Kleinert · 제17장 금융시장

$$ \begin{align*} y_{n}(\tau)= & \sqrt{\frac{m}{n!}} \sqrt{\Gamma(s-n) \Gamma(1+2 s-n)} \frac{2^{n-s}}{\Gamma(1+s-n)} \cosh ^{n-s}\left[m\left(\tau-\tau_{0}\right)\right] \\ & \times F\left(-n, 1+2 s-n ; s-n+1 ; \frac{1}{2}\left(1-\tanh \left[m\left(\tau-\tau_{0}\right)\right]\right)\right) \end{align*} $$

$$ V(x)=\frac{\omega^{2}}{8 a^{2}}(x-a)^{2}(x+a)^{2} $$

$$ I_{1}^{\prime}=\frac{97}{560} $$

$$ g=\omega^{2} / 2 a^{2} $$

$$ \begin{align*} \tilde{x}_{\mathrm{cl}}(\tau) & \equiv \sqrt{\frac{g}{2 \omega^{2}}} x_{\mathrm{cl}}(\tau)=\tanh (\tau / 2) \\ \tilde{y}_{0}(\tau) & \equiv \sqrt{\frac{8}{3 \omega}} y_{0}(\tau)=\frac{1}{\cosh ^{2}(\omega \tau / 2)} \end{align*} $$

$$ V(x)=\frac{\omega^{2}}{2}(x \mp a)^{2}\left(1 \pm \frac{x \mp a}{a}+\ldots\right) \equiv V_{ \pm}(x)+\Delta V_{ \pm}(x)+\ldots $$

$$ \begin{align*} x_{\mathrm{G}}(\tau) & \equiv \tilde{x}_{\mathrm{cl}}(\tau) G_{\mathcal{O}_{\omega}}(\tau, \tau) \\ x_{\mathrm{KG}}(\tau) & \equiv\left[\int_{-\infty}^{\tau} G_{\mathcal{O}_{\omega}}\left(\tau, \tau^{\prime}\right) x_{\mathrm{G}}\left(\tau^{\prime}\right) d \tau^{\prime}\right]_{\mathrm{A}} \\ x_{\mathrm{K} 3 \mathrm{G}}(\tau) & \equiv\left[\int_{-\infty}^{\tau} G_{\mathcal{O}_{\omega}}^{3}\left(\tau, \tau^{\prime}\right) x_{\mathrm{G}}\left(\tau^{\prime}\right) d t^{\prime}\right]_{\mathrm{A}}, \end{align*} $$

$$ V_{\max }=\frac{(\omega a)^{2}}{8} $$

$$ \begin{align*} I_{21} & =6 \int_{0}^{\infty} d \tau \tilde{x}_{\mathrm{cl}}(\tau) x_{\mathrm{K} 3 \mathrm{G}}(\tau) \\ I_{22} & =9 \int_{0}^{\infty} d \tau x_{\mathrm{G}}(\tau) x_{\mathrm{KG}}(\tau) \\ I_{3} & =24 \int_{0}^{\infty} d \tau y_{0}^{\prime}(\tau) x_{\mathrm{KG}}(\tau) \end{align*} $$

$$ \psi_{n}\left(\Delta x_{ \pm}\right) \rightarrow\left(\frac{\omega}{\pi \hbar}\right)^{1 / 4} \frac{1}{2^{n / 2} \sqrt{n!}} e^{-\omega\left(\Delta x_{ \pm}\right)^{2} / 2 \hbar} H_{n}\left(\Delta x_{ \pm} \sqrt{\omega / \hbar}\right) $$

$$ \Delta x_{ \pm} \equiv x \pm a $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & \xrightarrow{a \rightarrow \infty}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{-}+\left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{+} \\ & \equiv \int \mathcal{D} x(t) \exp \left\{\frac{i}{\hbar} \int_{t_{a}}^{t_{b}} d t \frac{1}{2}\left[\dot{x}^{2}-\omega^{2}(x+a)^{2}\right]\right\}+(a \rightarrow-a) \end{align*} $$

$$ x_{\mathrm{KG}}(\tau)=\frac{1}{4} \frac{12 \tau+\tanh (\tau / 2)}{12 \cosh ^{2}(\tau / 2)} $$

$$ L=\frac{\dot{x}^{2}}{2}-V(x) $$

$$ \int_{0}^{\infty} f(\tau / 2) \tanh \frac{\tau}{2} d \tau=-\int_{0}^{\infty} f^{\prime}(\tau / 2) \ln \left(2 \cosh \frac{\tau}{2}\right) d \tau $$

$$ \hat{H} \psi(x, t)=H\left(-i \partial_{x}, x\right) \psi(x, t)=i \hbar \partial_{t} \psi(x, t), $$

$$ \begin{align*} f_{n}^{\mathrm{S} / \mathrm{A}}(\tau) & \equiv \frac{1}{2}\left[H_{\mathrm{R}}^{n}(\tau) \pm H_{\mathrm{R}}^{n}(-\tau)\right] y_{0}^{3-n}(\tau) \\ F_{n}^{\mathrm{S} / \mathrm{A}}(\tau) & \equiv \int_{0}^{\tau} f_{n}^{\mathrm{S} / \mathrm{A}}\left(\tau^{\prime}\right) \tilde{x}_{\mathrm{cl}}\left(\tau^{\prime}\right) d \tau^{\prime} \\ N_{n} & \equiv \int_{0}^{\infty} H_{\mathrm{R}}^{n}(\tau) y_{0}^{3-n}\left(\tau^{\prime}\right) \tilde{x}_{\mathrm{cl}}\left(\tau^{\prime}\right) d \tau^{\prime} \end{align*} $$

$$ H(p, x)=\frac{p^{2}}{2}+V(x) $$

$$ \begin{align*} x_{\mathrm{K} 3 \mathrm{G}}(\tau)= & f_{3}^{\mathrm{A}}(\tau)\left[N_{0}-F_{0}^{\mathrm{S}}(\tau)\right]+3 f_{2}^{\mathrm{A}}(\tau)\left[N_{1}-F_{1}^{\mathrm{S}}(\tau)\right]+3 f_{1}^{\mathrm{A}}(\tau)\left[N_{2}-F_{2}^{\mathrm{S}}(\tau)\right] \\ & +3 f_{2}^{\mathrm{S}}(\tau) F_{1}^{\mathrm{A}}(\tau)+3 f_{1}^{\mathrm{S}}(\tau) F_{2}^{\mathrm{A}}(\tau)+f_{0}^{\mathrm{S}}(\tau) F_{3}^{\mathrm{A}}(\tau) \end{align*} $$

$$ \psi_{\mathrm{s}, \mathrm{a}} \approx \frac{1}{\sqrt{2}}\left[\psi_{0}(x-a) \pm \psi_{0}(x+a)\right] $$

$$ N_{0}=\frac{3}{32}, \quad N_{1}=-\frac{7}{128}, \quad N_{2}=\frac{203}{512}-\frac{\log 2}{2} $$

$$ E_{\mathrm{s}, \mathrm{a}}^{(0)}=\frac{1}{2} \hbar \omega+\Delta E_{\mathrm{s}, \mathrm{a}}^{(0)} $$

$$ \begin{align*} x_{\mathrm{K} 3 \mathrm{G}}(\tau) & =\frac{\operatorname{sech}^{7} \frac{\tau}{2}}{3 \cdot 2^{9}}\left[3 \tau\left(58 \cosh \frac{\tau}{2}-27 \cosh \frac{3 \tau}{2}-3 \cosh \frac{5 \tau}{2}\right)\right. \\ & -\left(753 \sinh \frac{\tau}{2}+48 \sinh \frac{3 \tau}{2}+22 \sinh \frac{5 \tau}{2}+\sinh \frac{7 \tau}{2}\right) \\ & +36 \cosh \frac{\tau}{2} \ln \left(2 \cosh \frac{\tau}{2}\right)(6 \tau+8 \sinh \tau+\sinh 2 \tau) \\ & \left.-108 \cosh \frac{\tau}{2}\left(\int_{0}^{\tau} d \tau^{\prime} \tau^{\prime} \tanh \frac{\tau^{\prime}}{2}\right)\right] . \end{align*} $$

$$ \Delta E_{\mathrm{s}, \mathrm{a}}=\int d x \psi_{\mathrm{s}, \mathrm{a}} \hat{H} \psi_{\mathrm{s}, \mathrm{a}} $$

$$ \begin{align*} -\int_{0}^{\infty} d \tau \tilde{x}_{\mathrm{cl}}(\tau) \operatorname{sech}^{6} \frac{\tau}{2} \int_{0}^{\tau} d \tau^{\prime} \tau^{\prime} \tanh \frac{\tau^{\prime}}{2} & =\frac{1}{3} \int_{0}^{\infty} d \tau \frac{d}{d \tau}\left[\operatorname{sech}^{6} \frac{\tau}{2}\right] \int_{0}^{\tau} d \tau^{\prime} \tau^{\prime} \tanh \frac{\tau^{\prime}}{2} \\ & =-\frac{1}{3} \int_{0}^{\infty} d \tau \operatorname{sech}^{6} \frac{\tau}{2} \tau \tanh \frac{\tau}{2}=-\frac{16}{135} \end{align*} $$

$$ \begin{align*} \left(x_{b} t_{b} \mid x_{a} t_{a}\right) & =\int \mathcal{D} x(t) e^{(i / \hbar) \int_{t_{a}}^{t_{b}} d t\left[\dot{x}^{2} / 2-V(x)\right]} \\ & =\sum_{n} \psi_{n}\left(x_{b}\right) \psi_{n}\left(x_{a}\right) e^{-i E_{n}\left(t_{b}-t_{a}\right) / \hbar} \end{align*} $$

$$ c_{1}=\frac{71}{24} \approx 2.958 $$

$$ \begin{align*} \left(x_{b} L / 2 \mid x_{a}-L / 2\right) & =\int \mathcal{D} x(\tau) e^{-(1 / \hbar) \int_{-L / 2}^{L / 2} d \tau\left[x^{\prime 2} / 2+V(x)\right]} \\ & =\sum_{n} \psi_{n}\left(x_{b}\right) \psi_{n}\left(x_{a}\right) e^{-E_{n} L / \hbar} \end{align*} $$

$$ C^{\prime}=\left[1-c_{1} \frac{g \hbar}{\omega^{3}}-c_{2}\left(\frac{g \hbar}{\omega^{3}}\right)^{2}-c_{3}\left(\frac{g \hbar}{\omega^{3}}\right)^{3}+\ldots\right] $$

$$ (a L / 2 \mid a-L / 2)=(-a L / 2 \mid-a-L / 2), $$

$$ c_{2}=\frac{315}{32} \approx 9.84376, \quad c_{3}=\frac{65953}{1152} \approx 57.2509 $$

$$ (a L / 2 \mid-a-L / 2)=(-a L / 2 \mid a-L / 2) . $$

$$ c_{k} \sim \frac{9}{\pi}\left(\frac{3}{2}\right)^{k} k![\ln (6 k)+\gamma] $$

$$ \sum_{\text {class. solutions }} \exp \left\{-\mathcal{A}_{\mathrm{cl}} / \hbar\right\} \times F $$

$$ x(\tau) \equiv \pm a $$

$$ x(\tau)=x_{\mathrm{cl}}^{ \pm}(\tau) \equiv \pm a \tanh \left[\omega\left(\tau-\tau_{0}\right) / 2\right] $$

$$ \ddot{x}(t)=-V^{\prime}(x(t)), $$

$$ x^{\prime \prime}(\tau)=V^{\prime}(x(\tau)) $$

$$ \frac{1}{2} \frac{d}{d \tau} x^{\prime 2}=\frac{d}{d \tau} V(x(\tau)) $$

$$ \frac{x^{\prime 2}}{2}+[-V(x(\tau))]=\mathrm{const} $$

$$ \text { const } \equiv E $$

$$ \tau-\tau_{0}= \pm \frac{1}{\sqrt{2}} \int_{x\left(\tau_{0}\right)}^{x(\tau)} \frac{d x}{\sqrt{E+V(x)}} $$

$$ \begin{align*} \tau-\tau_{0} & = \pm \frac{2 a}{\omega} \int_{0}^{x} \frac{d x^{\prime}}{\left(a-x^{\prime}\right)\left(x^{\prime}+a\right)}= \pm \frac{1}{\omega} \log \frac{a+x}{a-x} \\ & = \pm \frac{2}{\omega} \operatorname{arctanh} \frac{x}{a} \end{align*} $$

$$ x_{\mathrm{cl}}(\tau)= \pm a \tanh \left[\left(\tau-\tau_{0}\right) \omega / 2\right] $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}} & =\int_{-\infty}^{\infty} d \tau\left[\frac{x_{\mathrm{cl}}^{\prime 2}}{2}+V\left(x_{\mathrm{cl}}(\tau)\right)\right]=\int_{-\infty}^{\infty} d \tau\left(x_{\mathrm{cl}}^{\prime 2}-E\right) \\ & =-E L+\int_{-a}^{a} d x \sqrt{2[E+V(x)]} \end{align*} $$

$$ \sqrt{2[E+V(x)]}=\frac{\omega}{2 a}\left(a^{2}-x^{2}\right) $$

$$ \mathcal{A}_{\mathrm{cl}}=\frac{\omega}{2 a} \int_{-a}^{a} d x\left(a^{2}-x^{2}\right)=\frac{2}{3} a^{2} \omega=\frac{\omega^{3}}{3 g} $$

$$ \mathcal{A}_{\mathrm{cl}}=\int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{\prime 2} $$

$$ \begin{align*} \tau-\tau_{0} & =\mp \frac{2 a}{\omega} \int_{x}^{\infty} \frac{d x^{\prime}}{\left(x^{\prime}-a\right)\left(x^{\prime}+a\right)}= \pm \frac{1}{\omega} \log \frac{x+a}{x-a} \\ & = \pm \frac{2}{\omega} \operatorname{arccoth} \frac{x}{a} \end{align*} $$

$$ (a L / 2 \mid a-L / 2)=1 \times F_{\omega}(L) $$

$$ (a L / 2 \mid-a-L / 2)=e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} \times F_{\mathrm{cl}}(L) $$

$$ F_{\omega}(L)=\sqrt{\frac{\omega}{2 \pi \hbar \sinh \omega L}} \sim \sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2}+\mathcal{O}\left(e^{-3 \omega L / 2}\right) $$

$$ F_{\mathrm{cl}}(L)=\int \mathcal{D} y(\tau) e^{-(1 / \hbar) \int_{-L / 2}^{L / 2} d \tau(1 / 2)\left[y^{\prime 2}+V^{\prime \prime}\left(x_{\mathrm{cl}}(\tau)\right) y^{2}\right]} $$

$$ y(L / 2)=y(-L / 2)=0 $$

$$ \begin{align*} V^{\prime \prime}\left(x_{\mathrm{cl}}(\tau)\right) & =\frac{3}{2} \frac{\omega^{2}}{a^{2}} x_{\mathrm{cl}}^{2}(\tau)-\frac{1}{2} \omega^{2}=\omega^{2}\left(\frac{3}{2} \tanh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]-\frac{1}{2}\right) \\ & =\omega^{2}\left(1-\frac{3}{2} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]}\right) \end{align*} $$

$$ \mathcal{A}_{\mathrm{fl}}^{0}=\int_{-L / 2}^{L / 2} d \tau \frac{1}{2}\left[y^{\prime 2}+\omega^{2}\left(1-\frac{3}{2} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]}\right) y^{2}\right] $$

$$ \left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}\left(1-\frac{3}{2} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]}\right)\right] y_{n}(\tau)=\lambda_{n} y_{n}(\tau), $$

$$ V(\tau)=\omega^{2}\left(1-\frac{3}{2} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]}\right) . $$

$$ \int_{-\infty}^{\infty} d \tau y_{n}(\tau) y_{n^{\prime}}(\tau)=\delta_{n n^{\prime}} $$

$$ y^{\xi_{0}, \xi_{1}, \cdots}(\tau)=\sum_{n=0}^{\infty} \xi_{n} y_{n}(\tau) $$

$$ \mathcal{A}=\frac{1}{2} \sum_{n=0}^{\infty} \lambda_{n} \xi_{n}^{2} $$

$$ F_{\mathrm{cl}}(L)=\mathcal{N} \prod_{n=0}^{\infty}\left[\int_{-\infty}^{\infty} \frac{d \xi_{n}}{\sqrt{2 \pi \hbar}}\right] e^{-\sum_{n=0}^{\infty} \xi_{n}^{2} \lambda_{n} / 2 \hbar}=\mathcal{N} \frac{1}{\sqrt{\Pi_{n} \lambda_{n}}} $$

$$ V(\tau)=\omega^{2}-\frac{V_{0}}{\cosh ^{2}\left[m\left(\tau-\tau_{0}\right)\right]} $$

$$ s \equiv \frac{1}{2}\left[-1+\sqrt{1+4 \frac{V_{0}}{m^{2}}}\right] . $$

$$ V_{0}=m^{2} s(s+1) $$

$$ \lambda_{n}^{2}=\omega^{2}-m^{2}(s-n)^{2} . $$

$$ y_{0}(\tau)=-\sqrt{\frac{3 \omega}{8}} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]} $$

$$ \begin{align*} y_{1}(\tau) & =\sqrt{\frac{3 \omega}{4}} \frac{1}{\cosh \left[\omega\left(\tau-\tau_{0}\right) / 2\right]} F\left(-1,4 ; 2 ; \frac{1}{2}\left(1-\tanh \left[\omega\left(\tau-\tau_{0}\right) / 2\right]\right)\right) \\ & =\sqrt{\frac{3 \omega}{4}} \frac{\sinh \left[\omega\left(\tau-\tau_{0}\right) / 2\right]}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]} \end{align*} $$

$$ \int_{0}^{\infty} \frac{\sinh ^{\mu} x}{\cosh ^{\nu} x}=\frac{1}{2} B\left(\frac{\mu+1}{2}, \frac{\nu-\mu}{2}\right), $$

$$ \lambda_{0}=0, \quad \lambda_{1}=3 \omega^{2} / 4 $$

$$ \lambda_{k}=\omega^{2}+k^{2} $$

$$ y_{k}(\tau) \propto A e^{i k \tau} F\left(s+1,-s ; 1-i k / m ; \frac{1}{2}\left(1-\tanh \left[m\left(\tau-\tau_{0}\right)\right]\right)\right) $$

$$ \begin{align*} & F(a, b, c ; z)=\frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)} F(a, b ; a+b-c+1 ; 1-z) \\ & \quad+(1-z)^{c-a-b} \frac{\Gamma(c) \Gamma(-c+a+b)}{\Gamma(a) \Gamma(b)} F(c-a, c-b ; c-a-b+1 ; 1-z) \end{align*} $$

$$ F \xrightarrow{\tau \rightarrow \infty} 1, $$

$$ \psi(\tau) \rightarrow \begin{cases}e^{i k \tau}+R_{k} e^{-i k \tau}, & \tau \rightarrow-\infty, \\ T_{k} e^{i k \tau}, & \tau \rightarrow \infty .\end{cases} $$

$$ \begin{align*} T_{k} & =\frac{\Gamma(-s-i k / m) \Gamma(s+1-i k / m)}{\Gamma(-i k / m) \Gamma(1-i k / m)} \\ R_{k} & =\frac{\Gamma(-s-i k / m) \Gamma(s+1-i k / m)}{\Gamma(-s) \Gamma(1+s)} \frac{\Gamma(i k / m)}{\Gamma(-i k / m)}=T_{k} \frac{\Gamma(i k / m) \Gamma(1-i k / m)}{\Gamma(-s) \Gamma(1+s)} . \end{align*} $$

$$ \begin{align*} T_{k} & =\frac{\Gamma(s+1-i k / m)}{\Gamma(s+1+i k / m)} \frac{\Gamma(1+i k / m)}{\Gamma(1-i k / m)} \frac{\sin (i k / m)}{\sin (s+i k / m)} \\ R_{k} & =T_{k} \frac{\sin (\pi s)}{\sin (i k / m)} \end{align*} $$

$$ S_{k}=\left(\begin{array}{cc} T_{k} & R_{k} \\ R_{k} & T_{k} \end{array}\right) $$

$$ R_{k} T_{k}^{*}+R_{k}^{*} T_{k}=0, \quad\left|T_{k}\right|^{2}+\left|R_{k}\right|^{2}=1 $$

$$ \psi^{\mathrm{e}}=\frac{1}{\sqrt{2}}\binom{1}{1}, \quad \psi^{\mathrm{o}}=\frac{1}{\sqrt{2}}\binom{1}{-1} $$

$$ \begin{align*} T_{k} & =\frac{1}{2}\left(e^{2 i \delta_{k}^{\mathrm{e}}}+e^{2 i \delta_{k}^{\mathrm{o}}}\right) \\ R_{k} & =\frac{1}{2}\left(e^{2 i \delta_{k}^{\mathrm{e}}}-e^{2 i \delta_{k}^{\mathrm{o}}}\right) \end{align*} $$

$$ \psi^{\mathrm{r}}(\tau) \rightarrow \begin{cases}T_{k} e^{-i k \tau}, & \tau \rightarrow-\infty \\ e^{-i k \tau}+R_{k} e^{i k \tau}, & \tau \rightarrow \infty\end{cases} $$

$$ \psi^{\mathrm{e}}(\tau)=\psi(\tau)+\psi^{\mathrm{r}}(\tau) \rightarrow \begin{cases}e^{i k \tau}+\left(R_{k}+T_{k}\right) e^{-i k \tau}, & \tau \rightarrow-\infty \\ e^{-i k \tau}+\left(R_{k}+T_{k}\right) e^{i k \tau}, & \tau \rightarrow \infty\end{cases} $$

$$ \psi^{\mathrm{e}}(\tau) \rightarrow\left\{\begin{array}{lll} e^{i \delta_{k}^{\mathrm{e}}}\left[e^{i\left(k \tau-\delta_{k}^{\mathrm{e}}\right)}+e^{-i\left(k \tau-\delta_{k}^{\mathrm{e}}\right)}\right] & =2 e^{i \delta_{k}^{\mathrm{e}}} \cos \left(k|\tau|+\delta_{k}^{\mathrm{e}}\right), & \\ e^{i \delta_{k}^{\mathrm{e}}}\left[e^{-i\left(k \tau+\delta_{k}^{\mathrm{e}}\right)}+e^{i\left(k \tau+\delta_{k}^{\mathrm{e}}\right)}\right] & =2 e^{i \delta_{k}^{\mathrm{e}}} \cos \left(k|\tau|+\delta_{k}^{\mathrm{e}}\right), & \\ & \tau \rightarrow \infty . \end{array}\right. $$

$$ \begin{align*} e^{i k \tau}+\left(R_{k}-T_{k}\right) e^{-i k \tau}, & \tau \rightarrow-\infty \\ -e^{-i k \tau}-\left(R_{k}-T_{k}\right) e^{i k \tau}, & \tau \rightarrow \infty \end{align*} $$

$$ \begin{align*} e^{i \delta_{k}^{\mathrm{o}}}\left[e^{i\left(k \tau-\delta_{k}^{\mathrm{e}}\right)}-e^{-i\left(k \tau-\delta_{k}^{\mathrm{o}}\right)}\right] & =2 i e^{i \delta_{k}^{\mathrm{e}}} \sin \left(k|\tau|+\delta_{k}^{\mathrm{o}}\right), & & \tau \rightarrow-\infty, \\ -e^{i \delta_{k}^{\mathrm{e}}}\left[e^{-i\left(k \tau+\delta_{k}^{\mathrm{o}}\right)}-e^{i\left(k \tau+\delta_{k}^{\mathrm{e}}\right)}\right] & =-2 i e^{i \delta_{k}^{\mathrm{e}}} \sin \left(k|\tau|+\delta_{k}^{\mathrm{o}}\right), & & \tau \rightarrow \infty . \end{align*} $$

$$ \left|T_{k}\right|^{2}=\cos ^{2}\left(\delta_{k}^{\mathrm{e}}-\delta_{k}^{\mathrm{o}}\right), \quad\left|R_{k}\right|^{2}=\sin ^{2}\left(\delta_{k}^{\mathrm{e}}-\delta_{k}^{\mathrm{o}}\right), $$

$$ e^{2 i\left(\delta_{k}^{\mathrm{e}}+\delta_{k}^{\mathrm{o}}\right)}=\left(T_{k}+R_{k}\right)\left(T_{k}-R_{k}\right)=T_{k}^{2}+\frac{R_{k} R_{k}^{*} T_{k}}{T_{k}^{*}}=T_{k}^{2}+\left(1-T_{k} T_{k}^{*}\right) \frac{T_{k}}{T_{k}^{*}}=\frac{T_{k}}{T_{k}^{*}} $$

$$ \delta_{k}^{\mathrm{e}}+\delta_{k}^{\mathrm{o}}=\frac{1}{2} \arg \frac{T_{k}}{T_{k}^{*}}=\arg T_{k} $$

$$ -i \sin \left[2\left(\delta_{k}^{\mathrm{e}}-\delta_{k}^{\mathrm{o}}\right)\right]=T_{k} R_{k}^{*}-T_{k}^{*} R_{k}=2 T_{k} R_{k}^{*}=-2 \frac{R_{k}^{*}}{T_{k}^{*}}\left|T_{k}\right|^{2} $$

$$ -i \tan \left(\delta_{k}^{\mathrm{e}}-\delta_{k}^{\mathrm{o}}\right)=-\frac{R_{k}^{*}}{T_{k}^{*}}=-\frac{\sinh (i k / m)}{\sin (\pi s)} $$

$$ \delta_{k}^{\mathrm{e}}-\delta_{k}^{\mathrm{o}}=\arctan \frac{\sin (\pi s)}{\sinh (k / m)} \sin (\pi s) $$

$$ \delta_{k}^{\mathrm{e}}=\delta_{k}^{\mathrm{o}} \equiv \delta_{k}, $$

$$ 2 \delta_{k}=-i \log T_{k} $$

$$ y_{k}(\tau) \rightarrow e^{i\left(k \tau \pm \delta_{k}\right)}, \quad \tau \rightarrow \pm \infty $$

$$ e^{2 i \delta_{k}}=(-1)^{s} \frac{\Gamma(s+1-i k / m)}{\Gamma(1-i k / m)} \frac{\Gamma(1+i k / m)}{\Gamma(s+1+i k / m)} . $$

$$ e^{2 i \delta_{k}}=\frac{2-i k / m}{2+i k / m} \frac{1-i k / m}{1+i k / m}, $$

$$ \delta_{k}=\arctan [k / m]+\arctan [k / 2 m] . $$

$$ \begin{align*} y_{0} & =\alpha\left[\frac{x_{\mathrm{cl}}\left(\tau-\tau_{0}\right)-x_{\mathrm{cl}}\left(\tau-\tau_{1}\right)}{\tau_{0}-\tau_{1}}\right]_{\tau_{0} \rightarrow \tau_{1}}=\alpha x_{\mathrm{cl}}^{\prime}\left(\tau-\tau_{0}\right) \\ & =-\alpha \frac{a \omega / 2}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]} \end{align*} $$

$$ \alpha=\left[\int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{\prime 2}\right]^{-1 / 2} $$

$$ \alpha=\frac{1}{\sqrt{\mathcal{A}_{\mathrm{cl}}}} . $$

$$ \alpha=\sqrt{3 / 2 a^{2} \omega}=\sqrt{3 g / \omega^{3}} $$

$$ \frac{1}{\sqrt{\lambda_{0}}}=\text { const } \cdot L $$

$$ \mathcal{N} \int_{-\infty}^{\infty} \frac{d \xi_{0}}{\sqrt{2 \pi \hbar}} \prod_{n=1}^{\infty}\left[\int_{-\infty}^{\infty} \frac{d \xi_{n}}{\sqrt{2 \pi \hbar}}\right] $$

$$ \mathcal{N} \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} d \tau_{0} \prod_{n \neq 0} \int_{-\infty}^{\infty}\left[\frac{d \xi_{n}}{\sqrt{2 \pi \hbar}}\right]\left|\frac{\partial \xi_{0}}{\partial \tau_{0}}\left(\xi_{1}, \xi_{2}, \ldots\right)\right| $$

$$ \int d \tau_{0} \delta\left(\xi_{0}\right)\left|\frac{\partial \xi_{0}}{\partial \tau_{0}}\right|=1 $$

$$ \xi_{0}=\int_{-\infty}^{\infty} d \tau y^{\xi_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) y_{0}(\tau) $$

$$ x^{\xi_{0}, \xi_{1}, \xi_{2}, \cdots}(\tau)=x_{\mathrm{cl}}(\tau)+y^{\xi_{0}, \xi_{1}, \xi_{2}, \cdots}(\tau) $$

$$ \xi_{0}=\int_{-\infty}^{\infty} d \tau x^{\xi_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) y_{0}(\tau) $$

$$ \begin{align*} \int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}\left(\tau-\tau_{0}\right) y_{0}\left(\tau-\tau_{0}\right) & \propto \int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}\left(\tau-\tau_{0}\right) x_{\mathrm{cl}}^{\prime}\left(\tau-\tau_{0}\right) \\ & \left.\propto \frac{1}{2} x_{\mathrm{cl}}^{2}\right|_{-\infty} ^{\infty}=0 \end{align*} $$

$$ \delta\left(\xi_{0}\right)=\delta\left(\int_{-\infty}^{\infty} d \tau x^{\xi_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) y_{0}(\tau)\right) $$

$$ x^{\xi_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau)=x_{\mathrm{cl}}(\tau)+\sum_{n=0}^{\infty} \xi_{n} y_{n}(\tau) $$

$$ x^{\tau_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) \equiv x_{\mathrm{cl}}\left(\tau-\tau_{0}\right)+\sum_{n=1}^{\infty} \xi_{n} y_{n}\left(\tau-\tau_{0}\right) $$

$$ \int_{-\infty}^{\infty} d \tau_{0} \delta\left(\int_{-\infty}^{\infty} d \tau x^{\tau_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) y_{0}(\tau)\right)\left|\frac{\partial \xi_{0}}{\partial \tau_{0}}\right|=1 $$

$$ \int_{-\infty}^{\infty} d \tau x^{\tau_{0}, \xi_{1}, \xi_{2}, \ldots}(\tau) y_{0}(\tau)=-\alpha \tau_{0}\left[\int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{\prime 2}+\sum_{n=1}^{\infty} \xi_{n} \int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{\prime} y_{n}^{\prime}\right]+\mathcal{O}\left(\tau_{0}^{2}\right) $$

$$ r_{n} \equiv \int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{\prime} y_{n}^{\prime} $$

$$ -\alpha \tau_{0}\left[\mathcal{A}_{\mathrm{cl}}+\sum_{n=1}^{\infty} \xi_{n} r_{n}\right]+\mathcal{O}\left(\tau_{0}^{2}\right) $$

$$ \left|\frac{\partial \xi_{0}}{\partial \tau_{0}}\right|=\mathcal{A}_{\mathrm{cl}}^{1 / 2}\left(1+\mathcal{A}_{\mathrm{cl}}^{-1} \sum_{n=1}^{\infty} \xi_{n} r_{n}\right) . $$

$$ \begin{align*} F_{\mathrm{cl}}(L) & =\mathcal{N} \prod_{n=1}^{\infty}\left[\int_{-\infty}^{\infty} \frac{d \xi_{n}}{\sqrt{2 \pi \hbar}}\right] e^{-(1 / 2 \hbar) \sum_{n=1}^{\infty} \xi_{n}^{2} \lambda_{n}} \int_{-\infty}^{\infty} \frac{d \xi_{0}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \xi_{0}^{2} \lambda_{0}} \\ & =\mathcal{N} \prod_{n=1}^{\infty}\left[\int_{-\infty}^{\infty} \frac{d \xi_{n}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \sum_{n=1}^{\infty} \xi_{n}^{2} \lambda_{n}}\right] \int_{-\infty}^{\infty} \frac{d \tau_{0}}{\sqrt{2 \pi \hbar}} \mathcal{A}_{\mathrm{cl}}^{1 / 2}\left(1+\mathcal{A}_{\mathrm{cl}}^{-1} \sum_{n=1}^{\infty} \xi_{n} r_{n}\right) \\ & =\mathcal{N} \frac{1}{\sqrt{\prod_{n=1}^{\infty} \lambda_{n}}} \sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} \int_{-\infty}^{\infty} d \tau_{0} \end{align*} $$

$$ -\frac{\partial \xi_{0}}{\partial \tau_{0}}=\frac{\partial x(\tau) / \partial \tau_{0}}{\partial x(\tau) / \partial \xi_{0}}=\frac{\dot{x}_{\mathrm{cl}}(\tau)+\ldots}{\alpha \dot{x}_{\mathrm{cl}}(\tau)}=\frac{1}{\alpha}+\ldots=\sqrt{\mathcal{A}_{\mathrm{cl}}}+\ldots, $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{eff}}=-\hbar \log \left[1+\mathcal{A}_{\mathrm{cl}}^{-1} \sum_{n=1}^{\infty} \xi_{n} r_{n}\right]=-\hbar \log \left[1+\mathcal{A}_{\mathrm{cl}}^{-1} \int d \tau x_{\mathrm{cl}}^{\prime}(\tau) y^{\prime}(\tau)\right], $$

$$ \int_{-L / 2}^{L / 2} d \tau_{0}=L $$

$$ \frac{1}{\sqrt{\lambda_{0}}} \equiv \int \frac{d \xi_{0}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \lambda_{0} \xi_{0}^{2}} \longrightarrow \sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} \int_{-L / 2}^{L / 2} d \tau_{0}=\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} L $$

$$ \mathcal{N} \frac{1}{\sqrt{\prod_{n} \lambda_{n}^{0}}}=F_{\omega}(L)=\sqrt{\frac{\omega}{2 \pi \hbar \sinh \omega L}} \sim \sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2} $$

$$ F_{\mathrm{cl}}(L)=\mathcal{N} \frac{1}{\sqrt{\prod_{n} \lambda_{n}}}=\mathcal{N} \frac{1}{\sqrt{\prod_{n} \lambda_{n}^{0}}} \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}=F_{\omega}(L) \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}} $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}}=\exp \left[-\frac{1}{2} \int_{0}^{\infty} d k\left(\frac{\partial n}{\partial k}-\left.\frac{\partial n}{\partial k}\right|_{0}\right) \log \lambda_{n}\right] $$

$$ y(\tau+L)=y(\tau) $$

$$ e^{i\left(k L / 2+\delta_{k}\right)}=e^{-i\left(k L / 2+\delta_{k}\right)} $$

$$ k L+2 \delta_{k}=2 \pi n $$

$$ \frac{\partial n}{\partial k}=\frac{L}{2 \pi}+\frac{1}{\pi} \frac{d \delta_{k}}{d k} $$

$$ \left.\frac{\partial n}{\partial k}\right|_{0}=\frac{L}{2 \pi}, $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}}=\exp \left[-\frac{1}{2 \pi} \int_{0}^{\infty} d k \frac{d \delta_{k}}{d k} \log \left(\omega^{2}+k^{2}\right)\right] $$

$$ \frac{d \delta_{k}}{d k}=-\frac{1}{m}\left[\frac{2}{4+(k / m)^{2}}+\ldots+\frac{s}{s^{2}+(k / m)^{2}}\right] $$

$$ \frac{1}{2 \pi} \int_{-\infty}^{\infty} d x\left(\frac{1}{1+x^{2}}+\frac{2}{4+x^{2}}\right) \log \left[\omega^{2}\left(1+x^{2} m^{2} / \omega^{2}\right)\right] $$

$$ \int_{0}^{\infty} d k\left(\frac{\partial n}{\partial k}-\left.\frac{\partial n}{\partial k}\right|_{0}\right)=\frac{1}{\pi} \int_{0}^{\infty} d k \frac{d \delta_{k}}{d k}=s $$

$$ \log \omega^{2}+\int_{-\infty}^{\infty} \frac{d x}{2 \pi}\left(\frac{1}{1+x^{2}}+\frac{2}{4+x^{2}}\right) \log \left(1+x^{2} m^{2} / \omega^{2}\right) $$

$$ \int_{-\infty}^{\infty} \frac{d x}{2 \pi} \frac{\log \left(1+p^{2} x^{2}\right)}{r^{2}+s^{2} x^{2}}=\frac{1}{r s} \log \left(1+p \frac{r}{s}\right) $$

$$ \log \omega^{2}+\log \left(1+\frac{m}{\omega} 1\right)+\log \left(1+\frac{m}{\omega} 2\right) $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}}=\omega^{2}\left(1+\frac{m}{\omega}\right)\left(1+\frac{m}{\omega} 2\right) $$

$$ \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}=\frac{1}{\sqrt{3 \omega^{2} / 4}} 3 \omega^{2}=\sqrt{12} \omega \equiv K^{\prime} $$

$$ F_{\mathrm{cl}}(L)=F_{\omega}(L) K L $$

$$ K=\frac{1}{\sqrt{\lambda_{0}} L} K^{\prime}=\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} \sqrt{12} \omega $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}} $$

$$ \left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}-\frac{m^{2} s(s+1)}{\cosh ^{2} m\left(\tau-\tau_{0}\right)}\right] y_{n}(\tau)=\lambda_{n} y_{n}(\tau) $$

$$ \begin{align*} \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\text {cont }} & =\exp \left[-\frac{1}{2 \pi} \int_{0}^{\infty} d k \frac{d \delta_{k}}{d k} \log \left(\omega^{2}+k^{2}\right)\right] \\ & =\exp \left\{-\frac{1}{2 \pi} \int_{0}^{\infty} d(k / m) \sum_{n=1}^{s} \frac{n}{n^{2}+(k / m)^{2}} \log \left[\omega^{2}+(k / m)^{2} m^{2}\right]\right\} \end{align*} $$

$$ \frac{1}{\pi} \int_{-\infty}^{\infty} d(k / m) \sum_{n=1}^{s} \frac{n}{n^{2}+(k / m)^{2}}=s $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}}=\omega^{s} \exp \left[\frac{1}{2 \pi} \int_{-\infty}^{\infty} d x \sum_{n=1}^{s} \frac{n}{n^{2}+x^{2}} \log \left(1+x^{2} m^{2} / \omega^{2}\right)\right] $$

$$ \left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}\right|_{\mathrm{cont}}=\omega^{s} \prod_{n=1}^{s}\left(1+\frac{m}{\omega} n\right) $$

$$ \left[-\frac{d^{2}}{d \tau^{2}}+m^{2}\left(z-\frac{s(s+1)}{\cosh ^{2} m\left(\tau-\tau_{0}\right)}\right)\right] y_{n}(\tau)=\lambda_{n} y_{n}(\tau) . $$

$$ \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}=\exp \left[-\frac{1}{2 \pi} \int_{-\infty}^{\infty} d k \frac{d \delta_{k}}{d k} \log \left(m^{2} z+k^{2}\right)\right] . $$

$$ \sqrt{\frac{\Pi_{n} \lambda_{n}^{0}}{\Pi_{n} \lambda_{n}}}=\exp \left[-\frac{1}{2 \pi} \int_{C} d \epsilon \frac{d \delta \omega \sqrt{\epsilon}}{d \epsilon} \log (z+\epsilon)\right] $$

$$ \exp \left(\frac{1}{2 \pi} \int_{C} d \epsilon \delta_{m \sqrt{\epsilon}} \frac{1}{z+\epsilon}\right) $$

$$ \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}}=\exp \left[i \delta_{m \sqrt{-z}}\right] . $$

$$ \sqrt{\frac{\Pi_{n} \lambda_{n}^{0}}{\Pi_{n} \lambda_{n}}}=\left[\frac{\Gamma(\sqrt{z}-s) \Gamma(\sqrt{z}+s+1)}{\Gamma(\sqrt{z}) \Gamma(\sqrt{z}+1)}\right]^{1 / 2} $$

$$ \lambda_{0}=m^{2}\left[z-s^{2}\right] $$

$$ \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}=m\left[(\sqrt{z}+s) \frac{\Gamma(\sqrt{z}-s+1) \Gamma(\sqrt{z}+s+1)}{\Gamma(\sqrt{z}) \Gamma(\sqrt{z}+1)}\right]^{1 / 2} $$

$$ \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}=\sqrt{12} \omega $$

$$ \hat{\mathcal{O}}=-\frac{d^{2}}{d \tau^{2}}+\omega^{2}-\frac{m^{2} s(s+1)}{\cosh ^{2}\left[m\left(\tau-\tau_{0}\right)\right]} $$

$$ \operatorname{det} \hat{\mathcal{O}}=\mathcal{N} D(L / 2) $$

$$ D(-L / 2)=0, \quad \dot{D}(-L / 2)=1 $$

$$ y_{0}(\tau)=\alpha x_{\mathrm{cl}}^{\prime}(\tau) $$

$$ y_{0}(\tau) \rightarrow \frac{\omega}{2} e^{-\omega|\tau|} \quad \text { for } \quad \tau \rightarrow \pm \infty $$

$$ \xi(\tau) \rightarrow e^{-\omega|\tau|} \quad \text { for } \quad \tau \rightarrow \pm \infty $$

$$ \eta(\tau) \rightarrow \pm e^{\omega|\tau|} \quad \text { for } \tau \rightarrow \pm \infty . $$

$$ D(\tau)=\frac{1}{W}[\xi(-L / 2) \eta(\tau)-\eta(-L / 2) \xi(\tau)] $$

$$ W \equiv W[\xi(\tau) \eta(\tau)]=\xi(\tau) \dot{\eta}(\tau)-\eta(\tau) \dot{\xi}(\tau) $$

$$ W=2 \omega $$

$$ D(\tau)=\frac{1}{W}\left[e^{-\omega L / 2} \eta(\tau)+e^{\omega L / 2} \xi(\tau)\right] $$

$$ D(L / 2)=\frac{2}{W}=\frac{1}{\omega} $$

$$ D^{(0)}(\tau)=\frac{1}{\omega} \sinh [\omega(\tau+L / 2)] $$

$$ D^{(0)}(L / 2) \rightarrow \frac{1}{2 \omega} e^{\omega L} \quad \text { for large } L $$

$$ \frac{D^{(0)}(L / 2)}{D(L / 2)} \rightarrow \frac{1}{2} e^{\omega L} \quad \text { for large } L $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}=\lim _{L \rightarrow \infty} \frac{D^{(0)}(L / 2)}{D(L / 2)} \lambda_{0} $$

$$ \phi_{0}^{L}(\tau)=\phi_{0}(\tau)+\frac{\lambda_{0}}{W} \int_{-L / 2}^{\tau} d \tau^{\prime}\left[\xi(\tau) \eta\left(\tau^{\prime}\right)-\eta(\tau) \xi\left(\tau^{\prime}\right)\right] \phi_{0}\left(\tau^{\prime}\right) $$

$$ \lambda_{0}=-D(L / 2) W\left[\xi(L / 2) \int_{-L / 2}^{L / 2} d \tau \eta(\tau) D(\tau)-\eta(L / 2) \int_{-L / 2}^{L / 2} d \tau \xi(\tau) D(\tau)\right]^{-1} $$

$$ \lambda_{0}=-D(L / 2) W^{2}\left[\xi(-L / 2) \xi(L / 2) \int_{-L / 2}^{L / 2} d \tau \eta^{2}(\tau)+\eta(-L / 2) \eta(L / 2) \int_{-L / 2}^{L / 2} d \tau \xi^{2}(\tau)\right]^{-1} $$

$$ \lambda_{0}=-D(L / 2) W^{2}\left[e^{-\omega L} \int_{-L / 2}^{L / 2} d \tau \eta^{2}(\tau)-e^{\omega L} \int_{-L / 2}^{L / 2} d \tau \xi^{2}(\tau)\right]^{-1} $$

$$ \lambda_{0}=D(L / 2) e^{-\omega L} \frac{W^{2}}{\int_{-\infty}^{\infty} d \tau \xi^{2}(\tau)} $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}=\lim _{L \rightarrow \infty} 2 \omega \frac{1}{\int_{-\infty}^{\infty} d \tau \xi^{2}(\tau)} $$

$$ \xi(\tau)=\frac{1}{2 a \omega \alpha} \alpha x_{\mathrm{cl}}^{\prime}(\tau) $$

$$ \int_{-\infty}^{\infty} d \tau \xi(\tau)^{2}=\frac{\mathcal{A}_{\mathrm{cl}}}{4 a^{2} \omega^{2}} $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}=2 \omega \frac{4 a^{2} \omega}{\mathcal{A}_{\mathrm{cl}}} $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}=12 \omega^{2} $$

$$ (a L / 2 \mid a-L / 2), \quad(a L / 2 \mid-a-L / 2) $$

$$ \mathcal{A}_{\mathrm{cl}} \approx \frac{\omega}{2 \hbar \sinh \omega L}\left\{\left[\left(x_{a}-a\right)^{2}+\left(x_{b}-a\right)^{2}\right] \cosh \omega L-2\left(x_{b}-a\right)\left(x_{a}-a\right)\right\} $$

$$ \mathcal{A}_{\mathrm{cl}} \approx \frac{\omega}{2 \hbar}\left[\left(x_{b}-a\right)^{2}+\left(x_{a}-a\right)^{2}\right] $$

$$ \left(x_{b} L / 2 \mid x_{a} L / 2\right) \approx \psi_{0}\left(x_{b}-a\right) \psi_{0}\left(x_{a}-a\right) e^{-\omega L / 2 \hbar} $$

$$ \sqrt{\frac{\omega}{\pi \hbar}} e^{-(\omega / 2 \hbar)\left(x_{b}-a\right)^{2}} K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} e^{-(\omega / 2 \hbar)\left(x_{a}-a\right)^{2}} $$

$$ \left(x_{b} L / 2 \mid x_{a}-L / 2\right) \approx \psi_{0}\left(x_{b}+a\right) \psi_{0}\left(x_{a}-a\right) K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} e^{-\omega L / 2 \hbar} $$

$$ \mathcal{A}_{2 n} \approx 2 n \mathcal{A}_{\mathrm{cl}} $$

$$ \mathcal{A}_{2 n+1} \approx(2 n+1) \mathcal{A}_{\mathrm{cl}} $$

$$ \int_{-L / 2}^{L / 2} d \tau_{N} \int_{-L / 2}^{\tau_{N}} d \tau_{N-1} \cdots \int_{-L / 2}^{\tau_{1}} d \tau_{1}=\frac{L^{N}}{N!} $$

$$ {\sqrt{{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}}^{N}}}^{N} $$

$$ \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{\bar{L}_{N}}} \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{\bar{L}_{N-1}}} \times \ldots \times \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{\bar{L}_{1}}} $$

$$ L=\sum_{i=1}^{N} \bar{L}_{i} $$

$$ \frac{1}{\sqrt{\prod_{n} \lambda_{n}}} \rightarrow \frac{\psi_{0}\left(x_{i} \pm a\right)}{\psi_{0}(0)} \frac{1}{\sqrt{\prod_{n} \lambda_{n}}} \frac{\psi_{0}^{\dagger}\left(x_{i-1} \pm a\right)}{\psi_{0}^{\dagger}(0)} $$

$$ \begin{array}{r} \left.\frac{1}{\sqrt{\prod_{n} \lambda_{n}}}\right|_{L}=\left.\left.\int d x_{N} \cdots d x_{1} \frac{1}{\sqrt{\prod_{n} \lambda_{n}}}\right|_{L_{N}} \frac{\psi_{0}\left(x_{N-1}-a\right) \psi_{0}^{\dagger}\left(x_{N-1}-a\right)}{\left|\psi_{0}(0)\right|^{2}} \frac{1}{\sqrt{\prod_{n} \lambda_{n}}}\right|_{L_{N-1}} \\ \times \ldots \times\left.\frac{\psi_{0}\left(x_{1}-a\right) \psi_{0}^{\dagger}\left(x_{1}-a\right)}{\left|\psi_{0}(0)\right|^{2}} \frac{1}{\sqrt{\prod_{n} \lambda_{n}}}\right|_{L_{1}} .(17.18 \end{array} $$

$$ \frac{1}{\left|\psi_{0}(0)\right|^{2(N-1)}}=\sqrt{\frac{\omega}{\pi \hbar}}^{-(N-1)} $$

$$ \frac{1}{\left.\sqrt{\Pi_{n} \lambda_{n}^{0}}\right|_{L}}=\sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} $$

$$ \left.\sqrt{\frac{\omega}{\pi \hbar}}^{-(N-2)} e^{-\omega L / 2 \hbar} \sqrt{\prod_{n} \lambda_{n}^{0}}\right|_{L} \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{L_{1}}} \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{L_{2}}} \times \ldots \times \frac{1}{\left.\sqrt{\prod_{n}^{\prime} \lambda_{n}}\right|_{L_{N}}} $$

$$ \left.\sqrt{\prod_{n} \lambda_{n}^{0}}\right|_{L}=\left.\left.\sqrt{\frac{\omega}{\pi \hbar}}^{-(N-1)} \sqrt{\prod_{n} \lambda_{n}^{0}}\right|_{L_{1}} \cdots \sqrt{\prod_{n} \lambda_{n}^{0}}\right|_{L_{N}} $$

$$ \left.\sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} \sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}\right|_{L_{1}} \times \ldots \times\left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}\right|_{L_{N}} $$

$$ K^{\prime}=\left.\sqrt{\frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}}\right|_{L_{i}} $$

$$ K^{\prime}={\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}}}^{-1} K $$

$$ \sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} \frac{L^{N}}{N!} K^{N} e^{-N \mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ (a L / 2 \mid \pm a-L / 2)=\sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} \sum_{\substack{\text { even } \\ \text { odd }}} \frac{1}{N!}\left(K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right)^{N} $$

$$ \begin{align*} (a L / 2 \mid \pm a & -L / 2)=\sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} \\ & \times \frac{1}{2}\left[\exp \left(K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right) \pm \exp \left(-K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right)\right] \end{align*} $$

$$ \begin{align*} & \left(x_{b} L / 2 \mid x_{a}-L / 2\right)=e^{-\omega L / 2 \hbar} \\ & \quad \times\left\{\psi_{0}\left(x_{b}-a\right) \psi_{0}\left(x_{a}-a\right) \frac{1}{2}\left[\exp \left(K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} L\right)+\exp \left(-K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} L\right)\right]\right. \\ & \quad+\psi_{0}\left(x_{b}-a\right) \psi_{0}\left(x_{a}-a\right) \frac{1}{2}\left[\exp \left(K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} L\right)-\exp \left(-K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} L\right)\right] \\ & \left.\quad+\left(x_{b} \rightarrow-x_{b}\right)+\left(x_{a} \rightarrow-x_{a}\right)+\left(x_{b} \rightarrow-x_{b}, x_{a} \rightarrow-x_{a}\right)\right\} \end{align*} $$

$$ \begin{align*} \frac{1}{\sqrt{2}}\left[\psi_{0}\left(x_{b}-a\right)+\right. & \left.\psi_{0}\left(x_{b}+a\right)\right] \times \frac{1}{\sqrt{2}}\left[\psi_{0}\left(x_{a}-a\right)+\psi_{0}\left(x_{a}+a\right)\right] \\ & \times \exp \left[-\left(\frac{\omega}{2}-K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right) L\right] \\ +\frac{1}{\sqrt{2}}\left[\psi_{0}\left(x_{b}-a\right)-\right. & \left.\psi_{0}\left(x_{b}+a\right)\right] \times \frac{1}{\sqrt{2}}\left[\psi_{0}\left(x_{a}-a\right)-\psi_{0}\left(x_{a}+a\right)\right] \\ & \times \exp \left[-\left(\frac{\omega}{2}+K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right) L\right] \end{align*} $$

$$ \Psi_{0}(x)=\frac{1}{\sqrt{2}}\left[\psi_{0}(x-a)+\psi_{0}(x+a)\right] $$

$$ \mathcal{E}^{(0)}=E^{(0)}-\frac{\Delta E^{(0)}}{2}=\left(\omega / 2-K e^{-A_{\mathrm{cl}} / \hbar}\right) \hbar $$

$$ \Psi_{1}(x)=\frac{1}{\sqrt{2}}\left[\psi_{0}(x-a)-\psi_{0}(x+a)\right] $$

$$ \mathcal{E}^{(1)}=E^{(0)}+\frac{\Delta E^{(0)}}{2}=\left(\omega / 2+K e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right) \hbar $$

$$ \Delta E=2 K \hbar e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ \Delta E=4 \sqrt{3} \sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} \hbar \omega e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ \Delta E=4 \sqrt{3} \sqrt{\frac{8 V_{\max }}{3 \pi \omega \hbar}} \hbar \omega e^{-16 V_{\max } / 3 \hbar \omega} . $$

$$ \bar{N} \approx K L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}=\frac{\Delta E}{2 \hbar} L $$

$$ \Delta L \equiv 2 \hbar / \Delta E $$

$$ \frac{\text { distance }}{\text { size }} \approx \frac{\hbar \omega}{\Delta E} . $$

$$ V(x)=-\frac{\omega^{2}}{4} x^{2}+\frac{g}{4} x^{4}+\frac{1}{4 g} $$

$$ g \equiv \frac{\omega^{2}}{2 a^{2}} $$

$$ \mathcal{A}_{\mathrm{f}}^{0}=\frac{1}{2} \int d \tau d \tau^{\prime} y(\tau) \mathcal{O}_{\omega}\left(\tau, \tau^{\prime}\right) y\left(\tau^{\prime}\right) $$

$$ \mathcal{O}_{\omega}\left(\tau, \tau^{\prime}\right) \equiv\left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}\left(1-\frac{3}{2} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right) / 2\right]}\right)\right]^{\prime} \delta\left(\tau-\tau^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{eff}}=-\hbar \log \left[1+\mathcal{A}_{\mathrm{cl}}^{-1} \int d \tau x_{\mathrm{cl}}^{\prime}(\tau) y^{\prime}(\tau)\right] $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{eff}}=-\hbar \log \left[1-\sqrt{\frac{3 g}{\omega^{3}}} \int d \tau y_{0}^{\prime}(\tau) y(\tau)\right] $$

$$ \mathcal{A}_{\mathrm{fl}}^{\mathrm{int}}=\frac{g}{4} \int d \tau\left[y^{4}(\tau)+4 x_{\mathrm{cl}}(\tau) y^{3}(\tau)\right] $$

$$ C=\left[1-\left(I_{1}+I_{2}+I_{3}\right) \frac{g \hbar}{\omega^{3}}+\mathcal{O}\left(g^{2}\right)\right] $$

$$ \begin{align*} I_{1} & =\frac{\omega^{3}}{4 \hbar^{2}} \int d \tau\left\langle y^{4}(\tau)\right\rangle_{\mathcal{O}_{\omega}} \\ I_{2} & =-\frac{\omega^{3} g}{2 \hbar^{3}} \int d \tau d \tau^{\prime} x_{\mathrm{cl}}(\tau)\left\langle y^{3}(\tau) y^{3}\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\omega}} x_{\mathrm{cl}}\left(\tau^{\prime}\right) \\ I_{3} & =-\frac{\omega^{3}}{\hbar^{2}} \sqrt{\frac{3 g}{\omega^{3}}} \int d \tau d \tau^{\prime} y_{0}^{\prime}(\tau)\left\langle y(\tau) y^{3}\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\omega}} x_{\mathrm{cl}}\left(\tau^{\prime}\right) \end{align*} $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\left\langle y(\tau) y\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\omega}}=\hbar \mathcal{O}_{\omega}^{-1}\left(\tau, \tau^{\prime}\right) $$

$$ I_{1}=\frac{3 \omega^{3}}{4 \hbar^{2}} \int d \tau G_{\mathcal{O}_{\omega}}^{\prime 2}(\tau, \tau) $$

$$ I_{1}=L \frac{3 \omega}{16}+\frac{3 \omega^{3}}{4 \hbar^{2}} \int d \tau\left[G_{\mathcal{O}_{\omega}}^{\prime 2}(\tau, \tau)-\frac{\hbar^{2}}{4 \omega^{2}}\right] $$

$$ G_{\mathcal{O}_{\omega}}^{\prime 2}\left(\tau, \tau^{\prime}\right) \rightarrow \hbar G_{\omega}\left(\tau-\tau^{\prime}\right)=\frac{\hbar}{2 \omega} e^{-\omega\left|\tau-\tau^{\prime}\right|} $$

$$ C^{\prime}=\left[1-c_{1} \frac{g \hbar}{\omega^{3}}+\ldots\right]=\left[1-\left(I_{1}^{\prime}+I_{2}^{\prime}+I_{3}^{\prime}\right) \frac{g \hbar}{\omega^{3}}+\mathcal{O}\left(g^{2}\right)\right], $$

$$ \begin{align*} I_{2} \equiv I_{21}+I_{22} & =-\frac{g \omega^{3}}{2 \hbar^{3}} \int d \tau d \tau^{\prime} \\ & \times x_{\mathrm{cl}}(\tau)\left[6 G_{\mathcal{O}_{\omega}}^{\prime 3}\left(\tau, \tau^{\prime}\right)+9 G_{\mathcal{O}_{\omega}}^{\prime}(\tau, \tau) G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right) G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau^{\prime}, \tau^{\prime}\right)\right] x_{\mathrm{cl}}\left(\tau^{\prime}\right) \end{align*} $$

$$ I_{3}=I_{3}^{\prime}=-3 \frac{\omega^{3}}{\hbar^{2}} \sqrt{\frac{3 g}{\omega^{3}}} \int d \tau d \tau^{\prime} y_{0}^{\prime}(\tau) G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right) G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau^{\prime}, \tau^{\prime}\right) x_{\mathrm{cl}}\left(\tau^{\prime}\right) $$

$$ \left(-\frac{\hbar^{2}}{2 \mu} \frac{d^{2}}{d x^{2}}-E_{\mathcal{R} \mathcal{M}}+\frac{\hbar^{2}}{2 \mu}-\frac{E_{\mathcal{P} \mathcal{T}}}{\cosh ^{2} x}\right)\left(x_{b} \mid x_{a}\right)_{E_{\mathcal{R} \mathcal{M}}, E_{\mathcal{P} \mathcal{T}}}=-i \hbar \delta\left(x_{b}-x_{a}\right) . $$

$$ \begin{align*} \left(x_{b} \mid x_{a}\right)_{E_{\mathcal{R} \mathcal{M}}, E_{\mathcal{P} \mathcal{T}}}=\frac{-i \mu}{\hbar} & \Gamma\left(m\left(E_{\mathcal{R} \mathcal{M}}\right)-s\right) \Gamma\left(s+m\left(E_{\mathcal{R} \mathcal{M}}\right)+1\right) \\ & \times P_{s}^{-m\left(E_{\mathcal{R} \mathcal{M}}\right)}\left(\tanh x_{b}\right) P_{s}^{-m\left(E_{\mathcal{R} \mathcal{M}}\right)}\left(-\tanh x_{a}\right) \end{align*} $$

$$ m\left(E_{\mathcal{R} \mathcal{M}}\right)=\sqrt{1-2 \mu E_{\mathcal{R} \mathcal{M}} / \hbar^{2}} $$

$$ G_{\mathcal{O}_{\omega}}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{\omega} \Gamma(m-2) \Gamma(m+3) P_{2}^{-m}\left(\tanh \frac{\omega \tau}{2}\right) P_{2}^{-m}\left(-\tanh \frac{\omega \tau^{\prime}}{2}\right), $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\left.\frac{1}{2 m} \frac{d}{d m}\left(m^{2}-4\right) G_{\mathcal{O}_{\omega}}\left(\tau, \tau^{\prime}\right)\right|_{m=2} $$

$$ P_{2}^{-m}(z)=\frac{1}{\Gamma(1+m)}\left(\frac{1+z}{1-z}\right)^{-m / 2}\left[1-\frac{3}{1+m}(1-z)+\frac{3}{(1+m)(2+m)}(1-z)^{2}\right] $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\hbar\left[Y_{0}\left(\tau_{>}\right) y_{0}\left(\tau_{<}\right)+y_{0}\left(-\tau_{>}\right) Y_{0}\left(-\tau_{<}\right)\right], $$

$$ y_{0}(\tau)=-2 \sqrt{6 \omega} P_{2}^{-2}\left(-\tanh \frac{\omega \tau}{2}\right)=-\sqrt{\frac{3 \omega}{8}} \frac{1}{\cosh ^{2} \frac{\omega \tau}{2}} $$

$$ \begin{align*} Y_{0}(\tau)=\frac{1}{2 \sqrt{6 \omega}} \frac{1}{2 \omega m}\{ & \frac{1}{2}\left[\frac{d}{d m}\left(m^{2}-4\right) \Gamma(m-2) \Gamma(m+3)\right] P_{2}^{-m}\left(\tanh \frac{\omega \tau}{2}\right) \\ & \left.+\left[\left(m^{2}-4\right) \Gamma(m-2) \Gamma(m+3)\right] \frac{d}{d m} P_{2}^{-m}\left(\tanh \frac{\omega \tau}{2}\right)\right\}\left.\right|_{m=2} \end{align*} $$

$$ \left.\frac{d}{d m} P_{2}^{-m}\left(\tanh \frac{\omega \tau}{2}\right)\right|_{m=2}=\frac{\sqrt{6}}{144} y_{0}(\tau)\left[6(3-2 \gamma+\omega \tau)-e^{-\omega \tau}\left(8+e^{-\omega \tau}\right)\right], $$

$$ Y_{0}(\tau)=\frac{1}{12 \omega^{2}} y_{0}(\tau)\left[e^{-\omega \tau}\left(e^{-\omega \tau}+8\right)-2(2+3 \omega \tau)\right] $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}(\tau, \tau)=\frac{\hbar}{2 \omega} \frac{1}{\cosh ^{4} \frac{\omega \tau}{2}}\left(\cosh ^{4} \frac{\omega \tau}{2}+\cosh ^{2} \frac{\omega \tau}{2}-\frac{11}{8}\right) $$

$$ \mathcal{O}_{\omega}^{\prime} G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\hbar\left[\delta\left(\tau-\tau^{\prime}\right)-y_{0}(\tau) y_{0}\left(\tau^{\prime}\right)\right] $$

$$ \sum_{n \neq 0} y_{n}(\tau) y_{n}\left(\tau^{\prime}\right)=\delta\left(\tau-\tau^{\prime}\right)-y_{0}(\tau) y_{0}\left(\tau^{\prime}\right) $$

$$ I_{1}^{\prime}=\frac{97}{560}, \quad I_{21}^{\prime}=\frac{53}{420}, \quad I_{22}^{\prime}=\frac{117}{560}, \quad I_{3}=\frac{49}{20} $$

$$ C^{\prime}=\left[1-\frac{71}{24} \frac{g \hbar}{\omega^{3}}+\mathcal{O}\left(g^{2}\right)\right] $$

$$ \Delta E^{(0)}=4 \sqrt{3} \sqrt{\frac{\omega^{3} / 3 g}{2 \pi \hbar}} \hbar \omega e^{-\omega^{3} / 3 g \hbar-71 g \hbar / 24 \omega^{3}+\ldots} . $$

$$ \Delta V=-\epsilon \frac{x-a}{2 a} $$

$$ V^{\prime}\left(x_{\mathrm{ex}}\right)=\frac{\omega^{2} a}{2}\left[\left(\frac{x_{\mathrm{ex}}}{a}\right)^{3}-\frac{x_{\mathrm{ex}}}{a}-\frac{\epsilon}{\omega^{2} a^{2}}\right]=0 . $$

$$ \psi(\mathbf{x}) e^{-i E t / \hbar}=\psi(\mathbf{x}) e^{-i E^{\mathrm{re}} t / \hbar} e^{E^{\mathrm{im}} t / \hbar}=\psi(\mathbf{x}) e^{-i E^{\mathrm{re}} t / \hbar} e^{-\Gamma t / 2 \hbar} $$

$$ \int d^{3} x|\psi(\mathbf{x})|^{2}=e^{-\Gamma t / \hbar} $$

$$ \left(x_{+} L / 2 \mid x_{+}-L / 2\right) $$

$$ \left(x_{+} L / 2 \mid x_{+}-L / 2\right) \sim \psi_{0}(0) \psi_{0}(0) e^{-E^{\mathrm{re}} L / \hbar} e^{i \Gamma L / 2 \hbar} $$

$$ \left(x_{+} L / 2 \mid x_{+}-L / 2\right)=\sqrt{\frac{\omega}{\pi \hbar}} e^{-\omega L / 2 \hbar} \exp \left[\sqrt{\mathcal{A}_{\mathrm{cl}} / 2 \pi \hbar} K^{\prime} L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right] . $$

$$ K^{\prime}=\left.\sqrt{\frac{\prod_{n}^{0} \lambda_{n}}{\prod_{n}^{\prime} \lambda_{n}}}\right|_{L} $$

$$ E^{(0)}=\left(\frac{\omega}{2}-\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} K^{\prime} e^{-\mathcal{A}_{\mathrm{cl}} / \hbar}\right) $$

$$ y_{0}(\tau)=\frac{1}{\sqrt{\mathcal{A}_{\mathrm{cl}}}} x_{\mathrm{cl}}^{\prime}(\tau) $$

$$ \left[-\frac{d^{2}}{d \tau^{2}}+V^{\prime \prime}\left(x_{\mathrm{cl}}(\tau)\right)\right] y_{n}(\tau)=\lambda_{n} y_{n}(\tau) $$

$$ \int \frac{d \xi_{1}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \xi_{1}^{2} \lambda_{-1}} $$

$$ \int \frac{d \xi_{-1}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \xi_{-1}^{2} \lambda_{-1}}=\frac{1}{\sqrt{\lambda_{-1}}} $$

$$ \int \frac{d \xi_{-1}}{\sqrt{2 \pi \hbar}} e^{-(1 / 2 \hbar) \xi_{-1}^{2} \lambda_{-1}}= \pm \frac{i}{\sqrt{\left|\lambda_{-1}\right|}} $$

$$ \frac{1}{\hbar} \Gamma=-2 i \sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}} K^{\prime} e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} \quad \text { (wrong) } $$

$$ K^{\prime}=i\left|K^{\prime}\right|=\left.\sqrt{\frac{\prod_{n}^{0} \lambda_{n}}{\prod_{n \neq 0,-1} \lambda_{n}}}\right|_{L} \frac{i}{\sqrt{\left|\lambda_{-1}\right|}} $$

$$ x(\tau) \equiv x_{+} $$

$$ x(\tau)=x_{\mathrm{cl}}(\tau) $$

$$ Z=\int_{0}^{\infty} \frac{d \xi}{\sqrt{2 \pi}} e^{\lambda\left(\xi^{2}+\alpha \xi^{3}\right)} $$

$$ \xi_{m}=-\frac{2}{3 \alpha} $$

$$ \mathcal{A}=-\lambda\left[\frac{4}{27 \alpha^{2}}-\left(\xi-\xi_{m}\right)^{2}+\ldots\right] . $$

$$ \operatorname{Im} Z \sim e^{\lambda 4 / 27 \alpha^{2}} \frac{1}{2} \frac{1}{\sqrt{|\lambda|}}, $$

$$ Z=\frac{1}{\alpha} \int_{0}^{\infty} \frac{d t}{\sqrt{2 \pi}} \exp \left[\frac{\lambda}{\alpha^{2}}\left(t^{2}+t^{3}\right)\right] $$

$$ t=e^{i 2 \varphi / 3} t^{\prime}, \quad t^{\prime} \in(0, \infty) $$

$$ \Delta Z \equiv Z\left(|\alpha| e^{-i \pi}\right)-Z\left(|\alpha| e^{i \pi}\right)=\frac{1}{|\alpha|} \int_{C_{4}} \frac{d t}{\sqrt{2 \pi}} \exp \left\{\frac{\lambda}{\alpha^{2}}\left(t^{2}+t^{3}\right)\right\} $$

$$ \Delta Z \equiv \operatorname{disc} Z=Z(-|\alpha|-i \eta)-Z(-|\alpha|+i \eta) $$

$$ \begin{align*} \operatorname{disc} Z & \approx e^{\lambda 4 / 27 \alpha^{2}} \int_{-i \infty}^{i \infty} \frac{d \xi}{\sqrt{2 \pi}} e^{\lambda\left(\xi-\xi_{0}\right)^{2}} \\ & =e^{\lambda 4 / 27 \alpha^{2}} \frac{i}{\sqrt{-\lambda}} \end{align*} $$

$$ \operatorname{Im} Z(-|\alpha| \mp i \eta)= \pm e^{\lambda 4 / 27 \alpha^{2}} \frac{1}{2 \sqrt{-\lambda}} $$

$$ \begin{align*} \int_{0}^{\infty} \frac{d \xi}{\sqrt{2 \pi \hbar}} e^{-\mathcal{A}(\xi) / \hbar} & =\int_{0}^{1} \frac{d \xi}{\sqrt{2 \pi \hbar}} e^{-\mathcal{A}(\xi) / \hbar}+e^{-\mathcal{A}(1) / \hbar} \int_{1}^{1-i \infty} \frac{d \xi}{\sqrt{2 \pi \hbar}} e^{-\mathcal{A}^{\prime \prime}(1)(\xi-1)^{2} / 2 \hbar} \\ & \approx \int_{0}^{1} \frac{d \xi}{\sqrt{2 \pi \hbar}} e^{-\mathcal{A}(\xi) / \hbar}+\frac{i}{2} e^{-\mathcal{A}(1) / \hbar} \frac{1}{\sqrt{-\mathcal{A}^{\prime \prime}(1)}} \end{align*} $$

$$ \int \frac{d \xi_{-1}}{\sqrt{2 \pi \hbar}} e^{-\xi_{-1}^{2} \lambda_{-1} / 2 \hbar} $$

$$ \int \frac{d \xi_{-1}}{\sqrt{2 \pi \hbar}} e^{-\xi_{-1}^{2} \lambda_{-1} / 2 \hbar}=\frac{i}{2} \frac{1}{\sqrt{\left|\lambda_{-1}\right|}} $$

$$ \operatorname{Im} Z(-|g|-i \eta) \approx \frac{1}{2} \sqrt{\mathcal{A}_{\mathrm{cl}} / 2 \pi \hbar}\left|K^{\prime}\right| L e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ \operatorname{Re} Z+\operatorname{Im} Z=\operatorname{Re} Z(1+\operatorname{Im} Z / \operatorname{Re} Z) \underset{\text { infinite sum }}{\longrightarrow} \operatorname{Re} Z e^{\operatorname{Im} Z / \operatorname{Re} Z} $$

$$ \frac{1}{\hbar} \Gamma=\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}}\left|K^{\prime}\right| e^{-\mathcal{A}_{\mathrm{cl}} / \hbar} $$

$$ \omega_{\mathrm{att}}=\sqrt{\frac{\mathcal{A}_{\mathrm{cl}}}{2 \pi \hbar}}\left|K^{\prime}\right| $$

$$ \mathcal{A}=\int_{-L / 2}^{L / 2} d \tau\left[\frac{x^{\prime 2}}{2}-\frac{\omega^{2}}{2} x^{2}-\frac{g}{4} x^{4}\right] $$

$$ Z(g)=\int \mathcal{D} x(\tau) e^{\mathcal{A}} $$

$$ Z(g)=\sum_{n} e^{-E^{(n)}(g) L} $$

$$ Z(g) \rightarrow e^{-E^{(0)}(g) L} $$

$$ Z(g)=\sum_{k=0} Z_{k}\left(\frac{g}{\omega^{3}}\right)^{k} $$

$$ \begin{align*} Z_{k} & =\frac{(-g)^{k}}{k!} \int \mathcal{D} x(\tau)\left[\int_{-L / 2}^{L / 2} d \tau x^{4}(\tau)\right]^{k} \exp \left[-\int_{-L / 2}^{L / 2} d \tau\left(\frac{1}{2} \dot{x}^{2}+\frac{\omega^{2}}{2} x^{2}\right)\right] \\ & =Z^{-1} \frac{(-g)^{k}}{k!}\left\langle\int_{-L / 2}^{L / 2} d \tau x^{4}(\tau)\right\rangle_{\omega} \end{align*} $$

$$ E^{(n)}(g)=\sum_{k=0}^{\infty} E_{k}^{(n)}\left(\frac{g}{4}\right)^{k} $$

$$ E_{k}=\gamma p^{\beta+1} k^{\beta}(-4 a)^{k}(p k)!\left[1+\frac{\gamma_{1}}{k}+\frac{\gamma_{2}}{k^{2}}+\ldots\right], $$

$$ (p k)!=(k!)^{p}\left(p^{p}\right)^{k} k^{(1-p) / 2} \frac{\sqrt{p}}{(2 \pi)^{(p-1) / 2}}[1+\mathcal{O}(1 / k)] $$

$$ k \approx k_{\min } \equiv \frac{1}{p(a|g|)^{1 / p}} $$

$$ p \log k+\log \left(p^{p} a|g|\right)+(\beta+p / 2) / k+\ldots=0 $$

$$ E_{k}=\gamma p(-4 a)^{k} \Gamma(p k+\beta+1)\left[1+\frac{c_{1}}{p k+\beta}+\frac{c_{2}}{(p k+\beta)(p k+\beta-1)}+\ldots\right] $$

$$ E_{1}(g)=\int_{g}^{\infty} \frac{d t}{t} e^{-t} $$

$$ E(g) \equiv \frac{1}{g} e^{1 / g} E_{1}(1 / g)=\int_{0}^{\infty} d t \frac{1}{1+g t} e^{-t} $$

$$ E(g)=1-g+2!g^{2}-3!g^{3}+\ldots+(-1)^{N} N!g^{N}+\ldots $$

$$ B(t)=\frac{1}{1+t} $$

$$ B(t)=1-t+t^{2}-t^{3}+\ldots, $$

$$ F(g)=\int_{0}^{\infty} \frac{d t}{g} e^{-t / g} B(t) $$

$$ \alpha=1 / 137.035963(15) \approx 0.0073 . $$

$$ a_{\mathrm{e}}=\frac{\Delta \mu}{\mu}=\frac{1}{2} \frac{\alpha}{\pi}-0.3284789657\left(\frac{\alpha}{\pi}\right)^{2}+1.1765(13)\left(\frac{\alpha}{\pi}\right)^{3}+\ldots, $$

$$ a_{e}^{\text {theor }}=(1159652478 \pm 140) \cdot 10^{-12} . $$

$$ E(g)=\frac{1}{2 \pi i} \int_{0}^{\infty} d g^{\prime} \frac{\operatorname{disc} E\left(-g^{\prime}\right)}{g^{\prime}+g} $$

$$ \operatorname{disc} E(g) \equiv E(g-i \eta)-E(g+i \eta) $$

$$ E_{k}=(-4)^{k} \int_{0}^{\infty} \frac{d g^{\prime}}{2 \pi i} \frac{1}{g^{\prime k+1}} \operatorname{disc} E\left(-g^{\prime}\right) $$

$$ \int_{0}^{\infty} d g \frac{1}{|g|^{\alpha+1}} e^{-1 /(a|g|)^{(1 / p)}}=a^{\alpha} p \Gamma(p \alpha) $$

$$ E(g)=E(0)+\frac{g}{2 \pi i} \int_{0}^{\infty} \frac{d g^{\prime}}{g^{\prime}} \frac{\operatorname{disc} E\left(-g^{\prime}\right)}{g^{\prime}+g} $$

$$ x^{\prime \prime}(\tau)-V^{\prime}(x(\tau))=0 $$

$$ \frac{1}{2} x^{2}-\frac{1}{2} \omega^{2} x^{2}-\frac{g}{4} x^{4}=E=\mathrm{const}, $$

$$ \tau-\tau_{0}= \pm \frac{1}{\omega} \int d x \frac{1}{x \sqrt{1-\left(|g| / 2 \omega^{2}\right) x^{2}}}=\mp \frac{1}{\omega} \operatorname{arcosh}\left(\sqrt{\frac{2 \omega^{2}}{|g|}} \frac{1}{x}\right) $$

$$ x(\tau)=x_{\mathrm{cl}}(\tau) \equiv \pm \sqrt{\frac{2 \omega^{2}}{|g|}} \frac{1}{\cosh \left[\omega\left(\tau-\tau_{0}\right)\right]} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{cl}} & =\int_{-L / 2}^{L / 2} d \tau\left[\frac{1}{2} x_{\mathrm{cl}}^{\prime 2}(\tau)+V\left(x_{\mathrm{cl}}(\tau)\right)\right]=2 \int_{0}^{L / 2} d \tau\left[x_{\mathrm{cl}}^{\prime 2}(\tau)-E\right] \\ & =2 \int_{0}^{x_{m}} d x \sqrt{2(E+V)}-E L \end{align*} $$

$$ \mathcal{A}_{\mathrm{cl}}=2 \int_{0}^{x_{m}} d x \sqrt{2 V}=\frac{4 \omega^{3}}{3|g|} $$

$$ \begin{align*} \mathcal{O}_{\omega}\left(\tau, \tau^{\prime}\right) & =\left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}+3 g x_{\mathrm{cl}}^{2}(\tau)\right]^{\prime} \delta\left(\tau-\tau^{\prime}\right) \\ & =\left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}\left(1-\frac{6}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right)\right]}\right)\right]^{\prime} \delta\left(\tau-\tau^{\prime}\right) \end{align*} $$

$$ \begin{align*} y_{0}(\tau) & =-\sqrt{\frac{3 \omega}{2}} \frac{\sinh \left[\omega\left(\tau-\tau_{0}\right)\right]}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right)\right]} \quad \text { with } \quad \lambda_{0}=0, \\ y_{-1}(\tau) & =\sqrt{\frac{3 \omega}{4}} \frac{1}{\cosh ^{2}\left[\omega\left(\tau-\tau_{0}\right)\right]} \quad \text { with } \quad \lambda_{-1}=-3 \omega^{2} . \end{align*} $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n} \lambda_{n}}=\frac{\Gamma(\sqrt{z}-s) \Gamma(\sqrt{z}+s+1)}{\Gamma(\sqrt{z}) \Gamma(\sqrt{z}+1)} $$

$$ \frac{\prod_{n} \lambda_{n}^{0}}{\prod_{n}^{\prime} \lambda_{n}}=\lim _{z \rightarrow 1}(\sqrt{z}-1)(\sqrt{z}+1) \omega^{2}\left[\frac{\Gamma(\sqrt{z}-2) \Gamma(\sqrt{z}+3)}{\Gamma(\sqrt{z}) \Gamma(\sqrt{z}+1)}\right]=-12 \omega^{2} . $$

$$ \operatorname{Im} Z(-|g|-i \eta) \approx \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3|g|} \omega L e^{-4 \omega^{3} / 3|g|} e^{-\omega L / 2}} $$

$$ \operatorname{Im} E^{(0)}(-|g|-i \eta)=-\omega \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3|g|}} e^{-4 \omega^{3} / 3|g|} $$

$$ a=3 / 4 \omega^{3}, \quad \beta=-\frac{1}{2}, \quad \gamma=-\frac{\omega}{\pi} \sqrt{\frac{6}{\pi}}, \quad p=1 $$

$$ E_{k}^{(0)}=-\frac{\omega}{\pi} \sqrt{\frac{6}{\pi}}\left(-3 / \omega^{3}\right)^{k} \Gamma(k+1 / 2) $$

$$ \mathcal{A}_{\mathrm{cl}}^{\prime} \approx \frac{4 \omega^{3}}{3|g|}\left(1-12 e^{-\omega L}\right) $$

$$ e^{-\mathcal{A}_{\mathrm{cl}}^{\prime}}=e^{-\mathcal{A}_{\mathrm{cl}}} \sum_{n=0}^{\infty} \mathcal{A}_{\mathrm{cl}}^{n} \frac{12^{n}}{n!} e^{-n \omega L} $$

$$ \operatorname{Im} E^{(n)}(-|g|-i \eta)=-\frac{12^{n}}{n!} \omega \sqrt{\frac{6}{\pi}}{\sqrt{\frac{4 \omega^{3}}{3|g|}}}^{1+2 n} \quad e^{-4 \omega^{3} / 3|g|} $$

$$ E_{k}^{(n)}=-\frac{\omega}{\pi} \sqrt{\frac{6}{\pi}} \frac{12^{n}}{n!}\left(-3 / \omega^{3}\right)^{k} \Gamma(k+n+1 / 2) $$

$$ \begin{align*} Z(g) & =\int_{-i \infty}^{i \infty} \frac{d \lambda}{2 \pi i} \int_{0}^{\infty} \frac{d a}{4} e^{-(g a+\lambda a) / 4} \\ & \times \int \mathcal{D} x(\tau) \exp \left\{-\int_{L / 2}^{L / 2} d \tau\left[\frac{1}{2} x^{\prime 2}+\frac{\omega^{2}}{2} x^{2}-\frac{\lambda}{4} x^{4}\right]\right\} \end{align*} $$

$$ Z(g)=\int_{-i \infty}^{i \infty} \frac{d \lambda}{2 \pi i} \frac{1}{\lambda+g} Z(-\lambda) $$

$$ \operatorname{Im} Z(-|\lambda|-i \eta)=\sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} e^{-\omega L / 2} $$

$$ Z(g)=2 \omega \int_{0}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{\lambda+g} \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} e^{-\omega L / 2} $$

$$ E^{(0)}(g)=-2 \omega \int_{0}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{\lambda+g} \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} $$

$$ E^{(0)}(g)=\frac{\omega}{2}+2 \omega g \int_{0}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{\lambda(\lambda+g)} \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} $$

$$ \frac{1}{\lambda+g}=\sum_{k=0}^{\infty}(-1)^{k} g^{k} \lambda^{-k-1} $$

$$ E_{k}^{(0)}=-2 \omega(-4)^{k} \int_{0}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{\lambda^{k+1}} \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} $$

$$ \mathcal{O}_{\omega}\left(\tau, \tau^{\prime}\right)=\left[-\frac{d^{2}}{d \tau^{2}}+\omega^{2}\left(1-\frac{6}{\cosh ^{2} \omega\left(\tau-\tau_{0}\right)}\right)\right]^{\prime} \delta\left(\tau-\tau^{\prime}\right) $$

$$ \mathcal{A}_{\mathrm{e}}^{\mathrm{eff}}=-\hbar \log \left[1-\sqrt{\frac{3|g|}{4 \omega^{3}}} \int d \tau y_{0}^{\prime}(\tau) y(\tau)\right] $$

$$ C=\left[1+\left(I_{1}+I_{2}+I_{3}\right) \frac{|g| \hbar}{\omega^{3}}+\mathcal{O}\left(g^{2}\right)\right], $$

$$ \text { × n . } \sqrt{\frac{3|g|}{4 \omega^{3}}} y_{0}^{\prime}(\tau) $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\left\langle y(\tau) y\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\omega}}=\hbar \mathcal{O}_{\omega}^{-1}\left(\tau, \tau^{\prime}\right) $$

$$ I_{1}=\frac{3 \omega^{3}}{4 \hbar^{2}} \int d \tau G_{\mathcal{O}_{\omega}}^{\prime 2}(\tau, \tau)=L \frac{3 \omega}{16}+\frac{3 \omega^{3}}{4 \hbar^{2}} \int d \tau\left[G_{\mathcal{O}_{\omega}}^{\prime 2}(\tau, \tau)-\frac{\hbar^{2}}{4 \omega^{2}}\right] $$

$$ C^{\prime}=\left[1+\left(I_{1}^{\prime}+I_{2}+I_{3}\right) \frac{|g| \hbar}{\omega^{3}}+\mathcal{O}\left(g^{2}\right)\right] $$

$$ G_{\mathcal{O}_{\omega}}\left(\tau, \tau^{\prime}\right)=\frac{\hbar}{2 \omega} \Gamma(m-2) \Gamma(m+3) \times P_{2}^{-m}(\tanh \omega \tau) P_{2}^{-m}\left(-\tanh \omega \tau^{\prime}\right) $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\left.\frac{1}{2 m} \frac{d}{d m}\left(m^{2}-1\right) G_{\mathcal{O}_{\omega}}\left(\tau, \tau^{\prime}\right)\right|_{m=1} $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}\left(\tau, \tau^{\prime}\right)=\hbar\left[Y_{0}\left(\tau_{>}\right) y_{0}\left(\tau_{<}\right)+y_{0}\left(-\tau_{>}\right) Y_{0}\left(-\tau_{<}\right)\right] $$

$$ \begin{gather*} y_{0}(\tau)=2 \sqrt{\frac{3 \omega}{2}} P_{2}^{-1}(-\tanh \omega \tau)=-\sqrt{\frac{3 \omega}{2}} \frac{\sinh \omega \tau}{\cosh ^{2} \omega \tau} \\ Y_{0}(\tau)=\sqrt{\frac{2}{3 \omega}} \frac{1}{8 \omega m}\left\{\frac{1}{2}\left[\frac{d}{d m}\left(m^{2}-1\right) \Gamma(m-2) \Gamma(m+3)\right] P_{2}^{-m}(\tanh \omega \tau)\right. \\ \left.+\left[\left(m^{2}-1\right) \Gamma(m-2) \Gamma(m+3)\right] \frac{d}{d m} P_{2}^{-m}(\tanh \omega \tau)\right\}\left.\right|_{m=1} \\ =-\sqrt{\frac{2}{3 \omega^{3}}}\left[\frac{3}{4} \frac{1}{\cosh \omega \tau}+\left(-\frac{3}{4} \omega \tau-\frac{1}{8}\right) \frac{\sinh \omega \tau}{\cosh ^{2} \omega \tau}-\frac{1}{4} e^{-\omega \tau}\right] \end{gather*} $$

$$ G_{\mathcal{O}_{\omega}}^{\prime}(\tau, \tau)=\frac{\hbar}{2 \omega} \frac{1}{\cosh ^{2} \omega \tau}\left(\cosh ^{2} \omega \tau-1\right)\left(\cosh ^{2} \omega \tau-1 / 2\right) $$

$$ I_{1}^{\prime}=-\frac{11 \cdot 29}{2^{4} \cdot 5 \cdot 7}, \quad I_{21}=-\frac{71}{2^{5} \cdot 3 \cdot 7}, \quad I_{22}=\frac{3 \cdot 13}{2^{4} \cdot 7}, \quad I_{3}=-\frac{53}{2^{4} \cdot 5} $$

$$ C^{\prime}=\left[1-\frac{95}{72} \frac{3|g| \hbar}{4 \omega^{3}}+\mathcal{O}\left(g^{2}\right)\right] $$

$$ E_{k}^{(0)}=-\frac{\omega}{\pi} \sqrt{\frac{6}{\pi}}\left(-3 / \omega^{3}\right)^{k} \Gamma(k+1 / 2)[1-95 / 72 k+\ldots] . $$

$$ Z(g)=\int \mathcal{D} x(\tau) \exp \left\{-\int_{-L / 2}^{L / 2} d \tau\left[\frac{1}{2} x^{\prime 2}+\frac{\omega^{2}}{2} x^{2}+\frac{g}{4} x^{4}\right]\right\} $$

$$ W_{1}=\frac{\Omega}{2}+\frac{\omega^{2}-\Omega^{2}}{2} a^{2}+\frac{3 g}{4} a^{4}, $$

$$ \Omega=\frac{2 \omega}{\sqrt{3}} \cos \left[\frac{\pi}{3}-\frac{1}{3} \arccos \left(-g / g^{(0)}\right)\right] $$

$$ \Omega^{\mathrm{re}}=\frac{\omega}{\sqrt{3}} \cosh (\gamma / 3), \quad \Omega^{\mathrm{im}}=\omega \sinh (\gamma / 3) ; \quad \gamma=\operatorname{arcosh}\left(-g / g^{(0)}\right) $$

$$ \operatorname{Im} W_{1}=\frac{1}{4} \Omega^{\mathrm{i}}\left(1-1 /|\Omega|^{2}\right)-\frac{3 g}{4} \Omega^{\mathrm{re}} \Omega^{\mathrm{im}} / 2|\Omega|^{4} $$

$$ x(\tau)=x_{\mathrm{cl}}(\tau) \equiv \pm \sqrt{2 \Omega^{2} /|g|} \frac{1}{\cosh \left[\Omega\left(\tau-\tau_{0}\right)\right]} $$

$$ \operatorname{Im} Z(-|g|-i \eta)=\beta \Omega \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \Omega^{3}}{3|g|}} e^{-\beta \Omega / 2-4 \Omega^{3} / 3|g|} $$

$$ \begin{align*} \mathcal{A}_{\mathrm{fl}, \mathrm{var}}^{\mathrm{int}} & =\int_{-\infty}^{\infty} d \tau \frac{\omega^{2}-\Omega^{2}}{2} x^{2}(\tau) \\ & =\int_{-\infty}^{\infty} d \tau \frac{\omega^{2}-\Omega^{2}}{2}\left[x_{\mathrm{cl}}^{2}(\tau)+2 x_{\mathrm{cl}}(\tau) y(\tau)+y^{2}(\tau)\right] \end{align*} $$

$$ \int_{-\infty}^{\infty} d \tau x_{\mathrm{cl}}^{2}(\tau)=4 \Omega /|g| $$

$$ \begin{align*} \int_{-\infty}^{\infty} d \tau\left\langle y^{2}(\tau)\right\rangle_{\mathcal{O}_{\Omega}} & =L \frac{1}{2 \Omega}+\frac{1}{\Omega} \int_{-\infty}^{\infty} d \tau\left[G_{\mathcal{O}_{\Omega}}^{\prime}(\tau, \tau)-1 / 2\right] \\ & =L \frac{1}{2 \Omega}-\frac{7}{6 \Omega^{2}} \end{align*} $$

$$ A_{0}=\frac{95}{96} \frac{|g|}{\Omega^{3}}, \quad A_{1}=\frac{1}{2}\left(\omega^{2}-\Omega^{2}\right)\left(\frac{4 \Omega}{|g|}-\frac{7}{6 \Omega^{2}}\right) $$

$$ \exp \left\{\frac{1}{2}\left[\left\langle\mathcal{A}_{\mathrm{fl}, \text { tot }}^{\text {int } 2}\right\rangle_{\mathcal{O}_{\Omega}}-\left\langle\mathcal{A}_{\mathrm{fl}, \text { tot }}^{\text {int }}\right\rangle_{\mathcal{O}_{\Omega}}^{2}\right]\right\}=\exp \left(-A_{2}-A_{3}-A_{4}\right), $$

$$ \begin{align*} & A_{2}=-\frac{1}{2}\left(\omega^{2}-\Omega^{2}\right)^{2} \int d \tau d \tau^{\prime} x_{\mathrm{cl}}(\tau)\left\langle y(\tau) y\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\Omega}} x_{\mathrm{cl}}\left(\tau^{\prime}\right) \\ & A_{3}=-\left(\omega^{2}-\Omega^{2}\right) \sqrt{\frac{3|g|}{4 \Omega^{3}}} \int d \tau d \tau^{\prime} y_{0}^{\prime}(\tau)\left\langle y(\tau) y\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\Omega}} x_{\mathrm{cl}}\left(\tau^{\prime}\right) \\ & A_{4}=\left(\omega^{2}-\Omega^{2}\right)|g| \int d \tau d \tau^{\prime} x_{\mathrm{cl}}(\tau)\left\langle y(\tau) y^{3}\left(\tau^{\prime}\right)\right\rangle_{\mathcal{O}_{\Omega}} x_{\mathrm{cl}}\left(\tau^{\prime}\right) \end{align*} $$

$$ \begin{align*} & A_{2}=-\frac{1}{2}\left(\omega^{2}-\Omega^{2}\right)^{2} \frac{1}{\Omega|g|} a_{2} \\ & A_{3}=-\left(\omega^{2}-\Omega^{2}\right) \frac{1}{\Omega^{2}} a_{3} \\ & A_{4}=\left(\omega^{2}-\Omega^{2}\right) \frac{1}{\Omega^{2}} a_{4} \end{align*} $$

$$ \begin{align*} & a_{2}=|g| \Omega \int d \tau d \tau^{\prime} x_{\mathrm{cl}}(\tau) G_{\mathcal{O}_{\Omega}}^{\prime}\left(\tau, \tau^{\prime}\right) x_{\mathrm{cl}}\left(\tau^{\prime}\right) \\ & a_{3}=\Omega^{2} \sqrt{\frac{3|g|}{4 \Omega^{3}}} \int d \tau d \tau^{\prime} y_{0}^{\prime}(\tau) G_{\mathcal{O}_{\Omega}}^{\prime}\left(\tau, \tau^{\prime}\right) x_{\mathrm{cl}}\left(\tau^{\prime}\right) \\ & a_{4}=3|g| \Omega^{2} \int d \tau d \tau^{\prime} x_{\mathrm{cl}}(\tau) G_{\mathcal{O}_{\Omega}}^{\prime}\left(\tau, \tau^{\prime}\right) G_{\mathcal{O}_{\Omega}}^{\prime}\left(\tau^{\prime}, \tau^{\prime}\right) x_{\mathrm{cl}}\left(\tau^{\prime}\right) \end{align*} $$

$$ \begin{align*} & \operatorname{Im} E(-|g|-i \eta)=-\Omega \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \Omega^{3}}{3|g|}} e^{-4 \Omega^{3} / 3|g|-c_{1} 3|g| / 4 \Omega^{3}} \\ & \quad \times \exp \left[-\frac{\omega^{2}-\Omega^{2}}{2}\left(\frac{4 \Omega}{|g|}-\frac{7}{6 \Omega^{2}}-2 \frac{a_{3}-a_{4}}{\Omega^{2}}\right)+\frac{\left(\omega^{2}-\Omega^{2}\right)^{2}}{2 \Omega|g|} a_{2}\right] \end{align*} $$

$$ \operatorname{Im} E(-|g|-i \eta)=-\omega \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3|g|}} e^{-4 \omega^{3} / 3|g|} \varepsilon_{\mathrm{i}}(g), $$

$$ \begin{align*} \varepsilon_{\mathrm{i}}(g) & =\left(\frac{\Omega}{\omega}\right)^{5 / 2} \exp \left[-4 \frac{\Omega^{3}-\omega^{3}}{3|g|}-c_{1} \frac{3|g|}{4 \Omega^{3}}\right. \\ & \left.-\frac{\omega^{2}-\Omega^{2}}{2}\left(\frac{4 \Omega}{|g|}-\frac{7}{6 \Omega^{2}}-2 \frac{a_{3}-a_{4}}{\Omega^{2}}\right)+\frac{\left(\omega^{2}-\Omega^{2}\right)^{2}}{2 \Omega|g|} a_{2}\right] . \end{align*} $$

$$ \operatorname{Im} E(-|g|-i \eta)=-\omega \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3|g|}} e^{-4 \omega^{3} / 3|g|-3 c_{1}|g| / 4 \omega^{3}} $$

$$ E^{(0)}(g)=\frac{\omega}{2}+2 \omega g \int_{0}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{\lambda(\lambda+g)} \sqrt{\frac{6}{\pi}} \sqrt{\frac{4 \omega^{3}}{3 \lambda}} e^{-4 \omega^{3} / 3 \lambda} \varepsilon_{\mathrm{i}}(\lambda) $$

$$ W_{1}^{(n)}=\Omega n_{2}+\frac{\omega^{2}-\Omega^{2}}{2} \frac{n_{2}}{\Omega}+\frac{g}{4} \frac{n_{4}}{\Omega^{2}}, $$

$$ \operatorname{Im} W_{1}^{(n)}=\frac{1}{2} \Omega^{\mathrm{i}}\left(1-\frac{\omega^{2}}{|\Omega|^{2}}\right) n_{2}-\frac{g}{2} \Omega^{\mathrm{re}} \Omega^{\mathrm{im}} \frac{n_{4}}{|\Omega|^{4}} . $$

$$ \operatorname{Im} E^{(n)}(-|g|-i \eta)=-\frac{12^{n}}{n!} \omega \sqrt{\frac{6}{\pi}}{\sqrt{\frac{4 \omega^{3}}{3|g|}}}^{1+2 n} e^{-4 \omega^{3} / 3|g|-c_{1}^{(n)} 3|g| / 4 \omega^{3}} $$

$$ c_{1}^{(n)}=\frac{d \varepsilon_{\mathrm{i}}^{(n)}}{d\left(g / \omega^{3}\right)}=\left(\frac{95}{96}+\frac{29}{16} n+\frac{17}{16} n^{2}\right) \omega^{-3} . $$

$$ \operatorname{Im} E^{(n)}(-|g|-i \eta)=-\frac{12^{n}}{n!} \omega \sqrt{\frac{6}{\pi}}{\sqrt{\frac{4 \omega^{3}}{3|g|}}}^{1+2 n} e^{-4 \omega^{3} / 3|g|} \varepsilon_{\mathrm{i}}^{(n)}(g), $$

$$ \begin{align*} \varepsilon_{\mathrm{i}}^{(n)}(g)=\left(\frac{\Omega}{\omega}\right)^{3 n+5 / 2} \exp [ & -4 \frac{\Omega^{3}-\omega^{3}}{3|g|}-c_{1}^{(n)} \frac{3|g|}{4 \Omega^{3}} \\ & \left.-\frac{\omega^{2}-\Omega^{2}}{2}\left(\frac{4 \Omega}{|g|}-\frac{3 n+5 / 2}{\Omega^{2}}\right)-\frac{\left(\omega^{2}-\Omega^{2}\right)^{2}}{2 \Omega|g|}\right] . \end{align*} $$

$$ \log \varepsilon^{\mathrm{i}}(g)=-\frac{4}{3 g}+k_{1} \frac{g}{4}+k_{2}\left(\frac{g}{4}\right)^{2}+\ldots . $$

$$ E^{(0)}(g)=\frac{\omega}{2}-\frac{g}{2 \pi i} \int_{0}^{-\infty} \frac{d g^{\prime}}{g^{\prime}} \frac{\operatorname{disc} E^{(0)}\left(g^{\prime}\right)}{g^{\prime}-g} $$

$$ E^{(0)}(g)=\omega \sum_{k=0}^{N} E_{k}^{(0)}\left(\frac{g}{4 \omega^{3}}\right)^{k} $$

$$ W_{N}^{\Omega}(g)=\Omega \sum_{k=0}^{N} \varepsilon_{k}^{(0)}\left(\frac{g}{4 \Omega^{3}}\right)^{k} . $$

$$ \omega \longrightarrow \Omega(1-\sigma \hat{g})^{1 / 2} $$

$$ \sigma=\Omega\left(\Omega^{2}-1\right) / g $$

$$ \bar{g} \longrightarrow \tilde{g}(\hat{g}) \equiv \frac{\hat{g}}{(1-\sigma \hat{g})^{3 / 2}} $$

$$ \hat{E}(\hat{g}) \equiv E(g) / \Omega $$

$$ \hat{E}^{(0)}(g)=(1-\sigma \hat{g})^{1 / 2}\left[\frac{1}{2}+\frac{\tilde{g}(\hat{g})}{2 \pi i} \int_{0}^{-\infty} \frac{d \bar{g}^{\prime}}{\bar{g}^{\prime}} \frac{\operatorname{disc} \bar{E}^{(0)}\left(\bar{g}^{\prime}\right)}{\bar{g}^{\prime}-\tilde{g}(\hat{g})}\right] $$

$$ \hat{E}^{(0)}(\hat{g})=\frac{1}{2}+\frac{\hat{g}}{2 \pi i} \int_{C} \frac{d \hat{g}^{\prime}}{\hat{g}^{\prime}} \frac{\operatorname{disc}_{C} \hat{E}^{(0)}\left(\hat{g}^{\prime}\right)}{\hat{g}^{\prime}-\hat{g}} $$

$$ \varepsilon_{k}^{(0)}=-\frac{4^{k}}{2 \pi i} \int_{C} \frac{d \hat{g}}{\hat{g}^{k+1}} \operatorname{disc}_{C} \hat{E}^{(0)}(\hat{g}) $$

$$ \bar{D}(\bar{g}) \equiv \operatorname{disc} \bar{E}^{(0)}(\bar{g})=2 i \operatorname{Im} \bar{E}^{(0)}(\bar{g}-i \eta), \quad \bar{g} \leq 0 . $$

$$ \operatorname{disc}_{C_{1, \overline{1}, 2, \overline{2}}} \hat{E}^{(0)}(\hat{g})=(1-\sigma \hat{g})^{1 / 2} \bar{D}\left(\hat{g}(1-\sigma \hat{g})^{-3 / 2}\right), $$

$$ \begin{align*} \operatorname{disc}_{C_{3}} \hat{E}^{(0)}(\hat{g})= & -2 i(\sigma \hat{g}-1)^{1 / 2} \\ & \times\left[\frac{1}{2}-\int_{0}^{\infty} \frac{d \bar{g}^{\prime}}{2 \pi} \frac{\hat{g}(\sigma \hat{g}-1)^{-3 / 2}}{\bar{g}^{\prime 2}+\hat{g}^{2}(\sigma \hat{g}-1)^{-3}} \bar{D}\left(-\bar{g}^{\prime}\right)\right] \end{align*} $$

$$ \bar{D} \bar{g}) \approx-2 i \sqrt{\frac{6}{\pi}} \sqrt{\frac{4}{-3 \bar{g}}} e^{4 / 3 \bar{g}} $$

$$ \varepsilon_{k}^{(0)}\left(C_{1}\right) \approx-24^{k} \int_{C_{1}} \frac{d \hat{g}}{2 \pi} \frac{1}{\hat{g}^{k+1}} \sqrt{\frac{6}{\pi}} \sqrt{-\frac{4(1-\sigma \hat{g})^{5 / 2}}{3 \hat{g}}} e^{4(1-\sigma \hat{g})^{3 / 2} / 3 \hat{g}} $$

$$ S_{k} \propto\left[\int_{C_{\gamma}} \frac{d \gamma}{2 \pi} e^{f_{k}(\gamma)}\right](\sigma \hat{g})^{k} $$

$$ f_{k}(\gamma)=-\left(k+\frac{3}{2}\right) \log (-\gamma)+\frac{4 \sigma}{3 \gamma}(1-\gamma)^{3 / 2} $$

$$ -k+\frac{3}{2}=\frac{4 \sigma}{3 \gamma}(1-\gamma)^{1 / 2}\left(1+\frac{1}{2} \gamma\right) $$

$$ \gamma \underset{k \rightarrow \infty}{\longrightarrow} \gamma_{k}=-4 \sigma / 3 k $$

$$ f_{k} \underset{k \rightarrow \infty}{\longrightarrow} k \log (3 k / 4 e \sigma)-2 \sigma $$

$$ S_{k} \propto e^{-2 \sigma}\left(\frac{-3 k}{e}\right)^{k}\left(\frac{\hat{g}}{4}\right)^{k} $$

$$ \varepsilon_{k}^{(0)} \propto e^{-2 \sigma} E_{k}^{(0)} $$

$$ \sigma=\frac{\Omega\left(\Omega^{2}-1\right)}{g} $$

$$ \sigma \hat{g}=1-\frac{1}{\Omega^{2}} $$

$$ \sigma \approx \sigma_{N} \equiv c N $$

$$ f_{N}(\gamma) \approx N\left[-\log (-\gamma)+\frac{4 c}{3 \gamma}(1-\gamma)^{3 / 2}\right] $$

$$ 1+\frac{4 c}{3 \gamma}(1-\gamma)^{1 / 2}\left(1+\frac{1}{2} \gamma\right)=0 $$

$$ f_{N}(\gamma)=0 $$

$$ \gamma=-0.242964029973520 \ldots, \quad c=0.186047272987975 \ldots . $$

$$ S_{N} \propto\left(\sigma_{N} \hat{g}_{N}\right)^{N}=\left(1-\frac{1}{\Omega_{N}^{2}}\right)^{N} . $$

$$ \Omega_{N} \sim \sigma_{N}^{1 / 3} g^{1 / 3} \sim(c N g)^{1 / 3} $$

$$ S_{N}\left(C_{1}\right) \propto\left[1-\frac{1}{\left(\sigma_{N} g\right)^{2 / 3}}\right]^{N} \approx e^{-N /(\sigma g)^{2 / 3}} \approx e^{-N^{1 / 3} /(c g)^{2 / 3}} $$

$$ \sigma_{N} \sim c N\left(1+\frac{6.85}{N^{2 / 3}}\right) $$

$$ \begin{align*} e^{\Delta f_{N}} & \approx \exp \left[N \frac{4 c}{3} \frac{(1-\gamma)^{3 / 2}}{\gamma} \frac{6.85}{N^{2 / 3}}\right] \\ & =\exp \left[-N \log (-\gamma) \frac{6.85}{N^{2 / 3}}\right] \approx e^{-9.7 N^{1 / 3}} . \end{align*} $$

$$ S_{N}\left(C_{1}\right) \propto e^{-\left[9.7+(c g)^{-2 / 3}\right] N^{1 / 3}} $$

$$ \varepsilon_{k}^{(0)}\left(C_{2, \overline{2}, 3}\right) \sim \sigma^{k} $$

$$ S_{N}\left(C_{2, \overline{2}, 3}\right) \sim(\sigma \hat{g})^{N} $$

$$ \hat{E}^{(0)}(\hat{g})=\left(\frac{\hat{g}}{4}\right)^{1 / 3}\left\{\alpha_{0}+\alpha_{1}\left[\frac{\hat{g}}{4 \omega^{3}} \frac{1}{(1-\sigma \hat{g})^{3 / 2}}\right]^{-2 / 3}+\alpha_{2}\left[\frac{\hat{g}}{4 \omega^{3}} \frac{1}{(1-\sigma \hat{g})^{3 / 2}}\right]^{-4 / 3}+\ldots\right\} $$

$$ \Delta(\sigma \hat{g}) \sim\left(\frac{\hat{g}}{-\bar{g}_{\mathrm{s}}}\right)^{2 / 3}=\left\{\frac{1}{-\sigma \bar{g}_{\mathrm{s}}}[1+\Delta(\sigma \hat{g})]\right\}^{2 / 3} $$

$$ (\sigma \hat{g})^{-N} \sim[1+\Delta(\sigma \hat{g})]^{-N} $$

$$ S_{N}\left(C_{2, \overline{2}, 3}\right) \approx e^{-N^{1 / 3} a \cos \theta} \cos \left(N^{1 / 3} a \sin \theta\right) $$

$$ \left|\bar{g}_{\mathrm{s}}\right| \sim 0.160, \quad \theta \sim-0.467, $$

$$ \left|x_{\mathrm{s}}\right|=1 / 0.117, \quad \theta=-0.467, $$

$$ \psi(z)=\psi(z+L) $$

$$ \varepsilon(z)=\left|\partial_{z} \psi(z)\right|^{2}+m^{2}|\psi(z)|^{2}+\frac{g}{4}|\psi(z)|^{4} $$

$$ E\left[\psi^{*}, \psi\right]=\int_{-L / 2}^{L / 2} d z \varepsilon(z) $$

$$ m^{2} \approx m_{0}^{2}\left(\frac{T}{T_{c}}-1\right) $$

$$ Z=\int \mathcal{D} \psi^{*}(z) \mathcal{D} \psi(z) e^{-E\left[\psi^{*}, \psi\right] / k_{B} T} $$

$$ T_{c} k_{B}=\mu e^{-1 / g} $$

$$ M_{c}=\mu e^{-1 / g(\mu)} $$

$$ \xi(T)=\frac{\text { const. }}{T_{c}}\left(1-\frac{T}{T_{c}}\right)^{-1 / 2} \approx 1000 \AA\left(1-\frac{T}{T_{c}}\right)^{-1 / 2} $$

$$ \psi_{\mathrm{pair}}(\mathbf{x})=\psi_{\mathrm{e}}(\mathbf{x}) \psi_{\mathrm{e}}(\mathbf{x}) $$

$$ Z=\int \mathcal{D} \psi_{\mathrm{e}}^{*}(\mathbf{x}) \mathcal{D} \psi_{\mathrm{e}}(\mathbf{x}) e^{-\mathcal{A}\left[\psi_{\mathrm{e}}^{*}, \psi_{\mathrm{e}}\right]} $$

$$ Z=\int \mathcal{D} \psi_{\mathrm{pair}}^{*}(\mathbf{x}) \mathcal{D} \psi_{\mathrm{pair}}(\mathbf{x}) e^{-\mathcal{A}\left[\psi_{\mathrm{pair}}^{*}, \psi_{\mathrm{pair}}\right]} $$

$$ \begin{align*} & \mathcal{A}\left[\psi_{\text {pair }}^{*}, \psi_{\text {pair }}\right]=E / k_{B} T=\frac{1}{k_{B} T} \int d^{3} x \varepsilon(\mathbf{x}) \\ & \quad=\frac{1}{k_{B} T} \int d^{3} x\left[\left(-\log \frac{\mu}{T}+\frac{1}{g^{2}}\right)\left|\psi_{\text {pair }}\right|^{2}+\frac{1}{2 T_{c}^{2}}\left|\psi_{\text {pair }}\right|^{4}+\frac{1}{T_{c}^{2}}\left|\nabla \psi_{\text {pair }}\right|^{2}+\ldots\right] \end{align*} $$

$$ -\log \frac{T_{c}}{T}\left|\psi_{\mathrm{pair}}\right|^{2} \sim-\left(1-\frac{T}{T_{c}}\right)\left|\psi_{\mathrm{pair}}\right|^{2} $$

$$ \left|\psi_{\mathrm{pair}, 0}\right|=T_{c} \sqrt{1-\frac{T}{T_{c}}} $$

$$ \psi(\mathbf{x}) \equiv \psi_{\mathrm{pair}}(\mathbf{x}) \frac{1}{T_{c}\left(1-T / T_{c}\right)^{1 / 2}} $$

$$ \varepsilon(\mathbf{x})=|\boldsymbol{\nabla} \psi|^{2}-|\psi|^{2}+\frac{1}{2}|\psi|^{4} $$

$$ \varepsilon=\varepsilon_{c}=-1 / 2 $$

$$ Z=\int \mathcal{D} \psi^{*}(\mathbf{x}) \mathcal{D} \psi(\mathbf{x}) e^{-(1 / T) \int d^{3} x \varepsilon(\mathbf{x})} $$

$$ \mathbf{j}(\mathbf{x})=\frac{1}{2 i} \psi^{*}(\mathbf{x}) \stackrel{\leftrightarrow}{\nabla} \psi(\mathbf{x}) $$

$$ \psi(z)=\rho(z) e^{i \gamma(z)} $$

$$ \varepsilon(z)=-\rho^{2}+\frac{1}{2} \rho^{4}+\rho_{z}^{2}+\rho^{2} \gamma_{z}^{2} $$

$$ j(z)=\rho^{2}(z) \gamma_{z}(z)=\mathrm{const} $$

$$ \rho_{z z}=-\rho+\rho^{3}+\frac{j^{2}}{\rho^{3}} $$

$$ -V(\rho) \equiv \rho^{2}-\frac{1}{2} \rho^{4}+\frac{j^{2}}{\rho^{2}}, $$

$$ \gamma(z)=k z, \quad \rho(z) \equiv \rho_{0}=\sqrt{1-k^{2}} . $$

$$ k_{n}=\frac{2 \pi}{L} n, \quad n=0, \pm 1, \pm 2, \ldots $$

$$ j=\rho_{0}^{2} k=\left(1-k^{2}\right) k . $$

$$ |j|