\ & =\frac{1}{(2 \pi)^{D / 2}} \int_{0}^{\infty} d k k^{2 \nu} \frac{1}{(k R)^{\nu}} J_{\nu}(k R) e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} \end{align*} $$

$$ \frac{1}{(k R)^{\nu}} J_{\nu}(k R)=\frac{2^{\nu} \Gamma(\nu)}{\left(k^{2} r_{b} r_{a}\right)^{\nu}} \sum_{l=0}^{\infty}(\nu+l) J_{\nu+l}\left(k r_{b}\right) J_{\nu+l}\left(k r_{a}\right) C_{l}^{(\nu)}(\Delta \vartheta) $$

$$ \frac{1}{(k R)^{\nu}} J_{\nu}(k R)=\frac{(2 \pi)^{D / 2}}{\left(k^{2} r_{b} r_{a}\right)^{\nu}} \sum_{l=0}^{\infty} J_{\nu+l}\left(k r_{b}\right) J_{\nu+l}\left(k r_{a}\right) \sum_{\mathbf{m}} Y_{l \mathbf{m}}\left(\hat{\mathbf{x}}_{b}\right) Y_{l \mathbf{m}}^{*}\left(\hat{\mathbf{x}}_{a}\right) $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{l}=\sqrt{r_{b} r_{a}} \int_{0}^{\infty} d k k J_{\nu+l}\left(k r_{b}\right) J_{\nu+l}\left(k r_{a}\right) e^{-\frac{\hbar k^{2}}{2 M}\left(\tau_{b}-\tau_{a}\right)} $$

$$ \sqrt{z} J_{\mp 1 / 2}(z)=\frac{2}{\sqrt{2 \pi}}\left\{\begin{array}{c} \cos z \\ \sin z \end{array}\right\} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left(z_{b} \tau_{b} \mid z_{a} \tau_{a}\right)\left(\mathbf{x}_{b}^{\perp} \tau_{b} \mid \mathbf{x}_{a}^{\perp} \tau_{a}\right) $$

$$ \left(z_{b} \tau_{b} \mid z_{a} \tau_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right) / M}} \exp \left\{-\frac{M}{2 \hbar} \frac{\left(z_{b}-z_{a}\right)^{2}}{\tau_{b}-\tau_{a}}\right\} $$

$$ \left(\mathbf{x}_{b}^{\perp} \tau_{b} \mid \mathbf{x}_{a}^{\perp} \tau_{a}\right)=\frac{M}{2 \pi \hbar\left(\tau_{b}-\tau_{a}\right)} \frac{\omega\left(\tau_{b}-\tau_{a}\right) / 2}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]} \exp \left[-\mathcal{A}_{l}^{\perp} / \hbar\right], $$

$$ \mathcal{A}_{l}^{\perp}=\frac{M \omega}{2}\left\{\frac{1}{2} \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]\left(\mathbf{x}_{b}^{\perp}-\mathbf{x}_{a}^{\perp}\right)^{2}+\mathbf{x}_{a}^{\perp} \times \mathbf{x}_{b}^{\perp}\right\} . $$

$$ \mathbf{A}=\frac{1}{2} \mathbf{B} \times \mathbf{x} $$

$$ \mathbf{A}=(0, B x, 0) $$

$$ \tilde{\mathcal{A}}_{\mathrm{cl}}^{\perp}=\mathcal{A}_{\mathrm{cl}}^{\perp}+\frac{M \omega}{2}\left(x_{b} y_{b}-x_{a} y_{a}\right) $$

$$ \left(\mathbf{x}_{b}^{\perp} \tau_{b} \mid \mathbf{x}_{a}^{\perp} \tau_{a}\right)=\int \frac{d p_{y}}{2 \pi \hbar} e^{i p_{y}\left(y_{b}-y_{a}\right) / \hbar}\left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)_{x_{0}=p_{y} / M \omega} $$

$$ \left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right)_{x_{0}}=\sqrt{\frac{M \omega}{2 \pi \hbar \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}} \exp \left(-\frac{1}{\hbar} \mathcal{A}_{\mathrm{cl}}^{\mathrm{os}}\right) $$

$$ \begin{gather*} \mathcal{A}_{\mathrm{cl}}^{\mathrm{os}}=\frac{M \omega}{2 \sinh \left[\omega\left(\tau_{b}-\tau_{a}\right)\right]}\left\{\left[\left(x_{b}-x_{0}\right)^{2}+\left(x_{a}-x_{0}\right)^{2}\right] \cosh \left[\omega\left(\tau_{b}-\tau_{b}\right)\right]\right. \\ \left.-2\left(x_{b}-x_{0}\right)\left(x_{a}-x_{0}\right)\right\} \end{gather*} $$

$$ \left(x_{a} \tau_{b} \mid x_{a} \tau_{a}\right)_{x_{0}}=\sum_{n=0}^{\infty} \psi_{n}\left(x_{b}-x_{0}\right) \psi_{n}\left(x_{a}-x_{0}\right) e^{-\left(n+\frac{1}{2}\right) \omega\left(\tau_{b}-\tau_{a}\right)}, $$

$$ \begin{align*} \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)= & \int \frac{d p_{z}}{2 \pi \hbar} \int \frac{d p_{y}}{2 \pi \hbar} e^{i p_{z}\left(z_{b}-z_{a}\right) / \hbar} \\ & \times \sum_{n=0}^{\infty} \psi_{n}\left(x_{b}-p_{y} / M \omega\right) \psi_{n}\left(x_{a}-p_{y} / M \omega\right) e^{-\left(n+\frac{1}{2}\right) \omega\left(\tau_{b}-\tau_{a}\right)} \end{align*} $$

$$ E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}}^{\perp}=\frac{M}{2}\left\{\frac { \omega } { 2 } \operatorname { c o t h } [ \omega ( \tau _ { b } - \tau _ { a } ) / 2 ] \left[r_{b}{ }^{2}+r_{a}{ }^{2}-2 r_{b} r_{a}\right.\right. & \left.\cos \left(\varphi_{b}-\varphi_{a}\right)\right] \\ & \left.-i \omega r_{b} r_{a} \sin \left(\varphi_{b}-\varphi_{a}\right)\right\} \end{align*} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}}^{\perp} & =\frac{M}{2} \frac{\omega}{2} \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]\left(r_{b}^{2}+r_{a}^{2}\right) \\ & -\frac{M}{2} \frac{\omega}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]} \cos \left[\varphi_{b}-\varphi_{a}-i \omega\left(\tau_{b}-\tau_{a}\right) / 2\right] \end{align*} $$

$$ \begin{align*} e^{-\mathcal{A}_{\mathrm{cl}}^{\perp} / \hbar}= & \exp \left\{-\frac{M}{2 \hbar} \frac{\omega}{2} \operatorname{coth}\left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]\left(r_{b}^{2}+r_{a}^{2}\right)\right\} \\ & \times \sum_{m=-\infty}^{\infty} I_{m}\left(\frac{M \omega}{2 \hbar} \frac{r_{b} r_{a}}{\sinh \left[\omega\left(\tau_{b}-\tau_{a}\right) / 2\right]}\right) e^{m \omega\left(\tau_{b}-\tau_{a}\right) / 2} e^{i m\left(\varphi_{b}-\varphi_{a}\right)} \end{align*} $$

$$ \left(\mathbf{x}_{b}^{\perp} \tau_{b} \mid \mathbf{x}_{a}^{\perp} \tau_{a}\right)=\frac{1}{\sqrt{r_{b} r_{a}}} \sum_{m}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \frac{1}{2 \pi} e^{i m\left(\varphi_{b}-\varphi_{a}\right)}, $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m}=\sqrt{r_{b} r_{a}} \frac{M \omega}{2 \hbar \eta} \frac{\eta}{\sinh \eta} \exp \left[-\frac{M}{2 \hbar} \frac{\omega}{2} \operatorname{coth} \eta\left(r_{b}^{2}+r_{a}^{2}\right)\right] I_{m}\left(\frac{M \omega r_{b} r_{a}}{2 \hbar \sinh \eta}\right) e^{m \eta} $$

$$ \eta \equiv \omega\left(\tau_{b}-\tau_{a}\right) / 2 $$

$$ \begin{align*} \left(r_{b} \mid r_{a}\right)_{m, E} & =-i \int_{\tau_{a}}^{\infty} d \tau_{b} e^{E\left(\tau_{b}-\tau_{a}\right) / \hbar}\left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)_{m} \\ & =-i \sqrt{r_{b} r_{a}} \frac{M}{\hbar} \int_{0}^{\infty} d \eta e^{2 \nu \eta} \frac{1}{\sinh \eta} e^{-(M \omega / 4 \hbar) \operatorname{coth} \eta\left(r_{b}^{2}+r_{a}^{2}\right)} I_{m}\left(\frac{M \omega r_{b} r_{a}}{2 \hbar \sinh \eta}\right) \end{align*} $$

$$ \begin{align*} \left(r_{b} \mid r_{a}\right)_{m, E}= & -i \sqrt{r_{b} r_{a}} \frac{M}{\hbar} \frac{\Gamma\left(\frac{1}{2}-\nu+\frac{|m|}{2}\right)}{(M \omega / 2 \hbar) r_{b} r_{a} \Gamma(|m|+1)} \\ & \times W_{\nu,|m| / 2}\left(\frac{M \omega}{2 \hbar} r_{b}^{2}\right) M_{\nu,|m| / 2}\left(\frac{M \omega}{2 \hbar} r_{a}^{2}\right) \end{align*} $$

$$ \nu \equiv \frac{E}{\omega \hbar}+\frac{m}{2} $$

$$ \nu=\nu_{r} \equiv n_{r}+\frac{1}{2}+\frac{|m|}{2} $$

$$ \Gamma(1 / 2-\nu-|m| / 2) \approx-\frac{1}{n_{r}!} \frac{(-1)^{n_{r}}}{\nu-\nu_{r}} \sim-\frac{(-1)^{n_{r}}}{n_{r}!} \frac{\omega \hbar}{E-E_{n_{r} m}} $$

$$ E_{n_{r} m}=\hbar \omega\left(n_{r}+\frac{1}{2}+\frac{|m|}{2}-\frac{m}{2}\right) $$

$$ \begin{align*} M_{-\nu, m / 2}(z) & =e^{z / 2} z^{\frac{1+m}{2}} M\left(-n_{r}, 1+m,-z\right) \\ W_{\nu, m / 2}(z) & =e^{-z / 2} z^{\frac{1+m}{2}}(-)^{n_{r}} \frac{\left(n_{r}+m\right)!}{m!} M\left(-n_{r}, 1+m, z\right) \end{align*} $$

$$ \left(r_{b} \mid r_{a}\right)_{m, E} \sim \frac{i \hbar}{E-E_{n_{r} m}} R_{n_{r} m}\left(r_{b}\right) R_{n_{r} m}\left(r_{a}\right), $$

$$ \begin{align*} R_{n_{r} m}(r)= & \sqrt{r}\left(\frac{M \omega}{\hbar}\right)^{1 / 2} \sqrt{\frac{\left(n_{r}+|m|\right)!}{n_{r}!}} \frac{1}{|m|!} \\ & \times \exp \left(-\frac{M \omega}{4 \hbar} r^{2}\right)\left(\frac{M \omega}{2 \hbar} r^{2}\right)^{|m| / 2} M\left(-n_{r}, 1+|m|, \frac{M \omega}{2 \hbar} r^{2}\right) \end{align*} $$

$$ \begin{align*} R_{n_{r} m}= & \sqrt{r}\left(\frac{M \omega}{\hbar}\right)^{1 / 2} \sqrt{\frac{n_{r}!}{\left(n_{r}+|m|\right)!}} \exp \left(-\frac{M \omega}{4 \hbar} r^{2}\right) \\ & \times\left(\frac{M \omega}{2 \hbar} r^{2}\right)^{|m| / 2} L_{n_{r}}^{|m|}\left(\frac{M \omega}{2 \hbar} r^{2}\right) \end{align*} $$

$$ \int_{0}^{\infty} d r R_{n_{r} m}(r) R_{n_{r}^{\prime} m}(r)=\delta_{n_{r} n_{r}^{\prime}} $$

$$ \left(r_{b} \tau_{b} \mid r_{a} \tau_{a}\right)=\sum_{n_{r} m} R_{n_{r} m}\left(r_{b}\right) R_{n_{r} m}\left(r_{a}\right) e^{-E_{n_{r} m}\left(\tau_{b}-\tau_{a}\right) / \hbar} $$

$$ \psi_{n_{r} m}(\mathbf{x})=\frac{1}{\sqrt{r}} R_{n_{r} m}(r) \frac{e^{i m \varphi}}{\sqrt{2 \pi}} $$

$$ n \equiv n_{r}+\frac{|m|}{2}-\frac{m}{2} $$

$$ m=n^{\prime}-n $$

$$ a=\sqrt{\frac{2 \hbar}{M \omega}}=\sqrt{\frac{2 \hbar c}{e B}} $$

$$ z=(x+i y) / \sqrt{2} a, \quad z^{*}=(x-i y) / \sqrt{2} a . $$

$$ \psi_{n, n^{\prime}}\left(z, z^{*}\right)=N_{n, n^{\prime}} e^{z^{*} z}\left(-\frac{1}{\sqrt{2}} \partial_{z *}\right)^{n}\left(-\frac{1}{\sqrt{2}} \partial_{z}\right)^{n^{\prime}} e^{-2 z^{*} z} $$

$$ \begin{align*} e^{z^{*} z}\left(-\frac{1}{\sqrt{2}} \partial_{z^{*}}\right) e^{-z^{*} z} & =\frac{1}{\sqrt{2}}\left(-\partial_{z^{*}}+z\right) \\ e^{z^{*} z}\left(-\frac{1}{\sqrt{2}} \partial_{z}\right) e^{-z^{*} z} & =\frac{1}{\sqrt{2}}\left(-\partial_{z}+z^{*}\right) \end{align*} $$

$$ \begin{align*} & \hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(-\partial_{z^{*}}+z\right) \\ & \hat{b}^{\dagger}=\frac{1}{\sqrt{2}}\left(-\partial_{z}+z^{*}\right) \end{align*} $$

$$ \begin{align*} \hat{a} & =\frac{1}{\sqrt{2}}\left(\partial_{z}+z^{*}\right) \\ \hat{b} & =\frac{1}{\sqrt{2}}\left(\partial_{z^{*}}+z\right) \end{align*} $$

$$ \psi_{0,0}\left(z, z^{*}\right)=\left\langle z, z^{*} \mid 0\right\rangle \propto e^{-z^{*} z} $$

$$ \psi_{n, n^{\prime}}\left(z, z^{*}\right)=N_{n n^{\prime}} \hat{a}^{\dagger n} \hat{b}^{\dagger n^{\prime}} \psi_{0,0}\left(z, z^{*}\right) $$

$$ \begin{align*} \int d x & d y \psi_{n_{1}, n_{1}^{\prime}}\left(z, z^{*}\right) \psi_{n_{2}, n_{2}^{\prime}}\left(z, z^{*}\right) \\ & =N_{n_{1} n_{1}^{\prime}} N_{n_{2} n_{2}^{\prime}} \int d x d y\left[\left(a^{\dagger}\right)^{n_{1}}\left(b^{\dagger}\right)^{n_{1}^{\prime}} e^{-z^{*} z}\right]\left[\left(a^{\dagger}\right)^{n_{2}}\left(b^{\dagger}\right)^{n_{2}^{\prime}} e^{-z^{*} z}\right] \\ & =N_{n_{1} n_{1}^{\prime}} N_{n_{2} n_{2}^{\prime}} \int d x d y e^{-2 z^{*} z}\left(a^{n_{1}} b^{n_{1}^{\prime}} a^{\dagger n_{2}} b^{\dagger n_{2}^{\prime}}\right) \end{align*} $$

$$ n_{1}!n_{2}!\delta_{n_{1} n_{1}^{\prime}} \delta_{n_{2} n_{2}^{\prime}} $$

$$ \int d x d y e^{-2 z^{*} z}=\pi \int d r^{2} e^{-r^{2} / a^{2}}=\pi a^{2} $$

$$ N_{n, n^{\prime}}=\frac{1}{\sqrt{\pi a^{2} n!n^{\prime}!}} $$

$$ \frac{1}{2 M}\left(-i \hbar \nabla-\frac{e}{c} \mathbf{A}\right)^{2} \psi=E \psi $$

$$ -\frac{\hbar^{2}}{2 M}\left[\partial_{x}^{2}+\left(\partial_{y}-i \frac{e B}{c} x\right)^{2}+\partial_{z}^{2}\right] \psi=E \psi $$

$$ \mathbf{A}=(-B y / 2, B x / 2,0) $$

$$ \begin{gather*} {\left[-\frac{\hbar^{2}}{2 M}\left(\partial_{r}^{2}+\frac{1}{r} \partial_{r}+\frac{1}{r^{2}} \partial_{\varphi}^{2}+\partial_{z}^{2}\right)-\frac{i e \hbar B}{2 M c} \partial_{\varphi}+\frac{e^{2} B^{2}}{8 M c^{2}} r^{2}\right] \psi(r, z, \varphi)} \\ =E \psi(r, z, \varphi) \end{gather*} $$

$$ \left[\partial_{\rho}^{2}+\frac{1}{\rho} \partial_{\rho}+\frac{a^{2}}{\hbar^{2}}\left(2 M E-p_{z}^{2}\right)-\rho^{2}-2 i \partial_{\varphi}-\frac{1}{\rho^{2}} \partial_{\varphi}^{2}\right] \psi(r, \varphi)=0 $$

$$ \psi_{n_{r} m}(r, \varphi) \propto e^{i m \varphi} e^{-\rho^{2} / 2} \rho^{|m| / 2} M\left(-n_{r},|m|+\frac{1}{2}, \rho\right) $$

$$ n_{r}=n+\frac{1}{2} m-\frac{1}{2}|m|-\frac{1}{2} $$

$$ n+\frac{1}{2} \equiv \frac{a^{2}}{\hbar^{2}}\left(2 M E-p_{z}^{2}\right) $$

$$ \frac{2 M a^{2}}{\hbar^{2}}=\frac{1}{\hbar \omega} $$

$$ E=\left(n+\frac{1}{2}\right) \hbar \omega+\frac{p_{z}^{2}}{2 M} $$

$$ 4\left[-\left(a^{\dagger} a+1 / 2\right)+\frac{1}{\hbar \omega}\left(E-\frac{p_{z}^{2}}{2 M}\right)\right] \psi\left(z, z^{*}\right)=0 $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{\omega, E}-\frac{i}{\hbar} \int d^{D} x_{1}\left(\mathbf{x}_{b} \mid \mathbf{x}_{1}\right)_{\omega, E} V\left(\mathbf{x}_{1}\right)\left(\mathbf{x}_{1} \mid \mathbf{x}_{a}\right)_{\omega, E} \\ & +-\frac{1}{\hbar^{2}} \int d^{D} x_{2} \int d^{D} x_{1}\left(\mathbf{x}_{b} \mid \mathbf{x}_{2}\right)_{\omega, E} V\left(\mathbf{x}_{2}\right)\left(\mathbf{x}_{2} \mid \mathbf{x}_{1}\right)_{\omega, E} V\left(\mathbf{x}_{1}\right)\left(\mathbf{x}_{1} \mid \mathbf{x}_{a}\right)_{\omega, E} \\ & +\ldots \end{align*} $$

$$ V(\mathbf{x})=g \delta^{(D)}(\mathbf{x}-\mathbf{X}), \quad g \equiv \frac{\hbar^{2}}{M l^{2-D}} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{\omega, E}-i \frac{g}{\hbar} \frac{\left(\mathbf{x}_{b} \mid \mathbf{X}\right)_{\omega, E}\left(\mathbf{X} \mid \mathbf{x}_{a}\right)_{\omega, E}}{1+i \frac{g}{\hbar}(\mathbf{X} \mid \mathbf{X})_{\omega, E}} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =-2 i \frac{M}{\hbar} \frac{\kappa^{D-2}}{(2 \pi)^{D / 2}} \frac{K_{D / 2-1}(\kappa R)}{(\kappa R)^{D / 2-1}} \\ & -\frac{i g}{\hbar} \frac{i \frac{2 M}{\hbar} \frac{\kappa^{D-2}}{(2 \pi)^{D / 2}} \frac{K_{D / 2-1}\left(\kappa R_{b}\right)}{\left(\kappa R_{b}\right)^{D / 2-1}} \times i \frac{2 M}{\hbar} \frac{\kappa^{D-2}}{(2 \pi)^{D / 2}} \frac{K_{D / 2-1}\left(\kappa R_{a}\right)}{\left(\kappa R_{a}\right)^{D / 2-1}}}{1-\frac{g}{\hbar} \frac{2 M}{\hbar} \frac{\kappa^{D-2}}{\pi^{D / 2}} \frac{K_{D / 2-1}(\kappa \delta)}{(\kappa \delta)^{D / 2-1}}} \end{align*} $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=-i \frac{M}{\hbar} \frac{1}{\kappa} e^{-\kappa R}+i \frac{M}{\hbar \kappa} e^{\kappa\left(R_{b}+R_{a}\right)} \frac{1}{l \kappa+1} $$

$$ -i \frac{1}{\kappa l^{2}} \frac{\hbar}{E+\hbar^{2} / 2 M l^{2}} e^{\kappa\left(R_{b}+R_{a}\right)} $$

$$ \frac{2}{l} e^{-\left(R_{b}+R_{a}\right) / l} \frac{i \hbar}{E+\hbar^{2} / 2 M l^{2}} $$

$$ \psi_{B}(x)=\sqrt{\frac{2}{l}} e^{-|x-X| / l} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-i \frac{M}{\hbar} \frac{1}{2 \pi R} e^{-\kappa R}+i \frac{M}{\hbar} \frac{e^{\kappa R_{b}}}{2 \pi R_{b}} \frac{e^{\kappa R_{a}}}{2 \pi R_{a}} \frac{1}{1 / l+e^{-\kappa \delta} / 2 \pi \delta} $$

$$ \frac{1}{l_{r}} \equiv 1+\frac{1}{2 \pi \delta}, $$

$$ \frac{1}{1 / l_{r}-\kappa / 2 \pi} $$

$$ -l_{r} E_{B} \frac{1}{E-E_{B}} $$

$$ \psi_{B}\left(\mathbf{x}_{b}\right) \psi_{B}^{*}\left(\mathbf{x}_{a}\right) \frac{i \hbar}{E-E_{B}}, $$

$$ \psi_{B}(\mathbf{x})=\left(\frac{\kappa_{B}^{2}}{4 \pi^{2}}\right)^{1 / 4} \frac{e^{-\kappa_{B}|\mathbf{x}-\mathbf{X}|}}{r} $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =-i \frac{M}{\hbar} \frac{1}{\pi} \frac{K_{\epsilon / 2}(\kappa R)}{(2 \pi \kappa R)^{\epsilon / 2}} \\ & +i \frac{M^{2}}{\hbar^{2} \pi^{2}} \frac{1}{\left(2 \pi \kappa R_{b}\right)^{\epsilon / 2}} \frac{1}{\left(2 \pi \kappa R_{a}\right)^{\epsilon / 2}} \frac{K_{\epsilon / 2}\left(\kappa R_{b}\right) K_{\epsilon / 2}\left(\kappa R_{a}\right)}{\frac{\hbar}{g}+\frac{M}{\hbar \pi} \frac{1}{(2 \pi \kappa \delta)^{\epsilon / 2}} K_{\epsilon / 2}(\kappa \epsilon)} \end{align*} $$

$$ \frac{\hbar}{g}+\frac{M}{2 \hbar \pi} \frac{\Gamma(\epsilon / 2)}{(\pi \kappa \delta)^{\epsilon / 2}} \approx \frac{\hbar}{g}+\frac{M}{2 \hbar \pi} \frac{2}{\epsilon}\left[1-\frac{\epsilon}{2} \log (\pi \kappa \delta)\right] $$

$$ \frac{1}{g_{r}}=\frac{1}{g}+\frac{M}{\hbar^{2}} \frac{1}{\epsilon} $$

$$ \frac{\hbar}{g_{r}}-\frac{M}{2 \hbar \pi} \log \pi \kappa \delta $$

$$ \kappa_{B}=\frac{1}{\pi \delta} e^{2 \hbar^{2} \pi / M g_{r}} $$

$$ \psi_{B}\left(\mathbf{x}_{b}\right) \psi_{B}^{*}\left(\mathbf{x}_{a}\right) \frac{i \hbar}{E-E_{B}} $$

$$ \psi_{B}(\mathbf{x})=\frac{\kappa_{B}}{\sqrt{\pi}} K_{0}\left(\kappa_{B}|\mathbf{x}-\mathbf{X}|\right) $$