Kleinert · 제5장 쿨롱계

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

$$ \begin{align*} a^{2}= & \frac{\hbar}{\tilde{\omega}} \frac{1}{x}\left(\frac{x}{2} \operatorname{coth} \frac{x}{2}-1\right) \\ a_{2}^{4}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{2} \frac{1}{8 x} \frac{1}{\sinh ^{2} \frac{x}{2}}\left(4+x^{2}-4 \cosh x+x \sinh x\right) \\ a_{3}^{6}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{3} \frac{1}{64 x} \frac{1}{\sinh ^{3} \frac{x}{2}}\left(-3 x \cosh \frac{x}{2}+2 x^{3} \cosh \frac{x}{2}+3 x \cosh \frac{3 x}{2}\right. \\ & \left.+48 \sinh \frac{x}{2}+6 x^{2} \sinh \frac{x}{2}-16 \sinh \frac{3 x}{2}\right) \\ a_{2}^{8}= & \left(\frac{\hbar}{\omega}\right)^{4} \frac{1}{768 x^{3}} \frac{1}{\sinh ^{4} \frac{x}{2}}\left(-864+18 x^{4}+1152 \cosh x+32 x^{2} \cosh x\right. \\ & -288 \cosh 2 x-32 x^{2} \cosh 2 x-288 x \sinh x+24 x^{3} \sinh x \\ & \left.+144 x \sinh 2 x+3 x^{3} \sinh 2 x\right), \\ a_{3}^{10}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{5} \frac{1}{4096 x^{3}} \frac{1}{\sinh ^{5} \frac{x}{2}}\left(672 x \cosh \frac{x}{2}-8 x^{3} \cosh \frac{x}{2}+24 x^{5} \cosh \frac{x}{2}\right. \\ & -1008 x \cosh \frac{3 x}{2}+3 x^{3} \cosh \frac{3 x}{2}+336 x \cosh \frac{5 x}{2}+5 x^{3} \cosh \frac{5 x}{2} \\ & -7680 \sinh \frac{x}{2}-352 x^{2} \sinh \frac{x}{2}+72 x^{4} \sinh \frac{x}{2}+3840 \sinh \frac{3 x}{2} \\ & \left.+224 x^{2} \sinh \frac{3 x}{2}+12 x^{4} \sinh \frac{3 x}{2}-768 \sinh \frac{5 x}{2}-64 x^{2} \sinh \frac{5 x}{2}\right), \end{align*} $$

$$ \frac{d}{d \Omega} W_{N}^{\Omega}=\left(\frac{g}{\Omega^{3}}\right)^{N} p_{N}(\sigma), $$

$$ F[\rho] \equiv-2 \Gamma^{2} \bar{G}^{\Omega, \Gamma}(\rho, 0) $$

$$ \begin{align*} & B\left(x_{0}\right) \equiv e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) / k_{B} T}=\oint \mathcal{D}^{\prime} x e^{-\mathcal{A}_{\mathrm{e}} / \hbar}=\prod_{m=1}^{\infty}\left[\int \frac{d x_{m}^{\mathrm{re}} d x_{m}^{\mathrm{im}}}{\pi k_{B} T / M \omega_{m}^{2}}\right] \times \\ & \quad \times \exp \left[-\frac{M}{k_{B} T} \sum_{m=1}^{\infty} \omega_{m}^{2}\left|x_{m}\right|^{2}-\frac{1}{\hbar} \int_{0}^{\hbar / k_{B} T} d \tau V\left(x_{0}+\sum_{m=1}^{\infty}\left(x_{m} e^{-i \omega_{m} \tau}+\text { c.c. }\right)\right)\right] \end{align*} $$

$$ E_{N}(g)=\omega \sum_{l=0}^{N} E_{l}\left(\frac{g}{\omega^{(P+2) / 2}}\right)^{l}, $$

$$ E_{0}=\frac{3}{4 \Gamma}(\Gamma-\Omega)^{2}-\frac{1}{\sqrt{\pi}} \Gamma \int_{0}^{\infty} d \rho e^{-\rho} F^{-1 / 2}(\rho) $$

$$ \mathcal{A}_{\Omega}^{x_{0}}=\int_{0}^{\hbar / k_{B} T} d \tau M\left[\frac{\dot{x}^{2}}{2}+\Omega^{2}\left(x_{0}\right) \frac{\left(x-x_{0}\right)^{2}}{2}\right] $$

$$ W_{N}^{\Omega}=\Omega \sum_{l=0}^{N} \varepsilon_{l}(\sigma)\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{l}, $$

$$ \begin{gather*} \Delta E_{0}^{(2)}=-\frac{41^{2}}{\pi} \Gamma^{2} I+\frac{1}{2 \Gamma \sqrt{\pi}}\left(\Gamma^{2}-\Omega^{2}\right) \int_{0}^{\infty} d \rho e^{-\rho} F^{-1 / 2}(\rho)-\frac{3}{16 \Gamma^{3}}\left(\Gamma^{2}-\Omega^{2}\right)^{2} \\ -\frac{1}{4 \sqrt{\pi}}\left(\Gamma^{2}-\Omega^{2}\right) \int_{0}^{\infty} d \rho e^{-\rho} F^{-3 / 2}(\rho)\left\{\left(1+\frac{\Omega^{2}}{\Gamma}\right)\left(1-e^{-\Gamma \rho}\right)+\frac{\Gamma^{2}-\Omega^{2}}{\Gamma} \rho e^{-\Gamma \rho}\right\} \end{gather*} $$

$$ B_{\Omega}\left(x_{0}\right) \equiv e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) / k_{B} T}=\frac{\hbar \Omega\left(x_{0}\right) / 2 k_{B} T}{\sinh \left[\hbar \Omega\left(x_{0}\right) / 2 k_{B} T\right]} \equiv Z_{\Omega}^{x_{0}} $$

$$ \varepsilon_{l}(\sigma)=\sum_{j=0}^{l} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l-j}(-\sigma)^{l-j} . $$

$$ I=\frac{1}{4} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} d \rho_{1} d \rho_{2} d \rho_{3} e^{-\rho_{1}} F^{-1 / 2}\left(\rho_{1}\right) e^{-\rho_{2}} F^{-1 / 2}\left(\rho_{2}\right)\left(\frac{\arcsin Q}{Q}-1\right) $$

$$ F_{\Omega}^{x_{0}} \equiv-k_{B} T \log Z_{\Omega}^{x_{0}} $$

$$ \sigma \equiv \frac{\Omega^{(P-2) / 2}\left(\Omega^{2}-\omega^{2}\right)}{g} . $$

$$ \begin{align*} & Q=Q_{1} \text { for } \rho_{3}-\rho_{2}+\rho_{1} \geq 0 \text { and } \rho_{3}-\rho_{2} \geq 0, \\ & Q=Q_{2} \text { for } \rho_{3}-\rho_{2}+\rho_{1} \geq 0 \text { and } \rho_{3}-\rho_{2}<0, \\ & Q=Q_{3} \text { for } \rho_{3}-\rho_{2}+\rho_{1}<0 \text { and } \rho_{3}-\rho_{2}<0, \end{align*} $$

$$ V_{\Omega}^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right)=F_{\Omega}^{x_{0}}=k_{B} T \log \left\{\frac{\sinh \left[\hbar \Omega\left(x_{0}\right) / 2 k_{B} T\right]}{\hbar \Omega\left(x_{0}\right) / 2 k_{B} T}\right\} . $$

$$ \begin{align*} a_{3}^{12}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{6} \frac{1}{49152 x^{4}} \frac{1}{\sinh ^{6} \frac{x}{2}}\left(-107520-7360 x^{2}+624 x^{4}+96 x^{6}\right. \\ & +161280 \cosh x+12000 x^{2} \cosh x-777 x^{4} \cosh x+24 x^{6} \cosh x \\ & -64512 \cosh 2 x-5952 x^{2} \cosh 2 x+144 x^{4} \cosh 2 x+10752 \cosh 3 x \\ & -28800 x \sinh x+1312 x^{2} \cosh 3 x+9 x^{4} \cosh 3 x+1120 x^{3} \sinh x \\ & +324 x^{5} \sinh x+23040 x \sinh 2 x-320 x^{3} \sinh 2 x-5760 x \sinh 3 x \\ & \left.-160 x^{3} \sinh 3 x\right) \\ a_{2}^{6}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{3} \frac{1}{24 x^{2}} \frac{1}{\sinh ^{2} \frac{x}{2}}\left(-24-4 x^{2}+24 \cosh x+x^{2} \cosh x-9 x \sinh x\right), \\ a_{3}^{8}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{4} \frac{1}{288 x^{2}} \frac{1}{\sinh ^{3} \frac{x}{2}}\left(45 x \cosh \frac{x}{2}-6 x^{3} \cosh \frac{x}{2}-45 x \cosh \frac{3 x}{2}\right. \\ & \left.-432 \sinh \frac{x}{2}-54 x^{2} \sinh \frac{x}{2}+144 \sinh \frac{3 x}{2}+4 x^{2} \sinh \frac{3 x}{2}\right), \\ a_{3^{\prime}}^{10}= & \left(\frac{\hbar}{\tilde{\omega}}\right)^{5} \frac{1}{2304 x^{3}} \frac{1}{\sinh ^{4} \frac{x}{2}}\left(-3456-414 x^{2}-6 x^{4}+4608 \cosh x+\right. \\ & 496 x^{2} \cosh x-1152 \cosh 2 x-82 x^{2} \cosh 2 x-1008 x \sinh x- \\ & \left.16 x^{3} \sinh x+504 x \sinh 2 x+5 x^{3} \sinh 2 x\right) . \end{align*} $$

$$ \frac{d W_{N}^{\Omega}}{d \Omega}=\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{N} p_{N}(\sigma), $$

$$ \begin{align*} Q_{1}= & \frac{1}{2} F^{-1 / 2}\left(\rho_{1}^{2}\right) F^{-1 / 2}\left(\rho_{2}^{2}\right) \frac{\Gamma^{2}-\Omega^{2}}{\Gamma} e^{-\Gamma \rho_{3}}\left(1-e^{-\Gamma \rho_{1}}\right)\left(e^{\Gamma \rho}-1\right) \\ Q_{2}= & \frac{1}{2} F^{-1 / 2}\left(\rho_{1}\right) F^{-1 / 2}\left(\rho_{2}\right) \\ & \times\left\{\frac{\Gamma^{2}-\Omega^{2}}{\Gamma}\left[e^{-\Gamma\left(\rho_{2}-\rho_{3}\right)}-e^{-\Gamma \rho_{3}}\left(1+e^{-\Gamma\left(\rho_{1}-\rho_{2}\right)}-e^{-\Gamma \rho_{1}}\right)\right]-2 \Omega^{2}\left(\rho_{2}-\rho_{3}\right)\right\} \\ Q_{3}= & \frac{1}{2} F^{-1 / 2}\left(\rho_{1}\right) F^{-1 / 2}\left(\rho_{2}\right) \\ & \times\left\{\frac{\Gamma^{2}-\Omega^{2}}{\Gamma}\left[-e^{-\Gamma \rho_{3}}\left(1-e^{-\Gamma \rho_{1}}\right)-e^{-\Gamma\left(\rho_{2}-\rho_{3}\right)}\left(e^{\Gamma \rho_{1}}-1\right)\right]-2 \Omega^{2} \rho_{1}\right\} . \end{align*} $$

$$ \begin{align*} Z_{\Omega}^{x_{0}} & =\int \mathcal{D} x(\tau) \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-\mathcal{A}_{\Omega}^{x_{0}} / \hbar} \\ & =\prod_{m=1}^{\infty}\left[\int \frac{d x_{m}^{\mathrm{re}} d x_{m}^{\mathrm{im}}}{\pi k_{B} T / M \omega_{m}^{2}}\right] e^{-\frac{M}{k_{B} T} \sum_{m=1}^{\infty}\left[\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)\right]\left|x_{m}\right|^{2}} \end{align*} $$

$$ \begin{align*} p_{N}(\sigma) & =-2 \frac{d \varepsilon_{N+1}(\sigma)}{d \sigma} \\ & =2 \sum_{j=0}^{N} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{N+1-j}(N+1-j)(-\sigma)^{N-j} . \end{align*} $$

$$ \langle F[x(\tau)]\rangle_{\Omega}^{x_{0}} \equiv\left[Z_{\Omega}^{x_{0}}\right]^{-1} \int \mathcal{D} x \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-\mathcal{A}_{\Omega}^{x_{0}} / \hbar} F[x(\tau)] $$

$$ \frac{d W_{N}^{\Omega}}{d \Omega}=\sum_{l=0}^{N}\left[\varepsilon_{l}(\sigma)-\frac{P+2}{2} l \varepsilon_{l}(\sigma)+\Omega \frac{d \varepsilon_{l}}{d \Omega}\right]\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{l} $$

$$ \begin{align*} e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) / k_{B} T} & =Z^{x_{0}}=\int \mathcal{D} x \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-\mathcal{A} / \hbar} \\ & \equiv \int \mathcal{D} x \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-\mathcal{A}_{\Omega}^{x_{0}} / \hbar} e^{-\left(\mathcal{A}-\mathcal{A}_{\Omega}^{x_{0}}\right) / \hbar} \\ & =Z_{\Omega}^{x_{0}}\left\langle e^{-\left(\mathcal{A}-\mathcal{A}_{\Omega}^{x_{0}}\right) / \hbar}\right\rangle_{\Omega}^{x_{0}} \end{align*} $$

$$ \Omega \frac{d \varepsilon_{l}}{d \Omega}=\left[2 \frac{\Omega^{(P+2) / 2}}{g}+\frac{P-2}{2} \sigma\right] \frac{d \varepsilon_{l}}{d \sigma}, $$

$$ \left\langle e^{-\left(\mathcal{A}-\mathcal{A}_{\Omega}^{x_{0}}\right) / \hbar}\right\rangle_{\Omega}^{x_{0}} \geq e^{-\left\langle\mathcal{A} / \hbar-\mathcal{A}_{\Omega}^{x_{0}} / \hbar\right\rangle_{\Omega}^{x_{0}}} $$

$$ a_{V}^{2 L} \propto\left(\frac{\hbar}{\tilde{\omega}}\right)^{L} x^{V-1+L} $$

$$ \frac{d W_{N}^{\Omega}}{d \Omega}=\sum_{l=0}^{N}\left[\left(1-\frac{P+2}{2} l\right) \varepsilon_{l}(\sigma)+\left(2 \frac{\Omega^{(P+2) / 2}}{g}+\frac{P-2}{2} \sigma\right) \frac{d \varepsilon_{l}}{d \sigma}\right]\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{l} $$

$$ V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \leq F_{\Omega}^{x_{0}}\left(x_{0}\right)+k_{B} T\left\langle\mathcal{A} / \hbar-\mathcal{A}_{\Omega}^{x_{0}} / \hbar\right\rangle_{\Omega}^{x_{0}} . $$

$$ \begin{align*} & x / 12, \quad x^{3} / 720, \quad x^{5} / 30240, \\ & x^{5} / 241920, \quad x^{7} / 11404800, \quad 193 x^{8} / 47551795200, \\ & x^{4} / 30240, \quad x^{6} / 1814400, \quad x^{7} / 59875200, \end{align*} $$

$$ \begin{align*} \frac{d W_{N}^{\Omega}}{d \Omega} & =2 \frac{d \varepsilon_{0}}{d \sigma}\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{-1} \\ & +\sum_{l=0}^{N-1}\left[\left(1-\frac{P+2}{2} l\right) \varepsilon_{l}+\frac{P-2}{2} \sigma \frac{d \varepsilon_{l}}{d \sigma}+2 \frac{d \varepsilon_{l+1}}{d \sigma}\right]\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{l} \\ & +\left[\left(1-\frac{P+2}{2} N\right) \varepsilon_{N}+\frac{P-2}{2} \sigma \frac{d \varepsilon_{N}}{d \sigma}\right]\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{N} \end{align*} $$

$$ \frac{e^{-x_{1}}+e^{-x_{2}}}{2} \geq e^{-\frac{x_{1}+x_{2}}{2}} $$

$$ \frac{\hbar^{n}}{n!}\left(-\frac{\partial}{\partial \tilde{\omega}^{2}}\right)^{n}=\frac{\hbar^{n}}{n!}\left(-\frac{1}{2 \tilde{\omega}} \frac{\partial}{\partial \tilde{\omega}}\right)^{n}, $$

$$ \left(1-\frac{P+2}{2} l\right) \varepsilon_{l}+\frac{P-2}{2} \sigma \frac{d \varepsilon_{l}}{d \sigma}+2 \frac{d \varepsilon_{l+1}}{d \sigma}=0, \quad l=1 \ldots N-1 . $$

$$ \langle\mathcal{O}[x]\rangle \equiv \int \mathcal{D} \mu[x] \mathcal{O}[x] $$

$$ 2 \frac{d \varepsilon_{l+1}}{d \sigma}=2 \sum_{j=0}^{l} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l+1-j}(j-l-1)(-\sigma)^{l-j}, $$

$$ \left\langle e^{-\mathcal{O}}\right\rangle \geq e^{-\langle\mathcal{O}\rangle} $$

$$ 2\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l+1-j}=\frac{\frac{P-2}{2} j+2 l-1}{j-l-1}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l-j} $$

$$ \frac{e^{-\mathcal{O}\left(x_{1}\right)}+e^{-\mathcal{O}\left(x_{2}\right)}}{2} \geq e^{-\frac{\mathcal{O}\left(x_{1}\right)+\mathcal{O}\left(x_{2}\right)}{2}} $$

$$ 2 \frac{d \varepsilon_{l+1}}{d \sigma}=\sum_{j=0}^{l} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l-j}\left(\frac{P-2}{2} j+2 l-1\right)(-\sigma)^{l-j} $$

$$ \int d \mu(x) e^{-\mathcal{O}(x)} \geq e^{\int d \mu(x) e^{-\mathcal{O}(x)}} $$

$$ \frac{P-2}{2} \sigma \frac{d \varepsilon_{l}}{d \sigma}=\sum_{j=0}^{l} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{l-j} \frac{P-2}{2}(l-j)(-\sigma)^{l-j} . $$

$$ \int \mathcal{D} \mu[x] e^{-\mathcal{O}[x]} \geq e^{\int \mathcal{D} \mu[x] e^{-\mathcal{O}[x]}} $$

$$ \frac{d W_{N}^{\Omega}}{d \Omega}=-2\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{N} \frac{d \varepsilon_{N+1}(\sigma)}{d \sigma} $$

$$ V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) \leq F_{\Omega}^{x_{0}}+\frac{k_{B} T}{\hbar} \int_{0}^{\hbar / k_{B} T} d \tau\left\langle\left[V(x(\tau))-M \frac{\Omega^{2}\left(x_{0}\right)}{2}\left(x(\tau)-x_{0}\right)^{2}\right]\right\rangle_{\Omega}^{x_{0}} $$

$$ \frac{d W_{N}^{\Omega}}{d \Omega}=2\left(\frac{g}{\Omega^{(P+2) / 2}}\right)^{N} \sum_{j=0}^{N} E_{j}\binom{\left(1-\frac{P+2}{2} j\right) / 2}{N+1-j}(N+1-j)(-\sigma)^{N-j} $$

$$ \left\langle\eta(\tau) \eta\left(\tau^{\prime}\right)\right\rangle_{\Omega}^{x_{0}} \equiv a_{\tau \tau^{\prime}}^{2}\left(x_{0}\right)=\frac{\hbar}{M}\left[\frac{1}{2 \Omega\left(x_{0}\right)} \frac{\cosh \Omega\left(x_{0}\right)\left(\left|\tau-\tau^{\prime}\right|-\hbar \beta / 2\right)}{\sinh \left(\Omega\left(x_{0}\right) \hbar \beta / 2\right)}-\frac{1}{\hbar \beta \Omega^{2}}\right] $$

$$ Q_{N}(g)=\omega^{\alpha} \sum_{l=0}^{N} E_{l}\left(\frac{g}{\omega^{(P+2) / 2}}\right)^{l}, \quad \alpha \neq 1 $$

$$ Z_{\Omega}^{x_{0}}[j] \equiv \int \mathcal{D} x(\tau) \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-(1 / \hbar) \int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left\{\dot{x}^{2}(\tau)+\Omega^{2}\left(x_{0}\right)\left[x(\tau)-x_{0}\right]^{2}\right]-j(\tau)\left[x(\tau)-x_{0}\right]\right\}} $$

$$ Z_{\Omega}^{x_{0}}[j]=Z_{\Omega}^{x_{0}} \exp \left\{\frac{1}{2 M \hbar} \int_{0}^{\hbar \beta} d \tau \int_{0}^{\hbar \beta} d \tau^{\prime} j(\tau) G_{\Omega^{2}\left(x_{0}\right), \mathrm{e}}^{\mathrm{p} \prime}\left(\tau-\tau^{\prime}\right) j\left(\tau^{\prime}\right)\right\} $$

$$ G_{\Omega^{2}}^{R}\left(t, t^{\prime}\right)=\frac{M}{\hbar} \Theta\left(t-t^{\prime}\right)\left\langle x(t) x\left(t^{\prime}\right)\right\rangle $$

$$ \left\langle\eta^{2}(\tau)\right\rangle_{\Omega}^{x_{0}}=\left\langle\left(x(\tau)-x_{0}\right)^{2}\right\rangle_{\Omega}^{x_{0}}=a_{\tau \tau}^{2}\left(x_{0}\right) \equiv a_{\Omega\left(x_{0}\right)}^{2}, $$

$$ a_{\Omega\left(x_{0}\right)}^{2}=\frac{2}{M \beta} \sum_{m=1}^{\infty} \frac{1}{\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)}=\frac{1}{M \beta \Omega^{2}\left(x_{0}\right)}\left[\frac{\hbar \beta \Omega\left(x_{0}\right)}{2} \operatorname{coth} \frac{\hbar \beta \Omega\left(x_{0}\right)}{2}-1\right], $$

$$ a_{\Omega\left(x_{0}\right)}^{2} \underset{T \rightarrow \infty}{\longrightarrow} \hbar^{2} / 12 M k_{B} T $$

$$ a_{\mathrm{tot}}^{2} \equiv\left\langle\left(x(\tau)-x_{0}\right)^{2}\right\rangle_{\Omega\left(x_{0}\right)}=a_{\Omega\left(x_{0}\right)}^{2}+\frac{k_{B} T}{M \Omega^{2}\left(x_{0}\right)} $$

$$ V(x(\tau))=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k x(\tau)} \tilde{V}(k) $$

$$ \langle V(x(\tau))\rangle_{\Omega}^{x_{0}}=V_{a^{2}}\left(x_{0}\right) \equiv \int_{-\infty}^{\infty} \frac{d k}{2 \pi} \tilde{V}(k) e^{i k x_{0}-a^{2}\left(x_{0}\right) k^{2} / 2} $$

$$ \tilde{V}(k)=\int_{-\infty}^{\infty} d x V(x) e^{-i k x} $$

$$ \langle V(x(\tau))\rangle_{\Omega}^{x_{0}}=V_{a^{2}}\left(x_{0}\right) \equiv \int_{-\infty}^{\infty} \frac{d x_{0}^{\prime}}{\sqrt{2 \pi a^{2}\left(x_{0}\right)}} e^{-\left(x_{0}^{\prime}-x_{0}\right)^{2} / 2 a^{2}\left(x_{0}\right)} V\left(x_{0}^{\prime}\right) $$

$$ \left(x-x_{0}\right)_{a^{2}}^{2}=\int_{-\infty}^{\infty} \frac{d x^{\prime}}{\sqrt{2 \pi a^{2}}} e^{-\left(1 / 2 a^{2}\right)\left(x^{\prime}-x_{0}\right)^{2}}\left(x^{\prime}-x_{0}\right)^{2}=a^{2}\left(x_{0}\right) $$

$$ W_{1}^{\Omega}\left(x_{0}\right) \equiv F_{\Omega}^{x_{0}}+V_{a^{2}}\left(x_{0}\right)-\frac{M}{2} \Omega^{2}\left(x_{0}\right) a^{2}\left(x_{0}\right), $$

$$ W_{1}^{\Omega}\left(x_{0}\right) \geq V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) $$

$$ W_{1}\left(x_{0}\right)=F_{\omega}^{x_{0}}+M \omega^{2} \frac{x_{0}^{2}}{2} $$

$$ \frac{\partial W_{1}\left(x_{0}\right)}{\partial \Omega^{2}\left(x_{0}\right)}=\frac{M}{2}\left\{\frac{k_{B} T}{M \Omega^{2}\left(x_{0}\right)}\left[\frac{\hbar \Omega}{2 k_{B} T} \operatorname{coth}\left(\frac{\hbar \Omega}{2 k_{B} T}\right)-1\right]-a^{2}\left(x_{0}\right)\right\} . $$

$$ \frac{\partial W_{1}^{\Omega}\left(x_{0}\right)}{\partial a^{2}\left(x_{0}\right)}=0 $$

$$ \Omega^{2}\left(x_{0}\right)=\frac{2}{M} \frac{\partial V_{a^{2}}\left(x_{0}\right)}{\partial a^{2}\left(x_{0}\right)} $$

$$ \Omega^{2}\left(x_{0}\right)=\frac{1}{M}\left[\frac{\partial^{2}}{\partial x_{0}^{2}} V_{a^{2}}\left(x_{0}\right)\right]_{a^{2}=a^{2}\left(x_{0}\right)} $$

$$ Z_{1}=e^{-F_{1} / k_{B} T}=\int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} e^{-W_{1}\left(x_{0}\right) / k_{B} T} \leq Z $$

$$ Z \underset{T \rightarrow 0}{\longrightarrow} e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{\mathrm{m}}\right) / k_{B} T} \int \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} e^{-\left[V^{\mathrm{eff} \mathrm{cl}}\left(x_{\mathrm{m}}\right)\right]^{\prime \prime}\left(x_{0}-x_{\mathrm{m}}\right)^{2} / 2 k_{B} T} . $$

$$ F \underset{T \rightarrow 0}{\longrightarrow} V^{\mathrm{eff} \mathrm{cl}}\left(x_{\mathrm{m}}\right) $$

$$ \Delta F=k_{B} T \log \left\{\frac{\hbar}{k_{B} T} \sqrt{\frac{\left[V^{\mathrm{eff} \mathrm{cl}}\left(x_{\mathrm{m}}\right)\right]^{\prime \prime}}{M}}\right\}, $$

$$ E^{(0)}=V^{\mathrm{eff} \mathrm{cl}}\left(x_{\mathrm{m}}\right) . $$

$$ E_{1}^{(0)}=\left.\left.F_{1}\right|_{T=0} \equiv W_{1}\left(x_{\mathrm{m}}\right)\right|_{T=0} $$

$$ \lim _{T \rightarrow 0} W_{1}^{\Omega}\left(x_{0}\right)=\frac{1}{2}\left[\hbar \Omega\left(x_{0}\right)-M \Omega^{2}\left(x_{0}\right) a^{2}\left(x_{0}\right)\right]+V_{a^{2}}\left(x_{0}\right) . $$

$$ \lim _{T \rightarrow 0} a^{2}\left(x_{0}\right)=\frac{\hbar}{2 M \Omega\left(x_{0}\right)} $$

$$ \lim _{T \rightarrow 0} W_{1}^{\Omega}\left(x_{0}\right)=\frac{1}{4} \hbar \Omega\left(x_{0}\right)+V_{a^{2}}\left(x_{0}\right)=\frac{1}{8} \frac{\hbar^{2}}{M a^{2}\left(x_{0}\right)}+V_{a^{2}}\left(x_{0}\right) $$

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

$$ \psi(x)=\frac{1}{\left(2 \pi a^{2}\right)^{1 / 4}} \exp \left[-\frac{1}{4 a^{2}}\left(x-x_{0}\right)^{2}\right] $$

$$ \langle\hat{H}\rangle_{\psi} \equiv \int_{-\infty}^{\infty} d x \psi^{*}(x) \hat{H} \psi(x)=\frac{1}{8} \frac{\hbar^{2}}{M a^{2}}+V_{a^{2}}\left(x_{0}\right) $$

$$ E_{1}=\min _{x_{0}, a^{2}}\langle\hat{H}\rangle_{\psi} $$

$$ \lim _{T \rightarrow 0} F_{1}=E_{1} $$

$$ E_{1}=\min _{a}\left(\frac{3}{8} \frac{\hbar^{2}}{M a^{2}}-\frac{2}{\sqrt{\pi}} \frac{e^{2}}{\sqrt{2 a^{2}}}\right)=-\frac{3}{8} \frac{\hbar^{2}}{M a_{\min }^{2}} . $$

$$ W_{1}(0)=\frac{\Omega}{2}+V_{a^{2}}(0) $$

$$ V_{a^{2}}(0)=\int_{-\infty}^{\infty} \frac{d x_{0}^{\prime}}{\sqrt{2 \pi a^{2}}} e^{-x_{0}^{\prime 2} / 2 a^{2}} V\left(x_{0}^{\prime}\right) $$

$$ V_{a^{2}}(0) \approx \sqrt{\frac{\Omega}{\pi}} \int_{-\infty}^{\infty} d x_{0}^{\prime} V\left(x_{0}^{\prime}\right) $$

$$ E_{1}^{(0)} \approx-\frac{1}{2 \pi}\left[-\int_{-\infty}^{\infty} d x_{0}^{\prime} V\left(x_{0}^{\prime}\right)\right]^{2} $$

$$ V(x)=-g \delta(x), \quad g>0 $$

$$ E_{1}^{(0)}=-\frac{1}{2 \pi} g^{2} $$

$$ E^{(0)}=-\frac{1}{2} g^{2} $$

$$ H=-\frac{1}{2} \frac{d^{2}}{d x^{2}}-g \delta(x) $$

$$ \psi(x)=\sqrt{K} e^{-K|x|}, $$

$$ W_{1}=\frac{K^{2}}{2}-g K $$

$$ W(0)=\frac{K^{2}}{2}+V_{K}(0) $$

$$ V_{K}(0)=K \int_{-\infty}^{\infty} \frac{d x_{0}^{\prime}}{a} e^{-K\left|x_{0}\right|} V\left(x_{0}\right) $$

$$ E_{1}^{(0)} \approx-\frac{1}{2}\left[-\int_{-\infty}^{\infty} d x_{0}^{\prime} V\left(x_{0}\right)\right]^{2} $$

$$ \begin{align*} Z_{\bar{m}} \equiv & \int \mathcal{D} x(\tau) \exp \left\{-\frac{1}{\hbar} \int_{0}^{\hbar / k_{B} T} \frac{M}{2}\left[\frac{\dot{x}^{2}(\tau)}{2}+\Omega^{2}\left(x_{0}, \ldots, x_{\bar{m}}\right)\right.\right. \\ & \left.\left.\times\left(x(\tau)-x_{0}-\sum_{m=1}^{\bar{m}-1}\left(x_{m} e^{-i \omega_{m} \tau}+\text { c.c. }\right)\right)^{2}\right]\right\} e^{-\left(1 / k_{B} T\right) L_{\bar{m}}\left(x_{0}, \ldots, x_{\bar{m}}\right)} \\ = & \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \prod_{n=1}^{\bar{m}-1}\left[\int \frac{d x_{m}^{\mathrm{re}} d x_{m}^{\mathrm{im}}}{\pi k_{B} T / M \omega_{m}^{2}} \frac{\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)}{\omega_{m}^{2}}\right] \\ & \times \frac{\hbar \Omega\left(x_{0}\right) / 2 k_{B} T}{\sinh \left(\hbar \Omega\left(x_{0}\right) / 2 k_{B} T\right)} e^{-L_{\bar{m}}\left(x_{0}, \ldots, x_{\bar{m}}\right) / k_{B} T} \end{align*} $$

$$ \begin{align*} a_{\bar{m}}^{2} & =\frac{2 k_{B} T}{M} \sum_{m=\bar{m}}^{\infty} \frac{1}{\omega_{m}^{2}+\Omega^{2}} \\ & =\frac{k_{B} T}{M \Omega^{2}}\left(\frac{\hbar \Omega}{2 k_{B} T} \operatorname{coth} \frac{\hbar \Omega}{2 k_{B} T}-1\right)-\frac{2 k_{B} T}{M} \sum_{m=1}^{\bar{m}-1} \frac{1}{\omega_{m}^{2}+\Omega^{2}} \end{align*} $$

$$ \mathcal{A}[x]=\int_{0}^{\hbar / k_{B} T} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)+\frac{g}{4} x^{4}\right] $$

$$ V_{a^{2}}\left(x_{0}\right)=\frac{x_{0}^{2}}{2}+\frac{g}{4} x_{0}^{4}+\frac{a^{2}}{2}+\frac{3}{2} g x_{0}^{2} a^{2}+\frac{3 g}{4} a^{4} . $$

$$ \Omega^{2}\left(x_{0}\right)=\left[1+3 g x_{0}^{2}+3 g a^{2}\left(x_{0}\right)\right] . $$

$$ a^{2}\left(x_{0}\right)=\frac{1}{\beta \Omega^{2}\left(x_{0}\right)}\left[\frac{\beta \Omega\left(x_{0}\right)}{2} \operatorname{coth} \frac{\beta \Omega\left(x_{0}\right)}{2}-1\right] . $$

$$ V_{a^{2}}\left(x_{0}\right) \approx V\left(x_{0}\right)+\frac{1}{2} \frac{\hbar^{2} \beta}{12 M} V^{\prime \prime}\left(x_{0}\right)+\frac{1}{8}\left(\frac{\hbar^{2} \beta}{12 M}\right)^{2} V^{(4)}\left(x_{0}\right)+\ldots, $$

$$ E_{1}^{(0)}=\frac{1}{4 \Omega}+\frac{\Omega}{4}+\frac{3}{4} \frac{g}{4 \Omega^{2}} $$

$$ \Omega_{1}=\left(\frac{g}{4}\right)^{1 / 3}\left[6^{1 / 3}+\frac{1}{2^{1 / 3} 3^{4 / 3}} \frac{1}{(g / 4)^{2 / 3}}+\ldots\right] $$

$$ \begin{align*} E_{1}^{(0)}(g) & \approx\left(\frac{g}{4}\right)^{1 / 3}\left[\left(\frac{3}{4}\right)^{4 / 3}+\frac{1}{2^{7 / 3} 3^{1 / 3}} \frac{1}{(g / 4)^{2 / 3}}+\ldots\right] \\ & \approx\left(\frac{g}{4}\right)^{1 / 3}\left[0.681420+0.13758 \frac{1}{(g / 4)^{2 / 3}}+\ldots\right] \end{align*} $$

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

$$ \Omega^{2}\left(x_{0}\right)=-1+3 g x_{0}^{2}+3 g a^{2}\left(x_{0}\right) $$

$$ \begin{align*} a^{2}\left(x_{0}\right) & =\frac{2}{\beta} \sum_{m=1}^{\infty} \frac{1}{\omega_{m}^{2}+\Omega^{2}\left(x_{0}\right)} \\ & =\frac{1}{\beta \Omega^{2}\left(x_{0}\right)}\left(\frac{\beta\left|\Omega\left(x_{0}\right)\right|}{2} \cot \frac{\beta\left|\Omega\left(x_{0}\right)\right|}{2}-1\right) \end{align*} $$

$$ f\left(\Omega^{2}\left(x_{0}\right)\right) \equiv a^{2}\left(x_{0}\right)-\frac{1}{3 g}\left[1+\Omega^{2}\left(x_{0}\right)-3 g x_{0}^{2}\right] $$

$$ x_{m}=\sqrt{\frac{1-3 g a^{2}}{g}} $$

$$ \Omega^{2}\left(x_{m}\right)=2-6 g a^{2}\left(x_{m}\right) $$

$$ g>g_{c}=\frac{4}{9} \sqrt{\frac{2}{3}} \approx 0.3629 $$

$$ \omega_{m}^{2} \rightarrow \Omega_{m} \bar{\Omega}_{m}=\frac{1}{\epsilon^{2}}\left[2-2 \cos \left(\epsilon \omega_{m}\right)\right] $$

$$ \begin{align*} a^{2}\left(x_{0}\right) & =\frac{2 k_{B} T}{M} \frac{\partial}{\partial \Omega^{2}\left(x_{0}\right)} \log \prod_{m=1}^{m_{\max }}\left[\Omega_{m} \bar{\Omega}_{m}+\Omega^{2}\left(x_{0}\right)\right] \\ & =\frac{k_{B} T}{M} \frac{1}{\Omega} \frac{\partial}{\partial \Omega} \log \frac{\sinh \left(\hbar \Omega_{N}\left(x_{0}\right) / 2 k_{B} T\right)}{\hbar \Omega\left(x_{0}\right) / 2 k_{B} T} \\ & =\frac{k_{B} T}{M \Omega^{2}\left(x_{0}\right)}\left[\frac{\hbar \Omega\left(x_{0}\right)}{2 k_{B} T} \operatorname{coth} \frac{\hbar \Omega_{N}\left(x_{0}\right)}{2 k_{B} T} \frac{1}{\cosh \left(\epsilon \Omega_{N}\left(x_{0}\right) / 2\right)}-1\right] \end{align*} $$

$$ \sinh \left[\epsilon \Omega_{N}\left(x_{0}\right) / 2\right] \equiv \epsilon \Omega\left(x_{0}\right) / 2 $$

$$ W_{1}\left(x_{0}\right) \equiv k_{B} T \log \frac{\sinh \hbar \Omega_{N}\left(x_{0}\right) / 2 k_{B} T}{\hbar \Omega\left(x_{0}\right) / 2 k_{B} T}+V_{a^{2}\left(x_{0}\right)}\left(x_{0}\right)-\frac{M}{2} \Omega^{2}\left(x_{0}\right) a^{2}\left(x_{0}\right), $$

$$ \begin{align*} & \delta\left(x_{a}-x(\tau)\right)=\delta\left(x_{a}-x_{0}-\sum_{m=1}^{\infty}\left(x_{m} e^{-i \omega_{m} \tau}+\text { c.c. }\right)\right) \\ & \quad=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} \exp \left\{i k\left[x_{a}-x_{0}-\sum_{m=1}^{\infty}\left(x_{m} e^{-i \omega_{m} \tau}+\text { c.c. }\right)\right]\right\} \end{align*} $$

$$ \rho\left(x_{a}\right)=Z^{-1} \oint \mathcal{D} x \delta\left(x_{a}-x(\tau)\right) e^{-\mathcal{A} / \hbar} $$

$$ \begin{align*} \rho\left(x_{a}\right)= & Z^{-1} \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \int \frac{d k}{2 \pi} e^{i k\left(x_{a}-x_{0}\right)} \\ & \times \int \mathcal{D} x \tilde{\delta}\left(\bar{x}-x_{0}\right) e^{-i k \Sigma_{m=1}^{\infty}\left(x_{m}+\text { c.c. }\right)} e^{-\mathcal{A} / \hbar} \end{align*} $$

$$ \left\langle e^{-i k \Sigma_{m=1}^{\infty}\left(x_{m} e^{-i \omega_{m} \tau}+\text { c.c. }\right)}\right\rangle_{\Omega}^{x_{0}} \approx e^{\left.-\left.k^{2} \Sigma_{m=1}^{\infty}\langle | x_{m}\right|^{2}\right\rangle_{\Omega}^{x_{0}}} \equiv e^{-k^{2} a^{2} / 2}, $$

$$ \rho\left(x_{a}\right) \approx Z^{-1} \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \frac{e^{-\left(x_{a}-x_{0}\right)^{2} / 2 a^{2}\left(x_{0}\right)}}{\sqrt{2 \pi a^{2}\left(x_{0}\right)}} e^{-V^{\mathrm{eff} \mathrm{cl}}\left(x_{0}\right) / k_{B} T} $$

$$ \rho_{1}\left(x_{a}\right)=Z_{1}^{-1} \int_{-\infty}^{\infty} \frac{d x_{0}}{l_{\mathrm{e}}(\hbar \beta)} \frac{e^{-\left(x_{a}-x_{0}\right)^{2} / 2 a^{2}\left(x_{0}\right)}}{\sqrt{2 \pi a^{2}\left(x_{0}\right)}} e^{-W_{1}\left(x_{0}\right) / k_{B} T} $$

$$ \Omega_{i j}^{2}\left(\mathbf{x}_{0}\right)=\Omega_{L}^{2}\left(r_{0}\right) P_{L i j}\left(\mathbf{x}_{0}\right)+\Omega_{T}^{2}\left(r_{0}\right) P_{T i j}\left(\mathbf{x}_{0}\right), $$

$$ P_{L i j}\left(\hat{\mathbf{x}}_{0}\right)=x_{0 i} x_{0 j} / r_{0}^{2}, \quad P_{T i j}\left(\hat{\mathbf{x}}_{0}\right)=\delta_{i j}-x_{0 i} x_{0 j} / r_{0}^{2}, $$

$$ \begin{gather*} W_{1}^{\Omega}\left(r_{0}\right)=k_{B} T\left[\log \frac{\sinh \hbar \Omega_{L} / 2 k_{B} T}{\hbar \Omega_{L} / 2 k_{B} T}+(D-1) \log \frac{\sinh \hbar \Omega_{T} / 2 k_{B} T}{\hbar \Omega_{T} / 2 k_{B} T}\right] \\ -\frac{M}{2}\left[\Omega_{L}^{2} a_{L}^{2}+(D-1) \Omega_{T}^{2} a_{T}^{2}\right]+V_{a_{L}, a_{T}}, \end{gather*} $$

$$ \begin{align*} V_{a_{L}, a_{T}}\left(r_{0}\right) & =\frac{1}{\sqrt{2 \pi a_{L}^{2}}} \frac{1}{{\sqrt{2 \pi a_{T}^{2}}}^{D-1}} \prod_{i=1}^{D}\left[\int_{-\infty}^{\infty} d \delta x_{i}\right] \\ & \times \exp \left[-\frac{1}{2}\left(a_{L}^{-2} \delta x_{L}^{2}+a_{T}^{-2} \delta \mathbf{x}_{T}^{2}\right)\right] V\left(\mathbf{x}_{0}+\delta \mathbf{x}\right) \end{align*} $$

$$ V_{a_{L}, a_{T}}\left(r_{0}\right)=\frac{1}{\sqrt{(2 \pi)^{D} a_{L}^{2} a_{T}^{2 D-2}}} \int d \delta x_{L} \int d^{D-1} \delta x_{T} \exp \left[-\frac{\delta x_{L}^{2}}{2 a_{L}^{2}}-\frac{\delta x_{T}^{2}}{2 a_{T}^{2}}\right] V\left(\mathbf{x}_{0}+\delta \mathbf{x}\right) $$

$$ \begin{align*} V(\mathbf{x}) & =v\left(r_{0}^{2}+2 r_{0} \delta x_{L}+\delta x_{L}^{2}+\delta \mathbf{x}_{T}^{2}\right)=\sum_{n=0}^{\infty} \frac{1}{n!} v^{(n)}\left(r_{0}^{2}\right)\left(2 r_{0} \delta x_{L}+\delta x_{L}^{2}+\delta \mathbf{x}_{T}^{2}\right)^{n} \\ & =\left[v\left(r_{0}^{2}+\partial_{\lambda}\right) e^{\lambda\left(2 r_{0} \delta x_{L}+\delta x_{L}^{2}+\delta \mathbf{x}_{T}^{2}\right)}\right]_{\lambda=0} \end{align*} $$

$$ V_{a_{L}, a_{T}}\left(r_{0}\right)=\left[v\left(r_{0}^{2}+\partial_{\lambda}\right) \frac{1}{\sqrt{1-2 a_{L}^{2} \lambda}} \frac{1}{{\sqrt{1-2 a_{T}^{2} \lambda}}^{D-1}} e^{2 r_{0}^{2} \lambda^{2} a_{L}^{2} /\left(1-2 a_{L}^{2} \lambda\right)}\right]_{\lambda=0} $$

$$ \begin{align*} V_{a_{L}, a_{T}}\left(r_{0}\right) & =v\left(r_{0}^{2}\right)+v^{\prime}\left(r_{0}^{2}\right)\left[a_{L}^{2}+(D-1) a_{T}^{2}\right] \\ & +\frac{1}{2} v^{\prime \prime}\left(r_{0}^{2}\right)\left[3 a_{L}^{4}+2(D-1) a_{L}^{2} a_{T}^{2}+\left(D^{2}-1\right) a_{T}^{4}+4 r_{0}^{2} a_{L}^{2}\right]+\ldots \end{align*} $$

$$ V(\mathbf{x})=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k x}} V(\mathbf{k}) $$

$$ V_{a_{L}^{2}, a_{T}^{2}}\left(r_{0}\right)=\int \frac{d^{D} k}{(2 \pi)^{D}} V(\mathbf{k}) \exp \left(-\frac{a_{L}^{2}}{2} k_{L}^{2}-\frac{a_{T}^{2}}{2} k_{T}^{2}-i r_{0} k_{L}\right) $$

$$ \int \frac{d^{D} k}{(2 \pi)^{D}}=\frac{S_{D-1}}{(2 \pi)^{D}} \int_{-\infty}^{\infty} d k_{L} \int_{0}^{\infty} d k_{T} k_{T}^{D-2} $$

$$ S_{D}=\frac{2 \pi^{D / 2}}{\Gamma(D / 2)} $$

$$ \int \frac{d^{D} k}{(2 \pi)^{D}}=\frac{S_{D-1}}{(2 \pi)^{D}} \int_{-1}^{1} d \cos \phi \int_{0}^{\infty} d k k^{D-1} $$

$$ \begin{align*} V_{a_{L}^{2}, a_{T}^{2}}\left(r_{0}\right)= & \frac{S_{D-1}}{(2 \pi)^{D}} \int_{-1}^{1} d u \int_{-1}^{\infty} d k k^{D-1} v\left(k^{2}\right) \\ & \times \exp \left\{-\frac{1}{2}\left[\left(a_{L}^{2} u^{2}+a_{T}^{2}\left(1-u^{2}\right)\right) k^{2}-i r_{0} u\right]\right\} \end{align*} $$

$$ V(r)=-\left(e^{2} / r\right) e^{-m r} $$

$$ V(r)=4 \pi \int \frac{d^{3} k}{(2 \pi)^{3}} \frac{e^{i \mathbf{k x}}}{k^{2}+m^{2}}=4 \pi \int_{0}^{\infty} d \tau \int \frac{d^{3} k}{(2 \pi)^{3}} e^{-\tau\left(k^{2}+m^{2}\right)} $$

$$ \begin{align*} V_{a^{2}}\left(r_{0}\right) & =-e^{2} \int \frac{d^{3} x}{{\sqrt{2 \pi a^{2}}}^{3}} \exp \left[-\frac{1}{2 a^{2}}\left(\mathbf{x}_{0}-\mathbf{x}\right)^{2}\right] V(r) \\ & =-e^{2} 2 \pi e^{m^{2} a^{2} / 2} \int_{a^{2}}^{\infty} d a^{\prime 2} \frac{1}{{\sqrt{2 \pi a^{\prime 2}}}^{3}} \exp \left(-r_{0}^{2} / 2 a^{\prime 2}-m^{2} a^{\prime 2} / 2\right) \\ & =-e^{2} \frac{2 e^{m^{2} a^{2} / 2}}{\sqrt{\pi}} \frac{1}{r_{0}} \int_{0}^{r_{0} /{\sqrt{2 a^{2}}}^{2}} d t e^{-\left(t^{2}+m^{2} r_{0}^{2} / 4 t^{2}\right)} \end{align*} $$

$$ V_{a^{2}}\left(r_{0}\right)=-\frac{e^{2}}{r_{0}} \operatorname{erf}\left(r_{0} / \sqrt{2 a^{2}}\right), $$

$$ \operatorname{erf}(z) \equiv \frac{2}{\sqrt{\pi}} \int_{0}^{z} d x e^{-x^{2}} $$

$$ V_{a^{2}}(0)=-\frac{e^{2}}{a} \sqrt{\frac{2}{\pi}} e^{m^{2} a^{2} / 2} $$

$$ W_{1}^{\Omega}\left(\mathbf{x}_{0}\right)=\frac{3}{\beta} \ln \frac{\sinh \left[\hbar \beta \Omega\left(\mathbf{x}_{0}\right) / 2\right]}{\hbar \beta \Omega\left(\mathbf{x}_{0}\right) / 2}-\frac{e^{2}}{\left|\mathbf{x}_{0}\right|} \operatorname{erf}\left[\frac{\left|\mathbf{x}_{0}\right|}{\sqrt{2 a^{2}\left(\mathbf{x}_{0}\right)}}\right]-\frac{3}{2} M \Omega^{2}\left(\mathbf{x}_{0}\right) a^{2}\left(\mathbf{x}_{0}\right) $$

$$ \Omega^{2}\left(r_{0}\right)=e^{2} \frac{2}{3} \frac{1}{\sqrt{2 \pi}} \frac{1}{\left(a^{2}\right)^{3 / 2}} e^{-r_{0}^{2} / 2 a^{2}} $$

$$ V_{a_{L}^{2}, a_{T}^{2}}\left(r_{0}\right)=-\sqrt{\frac{1}{2 \pi}} \int_{-1}^{1} d u \frac{e^{2}}{\sqrt{a_{L}^{2} u^{2}+a_{T}^{2}\left(1-u^{2}\right)}} \exp \left\{-\frac{r_{0}^{2} u^{2}}{2\left[a_{L}^{2} u^{2}+a_{T}^{2}\left(1-u^{2}\right)\right]}\right\} $$

$$ V_{a_{L}^{2}, a_{T}^{2}}\left(r_{0}\right)=-e^{2} \sqrt{\frac{a_{L}^{2}}{2 \pi}} \int_{-1}^{1} d \lambda \frac{\exp \left[-\left(r_{0}^{2} / 2 a_{L}^{2}\right) \lambda^{2}\right]}{a_{L}^{2}\left(1-\lambda^{2}\right)+a_{T}^{2} \lambda^{2}} $$

$$ \begin{align*} \Omega_{T}^{2}\left(r_{0}\right) & \equiv \frac{\partial}{\partial a_{T}^{2}} V_{a_{T}^{2}, a_{L}^{2}}=e^{2} \sqrt{\frac{a_{L}^{2}}{2 \pi}} \int_{-1}^{1} d \lambda \frac{\lambda^{2} \exp \left[-\left(r_{0}^{2} / 2 a_{L}^{2}\right) \lambda^{2}\right]}{\left[a_{L}^{2}\left(1-\lambda^{2}\right)+a_{T}^{2} \lambda^{2}\right]^{2}} \\ \Omega^{2}\left(r_{0}\right) & \equiv \frac{1}{3}\left[\Omega_{L}^{2}+2 \Omega_{T}^{2}\right]=\frac{1}{3}\left[\frac{\partial}{\partial a_{L}^{2}}+2 \frac{\partial}{\partial a_{T}^{2}}\right] V_{a_{T}^{2}, a_{L}^{2}} \\ & =e^{2} \frac{2}{3} \frac{1}{\sqrt{2 \pi}} \frac{1}{\sqrt{a_{L}^{2} a_{T}^{4}}} \exp \left(-r_{0}^{2} / 2 a_{L}^{2}\right) \end{align*} $$

$$ a_{L, T}^{2}\left(r_{0}\right)=\frac{1}{\beta \Omega_{L, T}^{2}\left(r_{0}\right)}\left[\frac{\beta \Omega_{L, T}\left(r_{0}\right)}{2} \operatorname{coth} \frac{\beta \Omega_{L, T}\left(r_{0}\right)}{2}-1\right] . $$

$$ =2 \pi \int_{0}^{\infty} d r_{0} \int_{-1}^{1} d \lambda \frac{e^{-\left(\lambda r_{0}-r\right)^{2} / 2 a_{L}^{2}}}{\sqrt{2 \pi a_{L}^{2}\left(r_{0}\right)}} \frac{e^{-r_{0}^{2}\left(1-\lambda^{2}\right) / 2 a_{T}^{2}}}{2 \pi a_{T}^{2}\left(r_{0}\right)} \frac{e^{-\beta W_{1}\left(r_{0}\right)}}{(2 \pi \beta)^{3 / 2}} $$

$$ H(\mathbf{p}, \mathbf{x})=\frac{1}{2 M} \mathbf{p}^{2}+\frac{M}{2} \omega_{B}^{2} \mathbf{x}^{2}-\omega_{B} l_{z}(\mathbf{p}, \mathbf{x})-\frac{e^{2}}{|\mathbf{x}|}, $$

$$ \exp \left(-\frac{M}{2} \omega_{B} \mathbf{x}_{a}^{2}\right) . $$

$$ V_{a_{\|}^{2}, a_{\perp}^{2}}(0)=-e^{2} \sqrt{\frac{1}{2 \pi a_{\|}^{2}}} \int_{-1}^{1} d \lambda \frac{1}{\left(1-\lambda^{2}\right)+\lambda^{2} a_{\perp}^{2} / a_{\|}^{2}} $$

$$ V_{a_{\|}^{2}, a_{\perp}^{2}}(0)=-e^{2} \sqrt{\frac{1}{2 \pi\left(a_{\|}^{2}-a_{\perp}^{2}\right)}} 2 \operatorname{arccosh} \frac{a_{\|}}{a_{\perp}} . $$

$$ W_{1}(\mathbf{0})=\Omega_{\perp}+\frac{\Omega_{\|}}{2}+V_{a_{\|}^{2}, a_{\perp}^{2}}(\mathbf{0})+\frac{M}{2}\left(\omega_{\|}^{2}-\Omega_{\|}^{2}\right) a_{\|}^{2}+M\left(\omega_{B}^{2}-\Omega_{\perp}^{2}\right) a_{\perp}^{2}, $$

$$ W(\mathbf{0}) \approx \omega_{B}+\frac{\Omega_{\|}}{4}-e^{2} \sqrt{\frac{\Omega_{\|}}{\pi}} \log \frac{8 \omega_{B}}{\Omega_{\|}} $$

$$ \Omega_{\|} \approx \frac{4 e^{4}}{\pi} \log ^{2} \frac{2 \pi \omega_{B}}{e^{4}} $$

$$ E_{1}^{(0)} \approx \omega_{B}-\frac{e^{4}}{\pi} \log ^{2} \frac{2 \pi \omega_{B}}{e^{4}} $$

$$ \Omega_{\|} \approx \frac{B}{2}, \quad \sqrt{\Omega_{\perp}}=\frac{2}{\sqrt{\pi}}\left(\ln B-2 \ln \ln B+\frac{2 a}{\ln B}+\frac{a^{2}}{\ln ^{2} B}+b\right)+\mathcal{O}\left(\ln ^{-3} B\right), $$

$$ \begin{align*} E^{(0)}(B)=\frac{B}{2}-\frac{1}{\pi} & \left\{\ln ^{2} B-4 \ln B \ln \ln B+4 \ln ^{2} \ln B-4 b \ln \ln B+2(b+2) \ln B+b^{2}\right. \\ & \left.-\frac{1}{\ln B}\left[8 \ln ^{2} \ln B-8 b \ln \ln B+2 b^{2}\right]\right\}+\mathcal{O}\left(\ln ^{-2} B\right) \end{align*} $$

$$ \varepsilon(B) \equiv \frac{B}{2}-E^{(0)} $$

$$ w=\frac{1}{2} \log \frac{B}{w^{2}}+\sum_{n=1}^{N} a_{n}(B, w), $$

$$ a_{1} \equiv-\frac{1}{2}(\gamma+\log 2), a_{2} \equiv \frac{\pi^{2}}{12 w}, a_{3} \equiv-\sqrt{\frac{2}{B}}\left(\log \frac{B}{2 w^{2}}-\sqrt{\pi} w\right), a_{4}=\sqrt{\frac{2}{B}} \frac{\sqrt{\pi}}{2} D, $$

$$ W_{1}(\mathbf{0})=\frac{\Omega_{\perp}}{2}\left(1+\frac{\eta}{2}\right)+\frac{B^{2}}{8 \Omega_{\perp}}+\sqrt{\frac{\eta \Omega_{\perp}}{\pi}} \frac{1}{\sqrt{1-\eta}} \ln \frac{1-\sqrt{1-\eta}}{1+\sqrt{1-\eta}} $$

$$ W_{1}(\mathbf{0})=\sum_{n=0}^{\infty} \varepsilon_{n} B^{2 n} $$

$$ B\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right) \equiv e^{-\beta H^{\mathrm{eff} \mathrm{cl}}\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right)} \equiv \oint \mathcal{D}^{3} x \oint \frac{\mathcal{D}^{3} p}{(2 \pi \hbar)^{3}} \delta^{(3)}\left(\mathbf{x}_{0}-\overline{\mathbf{x}}\right)(2 \pi \hbar)^{3} \delta\left(\mathbf{p}_{0}-\overline{\mathbf{p}}\right) e^{-\mathcal{A}[\mathbf{p}, \mathbf{x}] / \hbar} $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{p}, \mathbf{x}]=\int_{0}^{\hbar \beta} d \tau[-i \mathbf{p}(\tau) \dot{\mathbf{x}}(\tau)+H(\mathbf{p}(\tau), \mathbf{x}(\tau))] $$

$$ \mathcal{A}_{\mathrm{e}}[\mathbf{p}, \mathbf{x}]=\beta H\left(\mathbf{p}_{0}, \mathbf{x}_{0}\right)+\mathcal{A}_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}[\mathbf{p}, \mathbf{x}]+\mathcal{A}_{\mathrm{int}}[\mathbf{p}, \mathbf{x}], $$

$$ \begin{align*} & \mathcal{A}_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}[\mathbf{p}, \mathbf{x}]=\int_{0}^{\hbar \beta} d \tau\left\{-i\left[\mathbf{p}(\tau)-\mathbf{p}_{0}\right] \cdot \dot{\mathbf{x}}(\tau)+\frac{1}{2 M}\left[\mathbf{p}(\tau)-\mathbf{p}_{0}\right]^{2}\right. \\ & \left.\quad+\Omega_{B} l_{z}\left(\mathbf{p}(\tau)-\mathbf{p}_{0}, \mathbf{x}(\tau)-\mathbf{x}_{0}\right)+\frac{M}{2} \Omega_{\perp}^{2}\left[\mathbf{x}_{\perp}(\tau)-\mathbf{x}_{0 \perp}\right]^{2}+\frac{M}{2} \Omega_{\|}^{2}\left[z(\tau)-z_{0}\right]^{2}\right\} \end{align*} $$

$$ \begin{align*} \langle\mathcal{O}[\mathbf{p}, \mathbf{x}]\rangle_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}} & =\frac{1}{Z_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}} \oint \mathcal{D}^{3} x \frac{\mathcal{D}^{3} p}{(2 \pi \hbar)^{3}} \mathcal{O}[\mathbf{p}, \mathbf{x}] \delta^{(3)}\left(\mathbf{x}_{0}-\overline{\mathbf{x}}\right)(2 \pi \hbar)^{3} \delta\left(\mathbf{p}_{0}-\overline{\mathbf{p}}\right) \\ & \times \exp \left\{-\frac{1}{\hbar} \mathcal{A}_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}[\mathbf{p}, \mathbf{x}]\right\} \end{align*} $$

$$ Z_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}} \equiv e^{-\beta F_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}}=l_{\mathrm{e}}^{3}(\hbar \beta) \frac{\hbar \beta \Omega_{+} / 2}{\sinh \hbar \beta \Omega_{+} / 2} \frac{\hbar \beta \Omega_{-} / 2}{\sinh \hbar \beta \Omega_{-} / 2} \frac{\hbar \beta \Omega_{\|} / 2}{\sinh \hbar \beta \Omega_{\|} / 2} $$

$$ \begin{align*} W_{\boldsymbol{\Omega}}^{(1)}\left(\mathbf{x}_{0}\right) & =F_{\boldsymbol{\Omega}}^{\mathbf{p}_{0}, \mathbf{x}_{0}}-\frac{M}{2} \Omega_{B}\left(\mathbf{x}_{0}\right)\left[\omega_{B}-\Omega_{B}\left(\mathbf{x}_{0}\right)\right] b_{\perp}^{2}\left(\mathbf{x}_{0}\right) a_{\perp}^{2}\left(\mathbf{x}_{0}\right) \\ & +\frac{M}{2}\left[\omega_{B}^{2}-\Omega_{\perp}^{2}\left(\mathbf{x}_{0}\right)\right]-\frac{1}{2} \Omega_{\|}^{2} a_{\|}^{2}\left(\mathbf{x}_{0}\right)-\left\langle\frac{e^{2}}{|\mathbf{x}|}\right\rangle_{\boldsymbol{\Omega}}^{\mathbf{p}_{0}, \mathbf{x}_{0}} \end{align*} $$

$$ a_{\|}^{2}\left(\mathbf{x}_{0}\right)=G_{z z}^{(2)} \mathbf{x}_{0}(\tau, \tau)=\frac{1}{\beta M \Omega_{\|}^{2}\left(\mathbf{x}_{0}\right)}\left[\frac{\hbar \beta \Omega_{\|}\left(\mathbf{x}_{0}\right)}{2} \operatorname{coth} \frac{\hbar \beta \Omega_{\|}\left(\mathbf{x}_{0}\right)}{2}-1\right] $$

$$ b_{\perp}^{2}\left(\mathbf{x}_{0}\right) \equiv \frac{1}{M \Omega_{B}}\left\langle l_{z}\right\rangle_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}} $$

$$ b_{\perp}^{2}\left(\mathbf{x}_{0}\right)=\frac{2}{M \Omega_{T 1}}\left\langle x(\tau) p_{y}(\tau)\right\rangle_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}} $$

$$ \left\langle x(\tau) p_{y}\left(\tau^{\prime}\right)\right\rangle_{\Omega}^{\mathbf{p}_{0}, \mathbf{x}_{0}}=i M \partial_{\tau^{\prime}} G_{\omega^{2}, B, x x}^{(2)}\left(\tau, \tau^{\prime}\right)-M \omega_{B} G_{\omega^{2}, B, x y}^{(2)}\left(\tau, \tau^{\prime}\right) $$

$$ W_{\boldsymbol{\Omega}}^{(1)}\left(\mathbf{x}_{0}\right) \rightarrow F_{\boldsymbol{\Omega}}^{\mathbf{p}_{0}, \mathbf{x}_{0}}-M \Omega_{B}\left(\omega_{\perp}-\Omega_{B}\right) b_{\perp}^{2}+M\left(\omega_{\perp}^{2}-\Omega_{\perp}^{2}\right) a_{\perp}^{2}-\frac{M}{2} \Omega_{\|}^{2} a_{\|}^{2} $$

$$ W^{(1)}\left(\mathbf{x}_{0}\right) \rightarrow-\frac{1}{\beta} \log \frac{\beta \hbar \omega_{B}}{\sinh \beta \hbar \omega_{B}} $$

$$ \mathcal{A}[x]=\int_{0}^{\hbar \beta} d \tau\left[\frac{M}{2}\left(\dot{x}^{2}+\omega^{2} x^{2}\right)+\frac{1}{4} g x^{4}\right] $$

$$ \mathcal{A}_{\Omega}^{x_{0}}=\int_{0}^{\hbar \beta} d \tau M\left[\frac{1}{2} \dot{x}^{2}+\frac{\Omega^{2}}{2}\left(x-x_{0}\right)^{2}\right] $$

$$ \mathcal{A}_{\mathrm{int}}^{x_{0}}=\int_{0}^{\beta} d \tau\left[M \frac{\omega^{2}-\Omega^{2}}{2}\left(x-x_{0}\right)^{2}+\frac{g}{4} x^{4}\right] $$

$$ \begin{align*} Z_{\Omega}^{(n)}\left(x_{0}\right) & \equiv \int d x_{b} d x_{a} \psi_{\Omega}^{(n) *}\left(x_{b}\right)\left(x_{b} \tau_{b} \mid x_{a} \tau_{a}\right) \psi_{\Omega}^{(n)}\left(x_{a}\right) \\ & =\int d x_{b} d x_{a} \psi_{\Omega}^{(n) *}\left(x_{b}\right)\left(\int_{\left(x_{a}, 0\right) \leadsto\left(x_{b}, \hbar \beta\right)} \mathcal{D} x e^{-\mathcal{A} / \hbar}\right) \psi_{\Omega}^{(n)}\left(x_{a}\right) \end{align*} $$

$$ Z^{(n)} \approx e^{-\beta E^{(n)}} $$

$$ Z^{(n)} \approx Z_{\Omega}^{(n)} e^{-\left\langle\mathcal{A}_{\mathrm{int}}^{x_{0}} / \hbar\right\rangle_{\Omega}^{(n)}} $$

$$ Z_{\Omega}^{(n)}=e^{-\beta \hbar \Omega(n+1 / 2)}, $$

$$ \left\langle x^{2 k}\right\rangle_{\Omega}^{(n)}=\frac{\hbar^{k}}{(M \Omega)^{k}} n_{2 k} $$

$$ \begin{align*} & n_{2}=(n+1 / 2), \quad n_{4}=\frac{3}{2}\left(n^{2}+n+1 / 2\right), \quad n_{6}=\frac{5}{4}\left(2 n^{3}+3 n^{2}+4 n+3 / 2\right) \\ & n_{8}=\frac{1}{16}\left(70 n^{4}+140 n^{3}+344 n^{2}+280 n+105\right), \quad \cdots \end{align*} $$

$$ Z^{(n)} \approx \exp \left\{-\beta\left[\frac{1}{2}\left(\Omega^{2}+\omega^{2}\right) \frac{\hbar n_{2}}{\Omega}+\frac{g}{4} \frac{\hbar^{2} n_{4}}{M^{2} \Omega^{2}}\right]\right\} $$

$$ W^{(n)}=\hbar\left[\frac{1}{2}\left(\Omega+\frac{\omega^{2}}{\Omega}\right) n_{2}+\frac{g^{\prime}}{4} \frac{\omega^{3}}{\Omega^{2}} n_{4}\right] $$

$$ \begin{align*} \frac{2}{\sqrt{3}} \omega \cosh \left[\frac{1}{3} \operatorname{arcosh}\left(g / g^{(n)}\right)\right] & \quad \text { for } \quad g>g^{(n)}, \\ \frac{2}{\sqrt{3}} \omega \cos \left[\frac{1}{3} \arccos \left(g / g^{(n)}\right)\right] & \quad g