Kleinert · 제6장 자기장

$$ \left\langle\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right\rangle=\left\langle\varphi_{b}\right| \exp \left[-\frac{i}{\hbar}\left(t_{b}-t_{a}\right) \hat{H}\right]\left|\varphi_{a}\right\rangle \equiv\left\langle\varphi_{b}\right| \prod_{n=1}^{N+1} \exp \left(-\frac{i}{\hbar} \epsilon \hat{H}\right)\left|\varphi_{a}\right\rangle . $$

$$ \int_{0}^{2 \pi} d \varphi_{n}\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|=1 $$

$$ \left\langle\varphi_{n} \mid \varphi_{n-1}\right\rangle=\delta\left(\varphi_{n}-\varphi_{n-1}\right), \quad \varphi_{n} \in[0,2 \pi) . $$

$$ \delta\left(\varphi_{n}-\varphi_{n-1}\right)=\sum_{m_{n}=-\infty}^{\infty} \frac{1}{2 \pi} \exp \left[i m_{n}\left(\varphi_{n}-\varphi_{n-1}\right)\right] $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{0}=\prod_{n=1}^{N}\left[\int_{0}^{2 \pi} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\sum_{m_{n}} \frac{1}{2 \pi}\right] \exp \left[i \sum_{n=1}^{N+1} m_{n}\left(\varphi_{n}-\varphi_{n-1}\right)\right] . $$

$$ \begin{align*} \left(\varphi_{n} t_{n} \mid \varphi_{n-1} t_{n-1}\right) & =\left\langle\varphi_{n}\right| \exp \left[-\frac{i}{\hbar} \epsilon \hat{H}(p, \varphi)\right]\left|\varphi_{n-1}\right\rangle \\ & =\exp \left[-\frac{i}{\hbar} \epsilon \hat{H}\left(-i \hbar \partial_{\varphi_{n}}, \varphi_{n}\right)\right] \sum_{m_{n}=-\infty}^{\infty} \frac{1}{2 \pi} \exp \left[i m_{n}\left(\varphi_{n}-\varphi_{n-1}\right)\right] \end{align*} $$

$$ \left(\varphi_{n} t_{n} \mid \varphi_{n-1} t_{n-1}\right)=\sum_{m_{n}=-\infty}^{\infty} \frac{1}{2 \pi} \exp \left[i m_{n}\left(\varphi_{n}-\varphi_{n-1}\right)-\frac{i \epsilon}{\hbar} H\left(\hbar m_{n}, \varphi_{n}\right)\right] . $$

$$ \begin{align*} \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right) \approx & \prod_{n=1}^{N}\left[\int_{0}^{2 \pi} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\sum_{m_{n}=-\infty}^{\infty} \frac{1}{2 \pi}\right] \\ & \times \exp \left\{i \sum_{n=1}^{N+1}\left[m_{n}\left(\varphi_{n}-\varphi_{n-1}\right)-\frac{1}{\hbar} \epsilon H\left(\hbar m_{n}, \varphi_{n}\right)\right]\right\} \end{align*} $$

$$ \sum_{l=-\infty}^{\infty} e^{2 \pi i k l}=\sum_{m=-\infty}^{\infty} \delta(k-m) $$

$$ \left\langle\varphi_{n} \mid \varphi_{n-1}\right\rangle=\sum_{l=-\infty}^{\infty} \delta\left(\varphi_{n}-\varphi_{n-1}+2 \pi l\right) $$

$$ \left\langle\varphi_{n} \mid \varphi_{n-1}\right\rangle=\sum_{l=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d k_{n}}{2 \pi} \exp \left[i k_{n}\left(\varphi_{n}-\varphi_{n-1}\right)+2 \pi i k_{n} l\right] $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{0}=\prod_{n=1}^{N}\left[\int_{0}^{2 \pi} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d k_{n}}{2 \pi} \sum_{l_{n}=-\infty}^{\infty}\right] e^{i \sum_{n=1}^{N+1}\left[k_{n}\left(\varphi_{n}-\varphi_{n-1}\right)+2 \pi k_{n} l_{n}\right]} $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{0}=\sum_{l=-\infty}^{\infty} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d k_{n}}{2 \pi}\right] e^{i \sum_{n=1}^{N+1} k_{n}\left(\varphi_{n}-\varphi_{n-1}+2 \pi l \delta_{n, N+1}\right)} $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{0, \text { noncyclic }}=\prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d k_{n}}{2 \pi}\right] e^{i \sum_{n=1}^{N+1} k_{n}\left(\varphi_{n}-\varphi_{n-1}\right)} $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{0}=\sum_{l=-\infty}^{\infty}\left(\varphi_{b}+2 \pi l, t_{b} \mid \varphi_{a} t_{a}\right)_{0, \text { noncyclic }} $$

$$ \begin{align*} \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right) & \approx \sum_{l=-\infty}^{\infty} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d \varphi_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \\ & \times \exp \left\{\frac{i}{\hbar} \sum_{n=1}^{N+1}\left[p_{n}\left(\varphi_{n}-\varphi_{n-1}+2 \pi l \delta_{n, N+1}\right)-\epsilon H\left(p_{n}, \varphi_{n}\right)\right]\right\} \end{align*} $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right) \xrightarrow{\epsilon \rightarrow 0} \sum_{l=-\infty}^{\infty} \int_{\varphi_{a} \leadsto \varphi_{b}+2 \pi l} \mathcal{D} \varphi(t) \int \frac{\mathcal{D} p(t)}{2 \pi \hbar} \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t[p \dot{\varphi}-H(p, \varphi)]\right\} $$

$$ H(p, \varphi)=\frac{p^{2}}{2 M} $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)_{\text {noncyclic }}=\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\varphi_{b}-\varphi_{a}\right)^{2}}{t_{b}-t_{a}}\right] . $$

$$ \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \sum_{l=-\infty}^{\infty} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\varphi_{b}-\varphi_{a}+2 \pi l\right)^{2}}{t_{b}-t_{a}}\right] . $$

$$ \psi_{m}(\varphi)=\frac{1}{\sqrt{2 \pi}} e^{i m \varphi} $$

$$ H=\frac{\hbar^{2}}{2 M} m^{2} $$

$$ \begin{align*} \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right) & =\left\langle\varphi_{b}\right| \exp \left[-\frac{i}{\hbar}\left(t_{b}-t_{a}\right) \hat{H}\right]\left|\varphi_{a}\right\rangle \\ & =\sum_{m=-\infty}^{\infty} \psi_{m}\left(\varphi_{b}\right) \psi_{m}^{*}\left(\varphi_{a}\right) \exp \left[-\frac{i}{\hbar} \frac{\hbar^{2} m^{2}}{2 M}\left(t_{b}-t_{a}\right)\right] \\ & =\sum_{m=-\infty}^{\infty} \frac{1}{2 \pi} \exp \left[i m\left(\varphi_{b}-\varphi_{a}\right)-i \frac{\hbar m^{2}}{2 M}\left(t_{b}-t_{a}\right)\right] . \end{align*} $$

$$ \begin{align*} \left(\varphi_{b} t_{b} \mid \varphi_{a} t_{a}\right) & =\sum_{l=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \exp \left\{\frac{i}{\hbar}\left[p\left(\varphi_{b}-\varphi_{a}+2 \pi l\right)-\frac{p^{2}}{2 M}\left(t_{b}-t_{a}\right)\right]\right\} \\ & =\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \sum_{l=-\infty}^{\infty} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(\varphi_{b}-\varphi_{a}+2 \pi l\right)^{2}}{t_{b}-t_{a}}\right] \end{align*} $$

$$ \int_{0}^{\infty} d r|r\rangle\langle r|=1 $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\delta\left(r-r^{\prime}\right) ; \quad r, r^{\prime}>0 . $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)=\left\langle r_{b}\right| \prod_{n=1}^{N+1} \exp \left(-\frac{i}{\hbar} \hat{H} \epsilon\right)\left|r_{a}\right\rangle . $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}=\prod_{n=1}^{N}\left[\int_{0}^{\infty} d r_{n}\right] \prod_{n=1}^{N+1}\left\langle r_{n} \mid r_{n-1}\right\rangle=\left\langle r_{b} \mid r_{a}\right\rangle $$

$$ \begin{align*} \left\langle r \mid r^{\prime}\right\rangle & =2 \int_{0}^{\infty} \frac{d k}{\pi} \sin k r \sin k r^{\prime} \\ & =\int_{-\infty}^{\infty} \frac{d k}{2 \pi}\left[\exp i k\left(r-r^{\prime}\right)-\exp i k\left(r+r^{\prime}\right)\right]=\delta\left(r-r^{\prime}\right)-\delta\left(r+r^{\prime}\right) \end{align*} $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\sum_{x= \pm r} \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \exp \left[\frac{i}{\hbar} p\left(x-x^{\prime}\right)+i \pi\left(\sigma(x)-\sigma\left(x^{\prime}\right)\right)\right]_{x^{\prime}=r^{\prime}} $$

$$ \sigma(x) \equiv \Theta(-x) $$

$$ \begin{align*} \left\langle x \mid x^{\prime}\right\rangle & =\sum_{x^{\prime \prime}= \pm x} \int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar} \exp \left[\frac{i}{\hbar} p\left(x^{\prime \prime}-x^{\prime}\right)+i \pi\left(\sigma\left(x^{\prime \prime}\right)-\sigma\left(x^{\prime}\right)\right)\right] \\ & =\delta\left(x-x^{\prime}\right)-\delta\left(x+x^{\prime}\right) \end{align*} $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\left.\left\langle x \mid x^{\prime}\right\rangle\right|_{x=r, x^{\prime}=r^{\prime}} $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}=\delta\left(r_{b}-r_{a}\right) $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}=\delta\left(r_{b}-r_{a}\right)-\delta\left(r_{b}+r_{a}\right) $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}=\prod_{n=1}^{N}\left[\int_{0}^{\infty} d r_{n}\right] \prod_{n=1}^{N+1}\left[\sum_{x_{n}= \pm r_{n}}\left(x_{n} \epsilon \mid x_{n-1} 0\right)_{0}\right] $$

$$ \left(x_{n} \epsilon \mid x_{n-1} 0\right)_{0}=\left\langle x_{n} \mid x_{n-1}\right\rangle, \quad x \in(-\infty, \infty) $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}= & \prod_{n=1}^{N}\left[\int_{0}^{\infty} d r_{n}\right] \prod_{n=1}^{N+1}\left[\sum_{x_{n}= \pm r_{n}} \int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \\ & \times \exp \left\{\sum_{n=1}^{N+1}\left[\frac{i}{\hbar} p\left(x_{n}-x_{n-1}\right)+i \pi\left(\sigma\left(x_{n}\right)-\sigma\left(x_{n-1}\right)\right)\right]\right\} \end{align*} $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}= & \sum_{x_{b}= \pm r_{b}} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \\ & \times \exp \left\{\sum_{n=1}^{N+1}\left[\frac{i}{\hbar} p\left(x_{n}-x_{n-1}\right)+i \pi\left(\sigma\left(x_{n}\right)-\sigma\left(x_{n-1}\right)\right)\right]\right\} \end{align*} $$

$$ \mathcal{A}_{0}^{\sigma}[p, x]=\int_{t_{a}}^{t_{b}} d t\left[p \dot{x}+\hbar \pi \partial_{t} \sigma(x)\right] \equiv \mathcal{A}_{0}[p, x]+\mathcal{A}_{\mathrm{topol}}^{\sigma} $$

$$ \mathcal{A}_{\text {topol }}^{\sigma}[x]=\hbar \pi\left(\sigma\left(x_{b}\right)-\sigma\left(x_{a}\right)\right) $$

$$ \mathcal{A}_{\text {topol }}^{\sigma}[x]=-\pi \hbar \int_{t_{a}}^{t_{b}} d t \dot{x}(t) \delta(x(t)) $$

$$ \sigma\left(x_{b}\right)-\sigma\left(x_{a}\right)=\int_{x_{a}}^{x_{b}} d x \sigma^{\prime}(x)=\int_{x_{a}}^{x_{b}} d x \Theta^{\prime}(-x)=-\int_{x_{a}}^{x_{b}} d x \delta(x) $$

$$ H=\frac{p^{2}}{2 M} . $$

$$ \mathcal{A}[p, x]=\int_{t_{a}}^{t_{b}} d t\left[p \dot{x}-p^{2} / 2 M-\hbar \pi \dot{x}(t) \delta(x(t))\right] $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right)= & \sum_{x_{b}= \pm r_{b}} \frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \\ & \times \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}+i \pi\left(\sigma\left(x_{b}\right)-\sigma\left(x_{a}\right)\right)\right] \\ = & \frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}\left\{\exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(r_{b}-r_{a}\right)^{2}}{t_{b}-t_{a}}\right]-\left(r_{b} \rightarrow-r_{b}\right)\right\} \end{align*} $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\frac{1}{2 d} \sum_{k_{\nu}}\left[e^{i k_{\nu}\left(r-r^{\prime}\right)}-e^{i k_{\nu}\left(r+r^{\prime}\right)}\right] $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right) & =\int_{-\infty}^{\infty} \frac{d p}{2 \pi \hbar}\left\{\exp \left[\frac{i}{\hbar} p\left(r_{b}-r_{a}\right)\right]-\left(r_{b} \rightarrow-r_{b}\right)\right\} e^{-i p^{2}\left(t_{b}-t_{a}\right) / 2 M \hbar} \\ & =2 \int_{0}^{\infty} \frac{d p}{2 \pi \hbar} \sin \left(p r_{b} / \hbar\right) \sin \left(p r_{a} / \hbar\right) \exp \left[-\frac{i}{\hbar} \frac{p^{2}}{2 M}\left(t_{b}-t_{a}\right)\right] \end{align*} $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}} \frac{1}{2} \sum_{\substack{x_{a}= \pm r_{a} \\ x_{b}= \pm r_{b}}} \exp \left[\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}+i \pi\left(\sigma\left(x_{b}\right)-\sigma\left(x_{a}\right)\right)\right] $$

$$ Y_{\mathrm{e}}(\hat{x})=\frac{1}{\sqrt{2}}, \quad Y_{\mathrm{o}}(\hat{x})=\frac{1}{\sqrt{2}} e^{i \pi \sigma(x)} $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)=\sum_{\substack{\hat{x}_{b}, \hat{x}_{a} \\\left|x_{b}\right|=r_{b},\left|x_{a}\right|=r_{a}}} Y_{\mathrm{o}}^{*}\left(\hat{x}_{b}\right)\left\langle x_{b} t_{b} \mid x_{a} t_{a}\right\rangle Y_{\mathrm{o}}\left(\hat{x}_{a}\right) $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\frac{2}{d} \sum_{k_{\nu}>0} \sin k_{\nu} r \sin k_{\nu} r^{\prime} $$

$$ k_{\nu}=\frac{\pi}{d} \nu, \quad \nu=1,2,3, \ldots $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\sum_{l=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d k}{2 \pi}\left[e^{i k\left(r-r^{\prime}+2 d l\right)}-e^{i k\left(r+r^{\prime}+2 d l\right)}\right] $$

$$ \left\langle r \mid r^{\prime}\right\rangle=\sum_{x= \pm r} \sum_{l=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k\left(x-x^{\prime}+2 d l\right)+i \pi\left(\sigma(x)-\sigma\left(x^{\prime}\right)\right)} $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0} & =\prod_{n=1}^{N}\left[\int_{0}^{d} d r_{n}\right]\left\langle r_{n} \mid r_{n-1}\right\rangle \\ & =\prod_{n=1}^{N}\left[\int_{0}^{d} d r_{n}\right] \prod_{n=1}^{N+1} \frac{2}{d} \sum_{k_{\nu}} \sin k_{\nu_{n}} r_{n} \sin k_{\nu_{n-1}} r_{n-1} \end{align*} $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)_{0}=\sum_{x_{b}= \pm r_{b}} \sum_{l=-\infty}^{\infty} \prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d x_{n}\right] \prod_{n=1}^{N+1}\left[\int_{-\infty}^{\infty} \frac{d p_{n}}{2 \pi \hbar}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}_{0}^{N}\right) $$

$$ \mathcal{A}_{0}^{N}=\sum_{n=1}^{N+1}\left[p_{n}\left(x_{n}-x_{n-1}\right)+\hbar \pi\left(\sigma\left(x_{n}\right)-\sigma\left(x_{n-1}\right)\right)\right] $$

$$ \mathcal{A}=\int_{t_{a}}^{t_{b}} d t[p \dot{x}-H(p, x)-\hbar \pi \dot{x} \delta(x)] $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)=\sum_{l=-\infty}^{\infty} \sum_{x_{b}= \pm r_{b}+2 d l} \int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi \hbar} \exp \left(\frac{i}{\hbar} \mathcal{A}\right) $$

$$ \mathcal{A}^{N}=\mathcal{A}_{0}^{N}-\epsilon \sum_{n=1}^{N+1} H\left(p_{n}, x_{n}\right) . $$

$$ \left(r_{b} t_{b} \mid r_{a} t_{a}\right)=\sum_{l=-\infty}^{\infty} \sum_{x_{b}= \pm r_{b}+2 d l} \frac{1}{\sqrt{2 \pi \hbar i\left(t_{b}-t_{a}\right) / M}}\left[e^{\frac{i}{\hbar} \frac{M}{2} \frac{\left(x_{b}-x_{a}+2 d l\right)^{2}}{t_{b}-t_{a}}}-\left(x_{b} \rightarrow-x_{b}\right)\right] . $$

$$ \begin{align*} \left(r_{b} t_{b} \mid r_{a} t_{a}\right) & =\left\langle r_{b}\right| \exp \left[-\frac{i}{\hbar}\left(t_{b}-t_{a}\right) \hat{H}\right]\left|r_{a}\right\rangle \\ & =\frac{2}{d} \sum_{\nu=1}^{\infty} \sin k_{\nu} r_{b} \sin k_{\nu} r_{a} \exp \left[-i \hbar \frac{k_{\nu}^{2}}{2 M}\left(t_{b}-t_{a}\right)\right] . \end{align*} $$

$$ E^{(\nu-1)}=\frac{\hbar^{2} k_{\nu}^{2}}{2 M}, \quad \nu=1,2,3 \ldots $$

$$ \psi^{(\nu-1)}(x)=\frac{2}{d} \sin k_{\nu} x $$

$$ Z=\int \mathcal{D} u(\tau) e^{\frac{1}{2} \int_{0}^{\hbar \beta} d \tau(\partial u)^{2}} $$

$$ \mathcal{A}_{\mathrm{e}}=\frac{1}{2} \int_{0}^{\hbar \beta} d \tau\left\{[\partial u(\tau)]^{2}+V(u(\tau))\right\} $$

$$ V(u)=\frac{\omega^{2}}{2}\left(\frac{d}{\pi} \tan \frac{\pi u}{d}\right)^{2}=\frac{\omega^{2}}{2}\left(u^{2}+\frac{2}{3} g u^{4}+\ldots\right) . $$

$$ Z_{\omega}=e^{-(1 / 2) \operatorname{Tr} \log \left(\partial^{2}+\omega^{2}\right)} $$

$$ F_{\omega}=\frac{\omega}{2} $$

$$ F=F_{\omega}+\omega \sum_{k=1}^{\infty} a_{k}\left(\frac{g}{\omega}\right)^{k} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{e}}^{\mathrm{int}} & =\frac{\omega^{2}}{2} \int d \tau\left(g v_{4} u^{4}+g^{2} v_{6} u^{6}+g^{3} v_{8} u^{8}+\ldots\right) \\ & =\frac{\omega^{2}}{2} \sum_{k=1}^{\infty} \int d \tau g^{k} v_{2 k+2}\left[u^{2}(\tau)\right]^{k+1} \end{align*} $$

$$ \left\langle u\left(\tau_{1}\right) u\left(\tau_{2}\right)\right\rangle=\int \frac{d k}{2 \pi} \frac{e^{i k\left(\tau_{1}-\tau_{2}\right)}}{k^{2}+\omega^{2}}=\frac{e^{-\omega\left|\tau_{1}-\tau_{2}\right|}}{2 \omega} $$

$$ \begin{align*} \langle u \partial u\rangle & =\int \frac{d k}{2 \pi} \frac{k}{k^{2}+\omega^{2}}=0 \\ \langle\partial u \partial u\rangle & =\int \frac{d k}{2 \pi} \frac{k^{2}}{k^{2}+\omega^{2}}=-\frac{\omega}{2}, \end{align*} $$

$$ \begin{align*} & F=\frac{\omega}{2}+\left(\frac{\omega^{2}}{2}\right)\left\{g v_{4} 3 \infty+g^{2} v_{6} 15 \ell+g^{3} v_{8} 105 \ell 0\right\} \\ & -\frac{1}{2!}\left(\frac{\omega^{2}}{2}\right)^{2}\left\{g^{2} v_{4}^{2}[72000+24 \bigcirc]+g^{3} 2 v_{4} v_{6}\left[540 \otimes^{0}+360 \bigcirc 0\right]\right\} \\ & +\frac{1}{3!}\left(\frac{\omega^{2}}{2}\right)^{3} g^{3} v_{4}^{3}\left\{25920000+17280^{0}+3456 \bigcirc 0+1728 \nabla\right\} . \end{align*} $$

$$ \begin{align*} \theta & =\beta \frac{1}{32 \omega^{5}}, & \phi & =\beta \frac{3}{128 \omega^{8}} \\ \sigma^{\infty} & =\beta \frac{1}{32 \omega^{6}}, & \phi & =\beta \frac{5}{8 \cdot 64 \omega^{8}} \\ \sigma^{0} & =\beta \frac{1}{32 \omega^{6}}, & \nabla & =\beta \frac{3}{8 \cdot 64 \omega^{8}} \end{align*} $$

$$ F_{3}=\omega\left\{\frac{1}{2}+\frac{3}{8} v_{4}\left(\frac{g}{\omega}\right)+\left[\frac{15}{16} v_{6}-\frac{21}{32} v_{4}^{2}\right]\left(\frac{g}{\omega}\right)^{2}+\left[\frac{105}{32} v_{8}-\frac{45}{8} v_{4} v_{6}+\frac{333}{128} v_{4}^{3}\right]\left(\frac{g}{\omega}\right)^{3}\right\} $$

$$ \begin{align*} & 2 p^{\prime} C_{n}^{p^{\prime}}=\left(p^{\prime}+1\right)\left(2 p^{\prime}+1\right) C_{n}^{p^{\prime}}+\frac{1}{2} \sum_{k=1}^{n} v_{2 k+2} C_{n-k}^{p^{\prime}-k-1}-\sum_{k=1}^{n-1} C_{k}^{1} C_{n-k}^{p^{\prime}}, 1 \leq p^{\prime} \leq 2 n, \\ & C_{0}^{0}=1, \quad C_{n}^{p^{\prime}}=0 \quad\left(n \geq 1, p^{\prime}<1\right) . \end{align*} $$

$$ \begin{align*} & a_{0}=\frac{1}{2}, a_{1}=\frac{1}{4}, a_{2}=\frac{1}{16}, a_{3}=0, a_{4}=-\frac{1}{256}, a_{5}=0, \\ & a_{6}=\frac{1}{2048}, a_{7}=0, a_{8}=-\frac{5}{65536}, a_{9}=0, \\ & a_{10}=\frac{7}{524288}, a_{11}=0, a_{12}=-\frac{21}{8388608}, a_{13}=0, \\ & a_{14}=\frac{33}{67108864}, a_{15}=0, a_{16}=-\frac{429}{4294967296}, \ldots . \end{align*} $$

$$ r^{2} \equiv 2 M\left(\Omega^{2}-\omega^{2}\right) / g $$

$$ \begin{align*} \left(N, f_{N}^{\min }\right)= & (2,0.466506),(3,0.492061),(4,0.497701), \\ & (5,0.499253),(6,0.499738),(7,0.499903), \\ & (8,0.499963),(9,0.499985),(10,0.499994), \\ & (11,0.499998),(12,0.499999),(13,0.5000), \\ & (14,0.50000),(15,0.50000),(16,0.5000) . \end{align*} $$

$$ \tilde{F}(\alpha)=\frac{1}{4}\left[1+\frac{1}{\sqrt{\alpha}} h(\alpha)\right], \quad h(\alpha) \equiv \sum_{n=0}^{N} 2^{2 n+1} a_{2 n} \alpha^{n} . $$

$$ h(\alpha)=1+\frac{\alpha}{2}-\frac{\alpha^{2}}{8}+\frac{\alpha^{3}}{16}-\frac{5}{128} \alpha^{4}+\frac{7}{256} \alpha^{5}-\frac{21}{1024} \alpha^{6}+\frac{33}{2048} \alpha^{7}-\frac{429}{32768} \alpha^{8}+\ldots . $$

$$ E^{(0)}=\frac{\pi^{2}}{4 d^{2}}\left(1+\sqrt{1+\frac{1}{\alpha}}\right)=\frac{\pi^{2}}{4 d^{2}}\left(1+\sqrt{1+4 \omega^{2} \frac{d^{4}}{\pi^{4}}}\right) . $$

$$ \frac{1}{2}\left\{-\frac{\partial^{2}}{\partial x^{2}}+\left[\frac{\lambda(1-\lambda)}{\cos ^{2} x}-1\right]\right\} \psi(x)=\frac{d^{2}}{\pi^{2}} E \psi(x) $$

$$ \lambda=\frac{1}{2}\left(1+\sqrt{1+4 \omega^{2} \frac{d^{4}}{\pi^{4}}}\right) $$

$$ \psi_{0}(x)=\text { const } \times \cos ^{\lambda} x $$

$$ E^{(0)}=\frac{1}{4}\left(g+\sqrt{g^{2}+4 \omega^{2}}\right) $$

$$ E^{(0)}=\frac{\Omega}{4}\left(\hat{g}+\sqrt{\hat{g}^{2}-4 \rho \hat{g}+4}\right) $$

$$ f_{N}(g)=\Omega \sum_{0}^{N} h_{k}(\rho) \hat{g}^{k} $$

$$ F_{N}=\frac{1}{2 \pi i} \oint_{C_{0}} \frac{d z}{z^{N+1}} \frac{1-z^{N+1}}{1-z} f(z) $$

$$ \begin{align*} F(z) & =\frac{\Omega}{4}\left(\frac{z}{r}+\sqrt{\frac{z^{2}}{r^{2}}-4 z+4}\right) \\ & =\frac{1}{4 r}\left(z+\sqrt{\left(z-z_{1}\right)\left(z-z_{2}\right)}\right) \end{align*} $$

$$ \begin{align*} & 1-z^{N+1}=(1-z)\left(1+z+\ldots+z^{N}\right)=(1-z)(N+1) \\ & -(1-z)^{2}\left[N+(N-1) z+\ldots+z^{N-1}\right] \end{align*} $$

$$ 1-z^{N+1}=(1-z)(N+1)+\mathcal{O}\left(|1-z|^{2} N^{2}\right) $$

$$ \begin{align*} \hat{F}_{N} & =\frac{(N+1)}{2 \pi i} \oint_{C_{0}} \frac{d z \hat{F}(z)}{z^{N+1}} \hat{F}(z)=\frac{(N+1)}{N!} \hat{F}^{(N)}(0) \\ & =(N+1) \sum_{i=1}^{2} \int_{z_{i}}^{\infty} \frac{d z}{z^{N+1}} \hat{F}(z) \end{align*} $$

$$ \Delta \hat{F}_{N} \approx \frac{(N+1)\left(N-\frac{3}{2}\right)!}{N!} \frac{1}{\left(r^{2}\right)^{N}} \frac{1}{\left(1+r^{2}\right)^{N}} $$

$$ \Delta \hat{F}_{N} \approx \frac{A}{\left(r^{2}\right)^{N} \sqrt{N}} e^{-r^{2} N} $$