Kleinert · 제4장 준고전 시간진화 진폭

$$ \tilde{V}(\mathbf{k})=\int d^{D} \eta V(\overline{\mathbf{x}}+\boldsymbol{\eta}) \exp [-i \mathbf{k}(\overline{\mathbf{x}}+\boldsymbol{\eta})] $$

$$ \int_{\tau_{a}}^{\tau_{b}} d \tau\left\langle e^{i \mathbf{k} \boldsymbol{\eta}(\tau)}\right\rangle=\int_{\tau_{a}}^{\tau_{b}} d \tau \int d^{D} \eta V(\overline{\mathbf{x}}+\boldsymbol{\eta}) \int \frac{d^{D} k}{(2 \pi)^{D}} e^{-(1 / 2) k_{i} G_{i j}^{\Delta x}(\tau, \tau) k_{j}-i k_{i} \eta_{i}(\tau)} $$

$$ \begin{array}{rll} V(x)E \\ \frac{2}{\sqrt{k}} \cos \left(\int_{x}^{b} d x^{\prime} k-\frac{\pi}{4}\right) & \longleftrightarrow & \frac{1}{\sqrt{\kappa}} e^{-\int_{b}^{x} d x^{\prime} \kappa} \\ -\frac{1}{\sqrt{k}} \sin \left(\int_{x}^{b} d x^{\prime} k-\frac{\pi}{4}\right) & \longleftrightarrow & \frac{2}{\sqrt{\kappa}} e^{\int_{b}^{x} d x^{\prime} \kappa} \end{array} $$

$$ \int_{\tau_{a}}^{\tau_{b}} d \tau\left\langle e^{i \mathbf{k} \boldsymbol{\eta}(\tau)}\right\rangle=\int d^{D} \eta V(\overline{\mathbf{x}}+\boldsymbol{\eta}) \int_{\tau_{a}}^{\tau_{b}} d \tau\left[\operatorname{det} G_{i j}^{\Delta x}(\tau, \tau)\right]^{-1 / 2} e^{-(1 / 2) \eta_{i}\left[G_{i j}^{\Delta x}(\tau, \tau)\right]^{-1} \eta_{j}} $$

$$ \int_{a}^{b} d x k(x)=(n+1 / 2) \pi, \quad n=0, \pm 1, \pm 2, \ldots $$

$$ P_{i j}^{T}=\delta_{i j}-\frac{\Delta x_{i} \Delta x_{j}}{(\Delta x)^{2}}, \quad P_{i j}^{L}=\frac{\Delta x_{i} \Delta x_{j}}{(\Delta x)^{2}} $$

$$ \oint d x q(x)=2 \pi n \hbar $$

$$ G_{i j}^{\Delta x}\left(\tau, \tau^{\prime}\right) \equiv G\left(\tau, \tau^{\prime}\right) P_{i j}^{T}+\left[G\left(\tau, \tau^{\prime}\right)+H\left(\tau, \tau^{\prime}\right)(\Delta x)^{2}\right] P_{i j}^{L} $$

$$ \oint\left\{\tilde{q}-\frac{\tilde{\hbar}^{2}}{32} \frac{\left(\tilde{q}^{2}\right)^{\prime 2}}{\tilde{q}^{5}}-\frac{\tilde{\hbar}^{4}}{2048}\left[\frac{49\left(\tilde{q}^{2}\right)^{4}}{\tilde{q}^{11}}-\frac{16\left(\tilde{q}^{2}\right)^{\prime}\left(\tilde{q}^{2}\right)^{\prime \prime \prime}}{\tilde{q}^{7}}\right]+\ldots\right\}=2 \pi\left(n+\frac{1}{2}\right) \tilde{\hbar} $$

$$ \operatorname{det} G_{i j}^{\Delta x}\left(\tau, \tau^{\prime}\right)=\left[G\left(\tau, \tau^{\prime}\right)\right]^{D-1}\left[G\left(\tau, \tau^{\prime}\right)+H\left(\tau, \tau^{\prime}\right)(\Delta x)^{2}\right] $$

$$ x_{c}= \pm \sqrt{\frac{2 E}{M \omega^{2}}} $$

$$ \left[G_{i j}^{\Delta x}\left(\tau, \tau^{\prime}\right)\right]^{-1}=\frac{1}{G\left(\tau, \tau^{\prime}\right)}\left[\delta_{i j}-\frac{\Delta x_{i} \Delta x_{j}}{(\Delta x)^{2}}\right]+\frac{1}{G\left(\tau, \tau^{\prime}\right)+H\left(\tau, \tau^{\prime}\right)(\Delta x)^{2}} \frac{\Delta x_{i} \Delta x_{j}}{(\Delta x)^{2}} . $$

$$ k(x)=\frac{p(x)}{\hbar}=\sqrt{\frac{2 M}{\hbar^{2}}\left(E-\frac{1}{2} M \omega^{2} x^{2}\right)}, $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left(\Delta \mathbf{x} / 2 \tau_{b} \mid-\Delta \mathbf{x} / 2 \tau_{a}\right) e^{-\int_{\tau_{a}}^{\tau_{b}} d \tau \bar{V}(\overline{\mathbf{x}}, \tau) / \hbar} $$

$$ \int_{-x_{E}}^{x_{E}} d x^{\prime} k(x)=\frac{E}{\hbar \omega} \pi=\left(n+\frac{1}{2}\right) \pi $$

$$ \bar{V}(\overline{\mathbf{x}}, \tau) \equiv\left[\operatorname{det} G_{i j}^{\Delta x}(\tau, \tau)\right]^{-1 / 2} \int d^{D} \eta V(\overline{\mathbf{x}}+\boldsymbol{\eta}) e^{-(1 / 2) \eta_{i}\left[G_{i j}^{\Delta x}(\tau, \tau)\right]^{-1} \eta_{j}} $$

$$ \frac{1}{\hbar} \int_{-x_{E}}^{x_{E}} d x \sqrt{2 M\left(E-g x^{4} / 4\right)} \equiv \nu(E) \pi=\left(n+\frac{1}{2}\right) \pi $$

$$ -\beta V(\overline{\mathbf{x}})-\frac{1}{2 \hbar} V_{i j}(\overline{\mathbf{x}}) \int_{\tau_{a}}^{\tau_{b}} d \tau G_{i j}^{\Delta x}(\tau, \tau)+\ldots $$

$$ \int_{0}^{1} d t t^{\mu-1}\left(1-t^{\lambda}\right)^{\nu-1}=\frac{1}{\lambda} B(\mu / \lambda, \nu) $$

$$ \int_{\tau_{a}}^{\tau_{b}} d \tau G_{i j}^{\Delta x}(\tau, \tau)=\frac{\hbar}{M} \frac{\Delta \tau^{2}}{6} \delta_{i j}+\frac{\Delta \tau}{12} \Delta x_{i} \Delta x_{j} $$

$$ E_{\mathrm{BS}}^{(n)}=\hbar \omega \kappa_{\mathrm{BS}}^{(n)}\left(\frac{g \hbar}{4 M^{2} \omega^{3}}\right)^{1 / 3} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left(\Delta \mathbf{x} / 2 \tau_{b} \mid-\Delta \mathbf{x} / 2 \tau_{a}\right) \times \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!}\left\langle\prod_{n=0}^{\infty} \int_{\tau_{a}}^{\tau_{b}} d \tau_{n} e^{\eta_{i}\left(\tau_{n}\right) \partial_{i}} V(\overline{\mathbf{x}})\right\rangle $$

$$ \kappa_{\mathrm{BS}}^{(n)}=\frac{1}{\pi^{2 / 3}}\left(\frac{3}{2}\right)^{4 / 3} \Gamma^{8 / 3}(3 / 4)\left(n+\frac{1}{2}\right)^{4 / 3} \approx 0.688253702 \times 2\left(n+\frac{1}{2}\right)^{4 / 3} $$

$$ \begin{align*} \left\langle e^{\eta(\tau) \partial_{i}}\right\rangle & =e^{\left\langle\eta_{i}(\tau) \eta_{j}(\tau)\right\rangle / 2}=e^{G_{i j}^{\Delta x}(\tau, \tau) \partial_{i} \partial_{j} / 2} \\ \left\langle e^{\eta_{i}(\tau) \partial_{i}} e^{\eta_{i}\left(\tau^{\prime}\right) \partial_{i}}\right\rangle & =e^{\left[\left\langle\eta_{i}(\tau) \eta_{j}(\tau)\right\rangle \partial_{i} \partial_{j}+\left\langle\eta_{i}\left(\tau^{\prime}\right) \eta_{j}\left(\tau^{\prime}\right)\right\rangle \partial_{i} \partial_{j}+2\left\langle\eta_{i}(\tau) \eta_{j}\left(\tau^{\prime}\right)\right\rangle \partial_{i} \partial_{j}\right] / 2} \\ & =e^{\left[G_{i j}^{\Delta x}(\tau, \tau) \partial_{i} \partial_{j}+G_{i j}^{\Delta x}\left(\tau^{\prime}, \tau^{\prime}\right) \partial_{i} \partial_{j}+2 G_{i j}^{\Delta x}\left(\tau, \tau^{\prime}\right) \partial_{i} \partial_{j}\right] / 2} \\ & \vdots \end{align*} $$

$$ \left\{-\frac{\hbar^{2}}{2 M} \partial_{r}^{2}-\left[E-V(r)-V_{\mathrm{cf}}(r)\right]\right\} R(r)=0 $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left\langle\mathbf{x}_{b}\right| e^{-\Delta \tau H\left(\hat{p}^{2}, \hat{\mathbf{x}}\right) / \hbar}\left|\mathbf{x}_{a}\right\rangle=\left\langle\mathbf{x}_{b}\right| e^{-\Delta \tau\left[\hat{p}^{2} / 2 M+V(\hat{\mathbf{x}})\right] / \hbar}\left|\mathbf{x}_{a}\right\rangle $$

$$ \left[-\frac{\hbar^{2}}{2 M} \partial_{\xi}^{2}-q^{2}(\xi)\right] \chi(\xi)=0 $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left\langle\mathbf{x}_{b}\right| e^{-\Delta \tau \hat{p}^{2} / 2 M \hbar} \hat{T} \exp \left\{-\int_{0}^{\Delta \tau} d \tau e^{\tau \hat{p}^{2} / 2 M \hbar} V(\mathbf{x}) e^{-\tau \hat{p}^{2} / 2 M \hbar}\right\}\left|\mathbf{x}_{a}\right\rangle $$

$$ \tilde{q}^{2}(\xi) \equiv e^{2 \xi}\left[E-V\left(e^{\xi}\right)-V_{\mathrm{cf}}\left(e^{\xi}\right)-\frac{\hbar^{2} e^{-2 \xi}}{8 M}\right]=e^{2 \xi}\left[E-V\left(e^{\xi}\right)-\frac{\left(l+\frac{1}{2}\right)^{2} \hbar^{2} e^{-2 \xi}}{2 M}\right] . $$

$$ e^{\tau \hat{p}^{2} / 2 M \hbar} V(\hat{\mathbf{x}}) e^{-\tau \hat{p}^{\prime 2} / 2 M \hbar} \equiv V\left(\hat{\mathbf{x}}_{H}(\tau)\right)=V(\hat{\mathbf{x}}-i \hat{\mathbf{p}} \tau / M) $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x e^{i \mathcal{A}[x] / \hbar} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\left\langle\mathbf{x}_{b}\right| e^{-\Delta \tau \hat{p}^{2} / 2 M \hbar} \hat{T} \exp \left\{-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau \Delta V(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau)\right\}\left|\mathbf{x}_{a}\right\rangle e^{-\Delta \tau V\left(\mathbf{x}_{a}\right) / \hbar} $$

$$ \int \frac{d x}{\sqrt{2 \pi i \hbar}} e^{i a(x) / \hbar} $$

$$ \Delta V(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau) \equiv V(\hat{\mathbf{x}}-i \hat{\mathbf{p}} \tau / M)-V(\hat{\mathbf{x}}) $$

$$ a^{\prime}\left(x_{\mathrm{cl}}\right)=0 $$

$$ \Delta V(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau)=\int \frac{d^{D} k}{(2 \pi)^{D}} \tilde{V}(\mathbf{k})\left(e^{i \mathbf{k}(\hat{\mathbf{x}}-i \hat{\mathbf{p}} \tau / M)}-e^{i \mathbf{k} \hat{\mathbf{x}}}\right) $$

$$ \int_{-\infty}^{\infty} \frac{d x}{\sqrt{2 \pi i \hbar}} e^{i a(x) / \hbar} \xrightarrow{\hbar \rightarrow 0} \text { const } \times e^{i a\left(x_{\mathrm{cl}}\right) / \hbar} $$

$$ \Delta V(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau)=\int d^{D} x^{\prime} V\left(\mathbf{x}^{\prime}\right) S\left(\hat{\mathbf{p}} \tau, \hat{\mathbf{x}}, \mathbf{x}^{\prime}\right) $$

$$ a(x)=a\left(x_{\mathrm{cl}}\right)+\frac{1}{2} a^{\prime \prime}\left(x_{\mathrm{cl}}\right)(\delta x)^{2}+\frac{1}{3!} a^{(3)}\left(x_{\mathrm{cl}}\right)(\delta x)^{3}+\ldots, $$

$$ S\left(\hat{\mathbf{p}} \tau, \hat{\mathbf{x}}, \mathbf{x}^{\prime} ; \tau\right) \equiv \int \frac{d^{D} k}{(2 \pi)^{D}}\left(e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}-i \hat{\mathbf{p}} \tau / M\right)}-e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)}\right) $$

$$ \int_{-\infty}^{\infty} \frac{d x}{\sqrt{2 \pi i \hbar}} e^{i a(x) / \hbar} \rightarrow e^{i a\left(x_{\mathrm{cl}}\right) / \hbar} \int_{-\infty}^{\infty} \frac{d \delta x}{\sqrt{2 \pi i \hbar}} e^{i a^{\prime \prime}\left(x_{\mathrm{cl}}\right)(\delta x)^{2} / 2 \hbar}=\frac{e^{i a\left(x_{\mathrm{cl}}\right) / \hbar}}{\sqrt{a^{\prime \prime}\left(x_{\mathrm{cl}}\right)}} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right) \equiv \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{i \mathbf{p} \cdot \mathbf{x}_{b} / \hbar} e^{-\Delta \tau H\left(p, \mathbf{x}_{a}\right) \hbar}\langle\mathbf{p}| \hat{T} \exp \left\{-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau \Delta V(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau)\right\}\left|\mathbf{x}_{a}\right\rangle $$

$$ e^{i a(x) / \hbar} \xrightarrow{\hbar \rightarrow 0} \frac{\sqrt{2 \pi i \hbar}}{\sqrt{a^{\prime \prime}\left(x_{\mathrm{cl}}\right)}} \delta\left(x-x_{\mathrm{cl}}\right) . $$

$$ \begin{align*} \langle\mathbf{p}| S\left(\hat{\mathbf{p}} \tau, \hat{\mathbf{x}}, \mathbf{x}^{\prime}\right)\left|\mathbf{x}_{a}\right\rangle & =\int \frac{d^{D} k}{(2 \pi)^{D}}\langle\mathbf{p}|\left(e^{\mathbf{k} \hat{\mathbf{p}} \tau / M} e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)} e^{-k^{2} \tau \hbar / 2 M}-e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)}\right)\left|\mathbf{x}_{a}\right\rangle \\ & =\int \frac{d^{D} k}{(2 \pi)^{D}}\left(e^{i \mathbf{k}\left(\mathbf{x}_{a}-\mathbf{x}^{\prime}-i \mathbf{p} \tau / M\right)} e^{-k^{2} \tau \hbar / 2 M}-e^{i \mathbf{k}\left(\mathbf{x}_{a}-\mathbf{x}^{\prime}\right)}\right) e^{-i \mathbf{p} \mathbf{x}_{a}} \end{align*} $$

$$ \begin{gather*} \exp \left\{\frac{i}{\hbar}\left[\frac{1}{3!} a^{(3)}\left(x_{\mathrm{cl}}\right)(\delta x)^{3}+\frac{1}{4!} a^{(4)}\left(x_{\mathrm{cl}}\right)(\delta x)^{4}+\ldots\right]\right\} \\ =1+\frac{i}{\hbar}\left[\frac{1}{3!} a^{(3)}\left(x_{\mathrm{cl}}\right)(\delta x)^{3}+\frac{1}{4!} a^{(4)}\left(x_{\mathrm{cl}}\right)(\delta x)^{4}+\ldots\right] \\ -\frac{1}{\hbar^{2}}\left[\frac{1}{72} a^{(3)}\left(x_{\mathrm{cl}}\right)^{2}(\delta x)^{6}+\ldots\right]+\ldots \end{gather*} $$

$$ \delta_{\tau}(\mathbf{x}) \equiv \int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k x}} e^{-k^{2} \tau \hbar / 2 M} $$

$$ \int_{-\infty}^{\infty} \frac{d \delta x}{\sqrt{2 \pi i \hbar}} e^{i a^{\prime \prime}\left(x_{\mathrm{cl}}\right)(\delta x)^{2} / 2 \hbar}(\delta x)^{n}=\left\{\begin{array}{cc} \frac{(n-1)!!}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{(1+n) / 2}}(i \hbar)^{n / 2}, & n=\text { even } \\ 0, & n=\text { odd } \end{array}\right. $$

$$ \int d^{D} x \delta_{\tau}(\mathbf{x})=1, \quad \int d^{D} x x_{i} x_{j} \delta_{\tau}(\mathbf{x})=\tau \frac{\hbar}{M} \delta_{i j}, \ldots $$

$$ 1-i a^{(4)}\left(x_{\mathrm{cl}}\right) \frac{3!!}{4!} \frac{\hbar}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{2}} $$

$$ V_{\tau}(\mathbf{x}) \equiv \int d^{D} x^{\prime} \delta_{\tau}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) V\left(\mathbf{x}^{\prime}\right)=\frac{1}{(2 \pi \tau \hbar / M)^{D / 2}} \int d^{D} x^{\prime} V\left(\mathbf{x}-\mathbf{x}^{\prime}\right) e^{-x^{\prime 2} M / 2 \tau \hbar} $$

$$ 1+i\left[a^{(3)}\left(x_{\mathrm{cl}}\right)\right]^{2} \frac{5!!}{72} \frac{\hbar}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{3}} $$

$$ V_{\tau}(\mathbf{x}) \equiv V_{\tau}(\mathbf{x})+\frac{\tau}{2} \frac{\hbar}{M} \nabla^{2} V(\mathbf{x})+\frac{\tau^{2}}{8}\left(\frac{\hbar}{M}\right)^{2} \nabla^{4} V(\mathbf{x})+\ldots $$

$$ \int_{-\infty}^{\infty} \frac{d x}{\sqrt{2 \pi i \hbar}} e^{i a(x) / \hbar}=\frac{e^{i a\left(x_{\mathrm{cl}}\right) / \hbar}}{\sqrt{a^{\prime \prime}\left(x_{\mathrm{cl}}\right)}}\left\{1-i \hbar\left[\frac{1}{8} \frac{a^{(4)}\left(x_{\mathrm{cl}}\right)}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{2}}-\frac{5}{24} \frac{\left[a^{(3)}\left(x_{\mathrm{cl}}\right)\right]^{2}}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{3}}\right]+\mathcal{O}\left(\hbar^{2}\right)\right\} . $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{i \mathbf{p} \cdot \Delta \mathbf{x} / \hbar} e^{-\Delta \tau H\left(p, \mathbf{x}_{a}\right) / \hbar}\left\{1-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau W_{\mathbf{p} \tau}+\ldots\right\} $$

$$ \begin{align*} \langle\delta x\rangle & =-i \hbar \frac{1}{2} \frac{a^{(3)}\left(x_{\mathrm{cl}}\right)}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{2}} \\ & -\hbar^{2}\left[\frac{2}{3} \frac{a^{(3)}\left(x_{\mathrm{cl}}\right) a^{(4)}\left(x_{\mathrm{cl}}\right)}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{4}}-\frac{5}{8} \frac{\left[a^{(3)}\left(x_{\mathrm{cl}}\right)\right]^{3}}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{5}}-\frac{1}{8} \frac{a^{(5)}\left(x_{\mathrm{cl}}\right)}{\left[a^{\prime \prime}\left(x_{\mathrm{cl}}\right)\right]^{3}}\right]+\mathcal{O}\left(\hbar^{3}\right) \end{align*} $$

$$ W_{\mathbf{p} \tau} \equiv\langle\mathbf{p}| V_{\tau}(\hat{\mathbf{x}}-i \hat{\mathbf{p}} \tau / M)-V(\hat{\mathbf{x}})\left|\mathbf{x}_{a}\right\rangle $$

$$ \int_{-\infty}^{\infty} d x e^{-i p x / \hbar} e^{i a(x) / \hbar} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=e^{-M \Delta \mathbf{x}^{2} / 2 \Delta \tau \hbar} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-\Delta \tau H\left(p, \mathbf{x}_{a}\right) / \hbar}\left\{1-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau W_{\mathbf{p} \tau+i M \Delta \mathbf{x}}+\ldots\right\} $$

$$ \int_{-\infty}^{\infty} d x e^{-i p x / \hbar} e^{i a(x) / \hbar} \rightarrow \sqrt{2 \pi i \hbar} \frac{e^{i\left[a\left(x_{\mathrm{cl}}\right)-p x_{\mathrm{cl}}\right] / \hbar}}{\sqrt{a^{\prime \prime}\left(x_{\mathrm{cl}}\right)}} $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{e^{-M \Delta \mathbf{x}^{2} / 2 \Delta \tau \hbar}}{(2 \pi \Delta \tau \hbar / M)^{D / 2}} e^{-\Delta \tau V\left(\mathbf{x}_{a}\right) / \hbar}\left\langle 1-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau W_{\mathbf{p} \tau+i M \Delta \mathbf{x}}+\ldots\right\rangle_{p} $$

$$ \int_{-\infty}^{\infty} d x e^{-i p x / \hbar} c(x) e^{i a(x) / \hbar} \rightarrow \sqrt{2 \pi i \hbar} c\left(x_{\mathrm{cl}}\right) \frac{e^{i\left[a\left(x_{\mathrm{cl}}\right)-p x_{\mathrm{cl}}\right] / \hbar}}{\sqrt{a^{\prime \prime}\left(x_{\mathrm{cl}}\right)}} $$

$$ \langle f(\mathbf{p})\rangle_{p} \equiv \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} f(\mathbf{p}) e^{-\Delta \tau p^{2} / 2 M \hbar} / \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-\Delta \tau p^{2} / 2 M \hbar} $$

$$ b(p)=a\left(x_{\mathrm{cl}}\right)-p x_{\mathrm{cl}}, \quad p=a^{\prime}\left(x_{\mathrm{cl}}\right) . $$

$$ \Delta V_{\tau}(\mathbf{x}) \approx V\left(\mathbf{x}_{b}\right)-V\left(\mathbf{x}_{a}\right)+\frac{\tau}{2} \frac{\hbar}{M} \nabla^{2} V\left(\mathbf{x}_{b}\right)-i \frac{\tau}{M} \mathbf{p} \nabla V\left(\mathbf{x}_{b}\right)-\frac{\tau^{2}}{2!M^{2}}(\mathbf{p} \nabla)^{2} V\left(\mathbf{x}_{b}\right) $$

$$ a(x)=b\left(p_{\mathrm{cl}}\right)+x p_{\mathrm{cl}}, \quad x=-b^{\prime}\left(p_{\mathrm{cl}}\right) . $$

$$ \begin{align*} & \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-\Delta \tau p^{2} / 2 M \hbar}\left\{1, p^{i} p^{j}, p^{i_{1}} p^{i_{2}} p^{i_{3}} p^{4}, \ldots \ldots, p^{i_{1}} p^{i_{2}} \cdots p^{i_{2 n-1}} p^{i_{2 n}}, \ldots\right\} \\ & \quad=\frac{1}{(2 \pi \Delta \tau \hbar / M)^{D / 2}}\left\{1, \frac{M \hbar}{\Delta \tau} \delta^{i j},\left(\frac{M \hbar}{\Delta \tau}\right)^{2} \delta^{i_{1} i_{2} i_{3} i_{4}}, \ldots,\left(\frac{M \hbar}{\Delta \tau}\right)^{n} \delta^{i_{1} i_{2} \ldots i_{2 n-1} i_{2 n}}, \ldots\right\}, \end{align*} $$

$$ \delta \mathcal{A}[x]=0 $$

$$ \left(\mathbf{x}_{b} \tau_{b} \mid \mathbf{x}_{a} \tau_{a}\right)=\frac{e^{-M \Delta \mathbf{x}^{2} / 2 \Delta \tau \hbar}}{(2 \pi \Delta \tau \hbar / M)^{D / 2}} e^{-\Delta \tau V\left(\mathbf{x}_{a}\right) / \hbar}\left(1+A_{1}+A_{2}+\ldots\right), $$

$$ \mathcal{A}[x]=\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{x}^{2}-V(x)\right] $$

$$ A_{1}=-\frac{1}{\hbar} \int_{0}^{\Delta \tau} d \tau\left[V\left(\mathbf{x}_{b}\right)-V\left(\mathbf{x}_{a}\right)+\left(\frac{\tau}{2}-\frac{\tau^{2}}{2 \Delta \tau}\right) \frac{\hbar}{M} \nabla^{2} V\left(\mathbf{x}_{b}\right)+\ldots\right] $$

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

$$ \begin{align*} & \langle\mathbf{p}| S\left(\hat{\mathbf{p}} \tau, \hat{\mathbf{x}}, \mathbf{x}^{\prime}\right) S\left(\hat{\mathbf{p}} \tau^{\prime}, \hat{\mathbf{x}}, \mathbf{x}^{\prime \prime}\right)\left|\mathbf{x}_{a}\right\rangle=\int \frac{d^{D} k}{(2 \pi)^{D}} \int \frac{d^{D} k^{\prime}}{(2 \pi)^{D}} \\ & \times\langle\mathbf{p}|\left(e^{\mathbf{k} \hat{\mathbf{p}} \tau / M} e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)} e^{-k^{2} \tau \hbar / 2 M}-e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)}\right)\left(e^{\mathbf{k}^{\prime} \hat{\mathbf{p}} \tau^{\prime} / M} e^{i \mathbf{k}^{\prime}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime \prime}\right)} e^{-k^{\prime 2} \tau \hbar / 2 M}-e^{i \mathbf{k}^{\prime}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime \prime}\right)}\right)\left|\mathbf{x}_{a}\right\rangle \end{align*} $$

$$ E=\frac{M}{2} \dot{x}_{\mathrm{cl}}^{2}+V\left(x_{\mathrm{cl}}\right)=\mathrm{const} . $$

$$ \begin{align*} & \langle\mathbf{p}| e^{\mathbf{k} \hat{\mathbf{p}} \tau / M} e^{i \mathbf{k}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)} e^{-k^{2} \tau \hbar / 2 M} e^{\mathbf{k}^{\prime} \hat{\mathbf{p}} \tau^{\prime} / M} e^{i \mathbf{k}^{\prime}\left(\hat{\mathbf{x}}-\mathbf{x}^{\prime}\right)} e^{-k^{\prime 2} \tau \hbar / 2 M}\left|\mathbf{x}_{a}\right\rangle \\ & \quad=e^{-\mathbf{k} \mathbf{k}^{\prime} \tau^{\prime} \hbar / M}\langle\mathbf{p}| e^{\mathbf{k} \mathbf{p} \tau / M} e^{i \mathbf{k}\left(\mathbf{x}_{a}-\mathbf{x}^{\prime \prime}\right)} e^{-k^{2} \tau \hbar / 2 M} e^{\mathbf{k}^{\prime} \mathbf{p} \tau^{\prime} / M} e^{i \mathbf{k}^{\prime}\left(\mathbf{x}_{a}-\mathbf{x}^{\prime \prime}\right)} e^{-k^{\prime 2} \tau \hbar / 2 M}\left|\mathbf{x}_{a}\right\rangle \end{align*} $$

$$ p_{\mathrm{cl}}(t) \equiv M \dot{x}_{\mathrm{cl}}(t) $$

$$ e^{-\mathbf{k k}^{\prime} \tau^{\prime} \hbar / M}=e^{\left(\tau^{\prime} \hbar / M\right) \nabla^{\prime} \nabla^{\prime \prime}} $$

$$ p_{\mathrm{cl}}(t)=p\left(x_{\mathrm{cl}}(t)\right) $$

$$ A_{2}=\left\langle\frac{1}{\hbar^{2}} \int_{0}^{\Delta \tau} d \tau \int_{0}^{\tau} d \tau^{\prime} W^{(2)}(\mathbf{p})\right\rangle_{p} $$

$$ t-t_{0}=\int_{0}^{x_{\mathrm{cl}}} d x \frac{M}{p(x)}=\int_{0}^{x_{\mathrm{cl}}} d x \frac{M}{\sqrt{2 M[E-V(x)]}} $$

$$ \begin{align*} W^{(2)}(\mathbf{p}) & \left.\equiv\left(e^{\left(\tau^{\prime} \hbar / M\right) \boldsymbol{\nabla}^{\prime} \nabla^{\prime \prime}}-1\right) V_{\tau}\left(\mathbf{x}^{\prime}-i \mathbf{p} \tau / M\right) V_{\tau^{\prime}}\left(\mathbf{x}^{\prime \prime}-i \mathbf{p} \tau^{\prime} / M\right)\right|_{\mathbf{x}^{\prime}=\mathbf{x}_{a}, \mathbf{x}^{\prime \prime}=\mathbf{x}_{a}} \\ + & {\left[V_{\tau}\left(\mathbf{x}_{b}-i \mathbf{p} \tau / M\right)-V\left(\mathbf{x}_{a}\right)\right]\left[V_{\tau^{\prime}}\left(\mathbf{x}_{b}-i \mathbf{p} \tau^{\prime} / M\right)-V\left(\mathbf{x}_{a}\right)\right] } \end{align*} $$

$$ E=E\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) $$

$$ \operatorname{Tr} \log \hat{H}=-\int d^{D} x \int_{0}^{\infty} \frac{d \tau}{\tau}\langle\mathbf{x}| e^{-\tau H\left(\hat{p}, \mathbf{x}_{a}\right) / \hbar} \hat{T} \exp \left\{-\frac{1}{\hbar} \int_{0}^{\tau} d \tau^{\prime} \Delta V\left(\hat{\mathbf{x}}, \hat{\mathbf{p}}, \tau^{\prime}\right)\right\}|\mathbf{x}\rangle $$

$$ \begin{align*} \mathcal{A}\left[x_{\mathrm{cl}}\right] & =\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2} \dot{x_{\mathrm{cl}}^{2}}-V\left(x_{\mathrm{cl}}\right)\right] \\ & =\int_{t_{a}}^{t_{b}} d t\left[p_{\mathrm{cl}}(t) \dot{x}_{\mathrm{cl}}-H\left(p_{\mathrm{cl}}, x_{\mathrm{cl}}\right)\right] \\ & =\int_{x_{a}}^{x_{b}} d x p(x)-\left(t_{b}-t_{a}\right) E \end{align*} $$

$$ A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) \equiv \int_{x_{a}}^{x_{b}} d x p(x)-\left(t_{b}-t_{a}\right) E\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) $$

$$ S\left(x_{b}, x_{a} ; E\right)=\int_{x_{a}}^{x_{b}} d x p(x) $$

$$ S_{E}[x] \equiv \int p(x) d x=\int d t p(x) \dot{x}=\int d t L_{E}(x, \dot{x})=\int d t \sqrt{2 M[E-V(x)]} \dot{x} $$

$$ \frac{d}{d t} \frac{\partial L_{E}}{\partial \dot{x}}=\frac{\partial L_{E}}{\partial x} $$

$$ S_{E}[\mathbf{x}]=\int d t L_{E}(\mathbf{x}, \dot{\mathbf{x}})=\int d t \sqrt{g_{i j}(\mathbf{x}) \dot{x}^{i}(t) \dot{x}^{j}(t)} $$

$$ g_{i j}(\mathbf{x})=p_{E}^{2}(\mathbf{x}) \delta_{i j} $$

$$ S_{E}[\mathbf{x}]=\int d \tau \sqrt{g_{i j}(\mathbf{x}) \dot{x}^{i}(\tau) \dot{x}^{j}(\tau)} $$

$$ E=\frac{M}{2} \dot{\mathbf{x}}^{2}+V(\mathbf{x}) . $$

$$ \dot{\mathbf{x}} \delta \dot{\mathbf{x}}=-\frac{1}{M} \nabla V(\mathbf{x}) \delta \mathbf{x} $$

$$ \delta \dot{\mathbf{x}}=\frac{d \mathbf{x}+d \delta \mathbf{x}}{d t+d \delta t}-\dot{\mathbf{x}}=\frac{d}{d t} \delta \mathbf{x}-\dot{\mathbf{x}} \frac{d}{d t} \delta t, $$

$$ \dot{\mathbf{x}}^{2} \frac{d}{d t} \delta t=\dot{\mathbf{x}} \frac{d}{d t} \delta \mathbf{x}+\frac{1}{M} \nabla V(\mathbf{x}) \delta \mathbf{x} . $$

$$ \delta S_{E}[\mathbf{x}]=\int d t\left(\frac{\partial L_{E}}{\partial \dot{\mathbf{x}}} \frac{d}{d t} \delta \mathbf{x}+\frac{\partial L_{E}}{\partial \mathbf{x}} \delta \mathbf{x}\right)+\int d t\left(L_{E}-\frac{\partial L_{E}}{\partial \dot{\mathbf{x}}} \dot{\mathbf{x}}\right) \frac{d}{d t} \delta t $$

$$ -L_{E}+\frac{\partial L_{E}}{\partial \dot{\mathbf{x}}} \dot{\mathbf{x}} \equiv H_{E} $$

$$ \frac{d}{d t} \frac{\partial L_{E}}{\partial \dot{\mathbf{x}}}=\frac{\partial L_{E}}{\partial \mathbf{x}} $$

$$ \begin{align*} \delta S_{E}[\mathbf{x}] & =\int d t\left[\frac{\partial L_{E}}{\partial \dot{\mathbf{x}}}-H_{E} \frac{\dot{\mathbf{x}}}{\dot{\mathbf{x}}^{2}}\right] \frac{d}{d t} \delta \mathbf{x} \\ & +\int d t\left[\frac{\partial L_{E}}{\partial \mathbf{x}}-H_{E} \frac{1}{\dot{\mathbf{x}}^{2}} \frac{1}{M} \nabla V(\mathbf{x})\right] \delta \mathbf{x} \end{align*} $$

$$ \frac{d}{d t}\left[\frac{\partial L_{E}}{\partial \dot{\mathbf{x}}}-H_{E} \frac{\dot{\mathbf{x}}}{\dot{\mathbf{x}}^{2}}\right]=\frac{\partial L_{E}}{\partial \mathbf{x}}-H_{E} \frac{1}{\dot{\mathbf{x}}^{2}} \frac{1}{M} \nabla V(\mathbf{x}) $$

$$ S_{E}[\mathbf{x}]=\int d t L_{E}^{\prime}(\mathbf{x}, \dot{\mathbf{x}})=M \int d t \dot{\mathbf{x}}^{2}(t) $$

$$ M \ddot{\mathrm{x}}=-\nabla V(\mathrm{x}) . $$

$$ \delta S_{E}[\mathbf{x}]=M \int \delta d t \dot{\mathbf{x}}^{2}+M \int d t \dot{\mathbf{x}} \delta \dot{\mathbf{x}}+M \int d t \dot{\mathbf{x}} \delta \dot{\mathbf{x}} $$

$$ \delta S_{E}[\mathbf{x}]=M\left[\int \delta d t \dot{\mathbf{x}}^{2}+\int d t \dot{\mathbf{x}} \delta \dot{\mathbf{x}}+\int d t \dot{\mathbf{x}} \frac{d}{d t} \delta \mathbf{x}-\int d t \dot{\mathbf{x}}^{2} \frac{d}{d t} \delta t\right] $$

$$ A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) \equiv S\left(x_{b}, x_{a} ; E\right)-\left(t_{b}-t_{a}\right) E $$

$$ \frac{\partial}{\partial x_{b, a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)= \pm p\left(x_{b, a}\right) . $$

$$ \frac{\partial A}{\partial x_{b}}=p\left(x_{b}\right)+\left[\int_{x_{a}}^{x_{b}} d x \frac{\partial p(x)}{\partial E}-\left(t_{b}-t_{a}\right)\right] \frac{\partial E}{\partial x_{b}} $$

$$ \frac{\partial p(x)}{\partial E}=\frac{M}{p(x)}=\frac{1}{\dot{x}} $$

$$ \int_{x_{a}}^{x_{b}} d x \frac{\partial p(x)}{\partial E}=\int_{t_{a}}^{t_{b}} d t=t_{b}-t_{a} $$

$$ \frac{\partial}{\partial E} S\left(x_{b}, x_{a} ; E\right)=t_{b}-t_{a} $$

$$ \frac{\partial}{\partial t_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=-E\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) . $$

$$ \frac{\partial}{\partial t_{b}} A=\left[\int_{x_{a}}^{x_{b}} d x \frac{\partial p}{\partial E}-\left(t_{b}-t_{a}\right)\right] \frac{\partial E}{\partial t_{b}}-E=-E $$

$$ \frac{1}{2 M}\left(\partial_{x} A\right)^{2}+V(x)=\partial_{t} A $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right) \xrightarrow{\hbar \rightarrow 0} \text { const } \times e^{i A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) / \hbar} . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{i A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) / \hbar} F\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) . $$

$$ \delta x(t)=x(t)-x_{c l}(t) $$

$$ \begin{align*} \mathcal{A}[x, \dot{x}]= & \mathcal{A}\left[x_{\mathrm{cl}}\right]+\int_{t_{a}}^{t_{b}} d t \frac{\delta \mathcal{A}}{\delta x(t)} \delta x(t) \\ & +\frac{1}{2} \int_{t_{a}}^{t_{b}} d t d t^{\prime} \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)} \delta x(t) \delta x\left(t^{\prime}\right) \\ & +\frac{1}{3!} \int_{t_{a}}^{t_{b}} d t d t^{\prime} d t^{\prime \prime} \frac{\delta^{3} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right) \delta x\left(t^{\prime \prime}\right)} \delta x(t) \delta x\left(t^{\prime}\right) \delta x\left(t^{\prime \prime}\right)+\ldots, \end{align*} $$

$$ \frac{1}{2} \int_{t_{a}}^{t_{b}} d t d t^{\prime} \frac{\delta^{2} \mathcal{A}}{\delta x(t) \delta x\left(t^{\prime}\right)} \delta x(t) \delta x\left(t^{\prime}\right)=\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}(\delta \dot{x})^{2}+\frac{1}{2} V^{\prime \prime}\left(x_{\mathrm{cl}}(t)\right)(\delta x)^{2}\right] $$

$$ \Omega^{2}(t)=\frac{1}{M} V^{\prime \prime}\left(x_{\mathrm{cl}}(t)\right) . $$

$$ \delta x\left(t_{a}\right)=0, \quad \delta x\left(t_{b}\right)=0 . $$

$$ \left(x_{b} t_{b} \mid x_{a} t_{a}\right)_{\mathrm{sc}}=e^{i A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) / \hbar} F_{\mathrm{sc}}\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) $$

$$ \begin{align*} F_{\mathrm{sc}}\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) & =\int \mathcal{D} \delta x(t) \exp \left\{\frac{i}{\hbar} \int_{t_{b}}^{t_{a}} d t \frac{M}{2}\left[\delta \dot{x}^{2}-\Omega^{2}(t) \delta x^{2}\right]\right\} \\ & =\frac{1}{\sqrt{2 \pi i \epsilon \hbar / M}} \operatorname{det}\left(-\bar{\nabla} \nabla-\Omega^{2}(t)\right)^{-1 / 2} \\ & =\frac{1}{\sqrt{2 \pi i \hbar\left(t_{b}-t_{a}\right) / M}} \sqrt{\frac{\operatorname{det}\left(-\partial_{t}^{2}\right)}{\operatorname{det}\left(-\partial_{t}^{2}-\Omega^{2}(t)\right)}} \end{align*} $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] y_{n}(t)=\left[-\partial_{t}^{2}-V^{\prime \prime}\left(x_{\mathrm{cl}}(t)\right) / M\right] y_{n}(t)=\lambda_{n} y_{n}(t) $$

$$ \frac{D^{0}}{D}=\frac{\operatorname{det}\left(-\partial_{t}^{2}\right)}{\operatorname{det}\left(-\partial_{t}^{2}-\Omega^{2}(t)\right)} $$

$$ -\partial_{t}^{2} y_{n}(t)=\lambda_{n}^{0} y_{n}(t) $$

$$ D_{a}\left(t_{a}\right)=0, \quad \dot{D}_{a}\left(t_{a}\right)=1 $$

$$ \left[-\partial_{t}^{2}-\Omega^{2}(t)\right] \xi(t)=0 $$

$$ D_{\mathrm{ren}}=\xi(t) \xi\left(t_{a}\right) \int_{t_{a}}^{t_{b}} \frac{d t^{\prime}}{\xi^{2}\left(t^{\prime}\right)} $$

$$ \xi(t)=\dot{x}_{\mathrm{cl}}(t) $$

$$ \partial_{t}\left[M \ddot{x}_{\mathrm{cl}}+V^{\prime}\left(x_{\mathrm{cl}}(t)\right)\right]=\left[M \partial_{t}^{2}+V^{\prime \prime}\left(x_{\mathrm{cl}}(t)\right)\right] \dot{x}_{\mathrm{cl}}(t)=0 $$

$$ D_{\mathrm{ren}}=\dot{x}_{\mathrm{cl}}\left(t_{b}\right) \dot{x}_{\mathrm{cl}}\left(t_{a}\right) \int_{t_{a}}^{t_{b}} \frac{d t}{\dot{x}_{\mathrm{cl}}^{2}(t)} $$

$$ D_{a}(t)=\dot{x}_{\mathrm{cl}}(t) \dot{x}_{\mathrm{cl}}\left(t_{a}\right) \int_{t_{a}}^{t} \frac{d t}{\dot{x}_{\mathrm{cl}}^{2}(t)}, \quad D_{b}(t)=\dot{x}_{\mathrm{cl}}\left(t_{b}\right) \dot{x}_{\mathrm{cl}}(t) \int_{t}^{t_{b}} \frac{d t}{\dot{x}_{\mathrm{cl}}^{2}(t)} $$

$$ D_{\mathrm{ren}}=-\left(\frac{\partial \dot{x}_{b}}{\partial x_{a}}\right)^{-1}=-M\left[\frac{\partial^{2}}{\partial x_{b} \partial x_{a}} \mathcal{A}_{\mathrm{cl}}\right]^{-1} $$

$$ \mathcal{A}_{\mathrm{qu}}[x, \dot{x}]=\mathcal{A}\left[x_{\mathrm{cl}}\right]+\int_{t_{a}}^{t_{b}} d t\left[\frac{M}{2}(\delta \dot{x})^{2}+\Omega^{2}(t)(\delta x)^{2}\right] $$

$$ \frac{\partial}{\partial x_{b}} \frac{\partial}{\partial x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{\partial}{\partial x_{a}} p\left(x_{b}\right)=\frac{M}{p\left(x_{b}\right)} \frac{\partial E}{\partial x_{a}} . $$

$$ \begin{align*} \frac{\partial E}{\partial x_{a}} & =-\frac{\partial}{\partial x_{a}} \frac{\partial A}{\partial t_{b}}=-\frac{\partial}{\partial t_{b}} \frac{\partial A}{\partial x_{a}} \\ & =\frac{\partial}{\partial t_{b}} p\left(x_{a}\right)=\frac{M}{p\left(x_{a}\right)} \frac{\partial E}{\partial t_{b}} \end{align*} $$

$$ \frac{\partial}{\partial x_{b}} \frac{\partial}{\partial x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{1}{\dot{x}\left(t_{b}\right) \dot{x}\left(t_{a}\right)} \frac{\partial E}{\partial t_{b}} . $$

$$ \begin{align*} \frac{\partial E}{\partial t_{b}} & =\left(\frac{\partial t_{b}}{\partial E}\right)^{-1}=\left[-\int_{x_{a}}^{x_{b}} d x \frac{M}{p^{2}} \frac{\partial p}{\partial E}\right]^{-1} \\ & =\left[-\int_{x_{a}}^{x_{b}} d x \frac{M^{2}}{p^{3}}\right]^{-1}=\left[-M \int_{t_{a}}^{t_{b}} \frac{d t}{p^{2}}\right]^{-1}=\left[-\frac{1}{M} \int_{t_{a}}^{t_{b}} \frac{d t}{\dot{x}_{\mathrm{cl}}^{2}(t)}\right]^{-1} \end{align*} $$

$$ \frac{\partial E}{\partial t_{b}}=\left(\frac{\partial^{2} S}{\partial E^{2}}\right)^{-1} $$

$$ D_{\mathrm{ren}}=-M \dot{x}_{\mathrm{cl}}\left(t_{b}\right) \dot{x}_{\mathrm{cl}}\left(t_{a}\right) \frac{\partial^{2} S}{\partial E^{2}} $$

$$ F\left(x_{b}, t_{a} ; t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}}\left[\frac{\partial \dot{x}_{b}}{\partial x_{a}}\right]^{1 / 2}=\frac{1}{\sqrt{2 \pi i \hbar}}\left[-\partial_{x_{b}} \partial_{x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)\right]^{1 / 2} $$

$$ F\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}^{D}}\left\{\operatorname{det}_{D}\left[-\partial_{x_{b}^{i}} \partial_{x_{a}^{j}} A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)\right]\right\}^{1 / 2} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}^{D}}\left\{\operatorname{det}_{D}\left[-\partial_{x_{b}^{i}} \partial_{x_{a}^{j}} A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)\right]\right\}^{1 / 2} e^{i A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ \partial_{x_{b}^{i}} \partial_{x_{a}^{j}} A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)=\frac{\partial p_{b}^{i}}{\partial x_{a}^{j}}, $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}}\left[\operatorname{det}_{D}\left(-\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}_{a}}\right)\right]^{1 / 2} e^{i A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}} $$

$$ D_{\mathrm{ren}}=t_{b}-t_{a} $$

$$ A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{M \omega}{2 \sin \omega\left(t_{b}-t_{a}\right)}\left[\left(x_{b}^{2}+x_{a}^{2}\right) \cos \omega\left(t_{b}-t_{a}\right)-2 x_{b} x_{a}\right] $$

$$ -\partial_{x_{b}} \partial_{x_{a}} A=\frac{M \omega}{\sin \omega\left(t_{b}-t_{a}\right)} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\prod_{n=1}^{N}\left[\int_{-\infty}^{\infty} d^{D} x_{n}\right] \prod_{n=1}^{N+1}\left(\mathbf{x}_{n} t_{n} \mid \mathbf{x}_{n-1} t_{n-1}\right) $$

$$ \frac{\partial}{\partial \mathbf{x}} A\left(\mathbf{x}_{b}, \mathbf{x} ; t_{b}-t\right)+\frac{\partial}{\partial \mathbf{x}} A\left(\mathbf{x}, \mathbf{x}_{a} ; t-t_{a}\right)=-\mathbf{p}^{\prime}\left(\mathbf{x}_{b}, \mathbf{x} ; t_{b}-t\right)+\mathbf{p}\left(\mathbf{x}, \mathbf{x}_{a} ; t-t_{a}\right)=0 . $$

$$ \begin{gather*} \operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}\right)\left\{\operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}+\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{a}}\right){ }_{\mathbf{p}^{\prime}=\mathbf{p}}\right\}^{-1} \operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}}\right) \\ =\operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}_{b}}\right) \end{gather*} $$

$$ \operatorname{det}_{D}^{-1}\left(-\left.\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}\right) \operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}_{b}}\right)=\operatorname{det}_{D}\left(\left.\frac{\partial \mathbf{x}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}_{b}}\right) $$

$$ \operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}}\right)=\operatorname{det}_{D}\left(-\left.\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}+\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{a}}\right)_{\mathbf{p}^{\prime}=\mathbf{p}} \operatorname{det}_{D}\left(\left.\frac{\partial \mathbf{x}}{\partial \mathbf{x}_{a}}\right|_{\mathbf{x}_{b}}\right) $$

$$ \left.\frac{\partial \mathbf{p}^{\prime}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}=\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}=\left.\frac{\partial \mathbf{p}\left(\mathbf{x}\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right), \mathbf{x}_{a} ; t-t_{a}\right)}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}=\left.\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}}\right|_{\mathbf{x}} \frac{\partial \mathbf{x}_{a}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{b}}+\left.\frac{\partial \mathbf{p}}{\partial \mathbf{x}}\right|_{\mathbf{x}_{a}} . $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\int d t\left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{\mathrm{cl}}(t) t\right) \dot{\mathbf{x}}_{\mathrm{cl}}(t)\left(\mathbf{x}_{\mathrm{cl}}(t) t \mid \mathbf{x}_{a} t_{a}\right) $$

$$ \begin{align*} \left(x_{b} \mid x_{a}\right)_{E}=\frac{1}{\sqrt{2 \pi i \hbar}} \int_{t_{a}}^{\infty} d t_{b}[- & \left.\partial_{x_{b}} \partial_{x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)\right]^{1 / 2} \\ & \times e^{i\left[A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)+\left(t_{b}-t_{a}\right) E\right] / \hbar} \end{align*} $$

$$ \frac{\partial}{\partial t} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=-E . $$

$$ t_{b}-t_{a}=t\left(x_{b}, x_{a} ; E\right) $$

$$ \frac{\partial}{\partial E} S\left(x_{b}, x_{a} ; E\right)=t\left(x_{b}, x_{a} ; E\right) . $$

$$ \frac{i}{\hbar} \frac{\partial^{2} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)}{\partial t_{b}^{2}}\left[t_{b}-t_{a}-t\left(x_{b}, x_{a} ; E\right)\right]^{2} . $$

$$ \sqrt{2 \pi i \hbar}\left[\frac{\partial^{2} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)}{\partial t_{b}^{2}}\right]^{-1 / 2} . $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\left[-\partial_{x_{b}} \partial_{x_{a}} A\left(x_{b}, x_{a} ; t\right) / \partial_{t}^{2} A\left(x_{b}, x_{a} ; t\right)\right]^{1 / 2} e^{i S\left(x_{b}, x_{a} ; E\right) / \hbar} $$

$$ \frac{\partial^{2} A}{\partial t^{2}}=-\frac{\partial E}{\partial t}=-\left(\frac{\partial t}{\partial E}\right)^{-1}=-\left(\frac{\partial^{2} S}{\partial E^{2}}\right)^{-1} $$

$$ \frac{\partial t}{\partial x_{b}}=0 $$

$$ \left.\frac{\partial^{2} S}{\partial E \partial x_{b}}\right|_{t}=\frac{\partial^{2} S}{\partial E \partial x_{b}}+\frac{\partial^{2} S}{\partial E^{2}} \frac{\partial E}{\partial x_{b}}=0 $$

$$ \left.\frac{\partial A}{\partial x_{b}}\right|_{t}=\left.\frac{\partial S}{\partial x_{b}}\right|_{t}-\left.\frac{\partial E}{\partial x_{b}}\right|_{t} t=\frac{\partial S}{\partial x_{b}}+\left.\frac{\partial S}{\partial E} \frac{\partial E}{\partial x_{b}}\right|_{t}-\left.\frac{\partial E}{\partial x_{b}}\right|_{t} t=\left.\frac{\partial S}{\partial x_{b}}\right|_{E} $$

$$ \begin{align*} \left.\frac{\partial^{2} A}{\partial x_{b} \partial x_{a}}\right|_{t} & =\frac{\partial^{2} S}{\partial x_{b} \partial x_{a}}+\frac{\partial^{2} S}{\partial x_{b} \partial E} \frac{\partial E}{\partial x_{a}} \\ & =\frac{\partial^{2} S}{\partial x_{b} \partial x_{a}}-\frac{\partial^{2} S}{\partial x_{a} \partial E} \frac{\partial^{2} S}{\partial x_{b} \partial E} / \frac{\partial^{2} S}{\partial E^{2}} \end{align*} $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=D_{S}^{1 / 2} e^{i S\left(x_{b}, x_{a} ; E\right) / \hbar} $$

$$ D_{S}=\left|\begin{array}{cc} \frac{\partial^{2} S}{\partial x_{b} \partial x_{a}} & \frac{\partial^{2} S}{\partial E \partial x_{a}} \\ \frac{\partial^{2} S}{\partial x_{b} \partial E} & \frac{\partial^{2} S}{\partial E^{2}} \end{array}\right| $$

$$ H\left(\frac{\partial S}{\partial x_{b}}, x_{b}\right)=E $$

$$ \frac{\partial H}{\partial p_{b}} \frac{\partial^{2} S}{\partial x_{b} \partial x_{a}}=\dot{x}_{b} \frac{\partial^{2} S}{\partial x_{b} \partial x_{a}}=0 . $$

$$ D_{S}=-\frac{\partial^{2} S}{\partial x_{b} \partial E} \frac{\partial^{2} S}{\partial x_{a} \partial E} $$

$$ D_{S}=\frac{1}{\dot{x}_{b} \dot{x}_{a}} $$

$$ A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}} $$

$$ E\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=-\frac{\partial}{\partial t_{b}} \frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{t_{b}-t_{a}}=\frac{M}{2} \frac{\left(x_{b}-x_{a}\right)^{2}}{\left(t_{b}-t_{a}\right)^{2}} . $$

$$ S\left(x_{b}, x_{a} ; E\right)=\sqrt{2 M E}\left|x_{b}-x_{a}\right| $$

$$ D_{s}=\frac{M}{2 E} $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\sqrt{\frac{M}{2 E}} e^{i \sqrt{2 M E}\left|x_{b}-x_{a}\right| / \hbar} $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)_{\mathrm{sc}}=\int d x_{b} d x_{a} e^{-i\left(p_{b} x_{b}-p_{a} x_{a}\right) / \hbar} e^{i A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) / \hbar} F\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)_{\mathrm{sc}}=\frac{\sqrt{2 \pi i \hbar}}{\sqrt{\operatorname{det} H}}\left[-\partial_{x_{b}} \partial_{x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)\right]^{1 / 2} e^{i\left[A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)-p_{b} x_{b}+p_{a} x_{a}\right] / \hbar},(4 $$

$$ H=\left(\begin{array}{rr} \partial_{x_{b}}^{2} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) & \partial_{x_{b}} \partial_{x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) \\ \partial_{x_{a}} \partial_{x_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) & \partial_{x_{a}}^{2} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) \end{array}\right) . $$

$$ p_{b}=\partial_{x_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right), \quad p_{a}=-\partial_{x_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) . $$

$$ A\left(p_{b}, p_{a} ; t_{b}-t_{a}\right)=A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)-p_{b} x_{b}+p_{a} x_{a} $$

$$ x_{b}=-\partial_{p_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right), \quad x_{a}=\partial_{x_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right) . $$

$$ H=\left(\begin{array}{cc} \frac{\partial p_{b}}{\partial x_{b}} & \frac{\partial p_{b}}{\partial x_{a}} \\ -\frac{\partial p_{a}}{\partial x_{b}} & -\frac{\partial p_{a}}{\partial x_{a}} \end{array}\right), \quad H^{-1}=\left(\begin{array}{cc} \frac{\partial x_{b}}{\partial p_{b}} & -\frac{\partial x_{b}}{\partial p_{a}} \\ \frac{\partial x_{a}}{\partial p_{b}} & -\frac{\partial x_{a}}{\partial p_{a}} \end{array}\right), $$

$$ H_{12}^{-1}=\frac{\partial x_{a}}{\partial p_{b}}=\frac{\partial^{2} A\left(p_{b}, p_{a} ; t_{b}-t_{a}\right)}{\partial p_{b} \partial p_{a}} $$

$$ \left(p_{b} t_{b} \mid p_{a} t_{a}\right)_{\mathrm{sc}}=\frac{2 \pi \hbar}{\sqrt{2 \pi i \hbar}}\left[-\partial_{p_{b}} \partial_{p_{a}} A\left(p_{b}, p_{a} ; t_{b}-t_{a}\right)\right]^{1 / 2} e^{i A\left(p_{b}, p_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ \left(\mathbf{p}_{b} t_{b} \mid \mathbf{p}_{a} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}^{D}}\left\{\operatorname{det}_{D}\left[-\partial_{p_{b}^{i}} \partial_{p_{a}^{j}} A\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; t_{b}-t_{a}\right)\right]\right\}^{1 / 2} e^{i A\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ \left(\mathbf{p}_{b} t_{b} \mid \mathbf{p}_{p} t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}^{D}}\left\{\operatorname{det}_{D}\left[-\frac{\partial \mathbf{p}_{b}}{\partial \mathbf{x}_{a}}\right]\right\}^{1 / 2} e^{i A\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ A\left(p_{b}, p_{a} ; t_{b}-t_{a}\right)=-\frac{p^{2}}{2}\left(t_{b}-t_{a}\right) $$

$$ Z_{\mathrm{QM}}^{\mathrm{sc}}\left(t_{b}-t_{a}\right)=\int d x_{a}\left(x_{a} t_{b} \mid x_{a} t_{a}\right)_{\mathrm{sc}}=\int d x_{a} F\left(x_{a}, x_{a} ; t_{b}-t_{a}\right) e^{i A\left(x_{a}, x_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ \begin{align*} \frac{\partial}{\partial x_{a}} A\left(x_{a}, x_{a} ; t_{b}-t_{a}\right) & =\left.\frac{\partial}{\partial x_{b}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)\right|_{x_{b}=x_{a}}+\left.\frac{\partial}{\partial x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)\right|_{x_{b}=x_{a}} \\ & =p_{b}-p_{a}=0 \end{align*} $$

$$ Z_{\mathrm{QM}}^{\mathrm{sc}}\left(t_{b}-t_{a}\right)=\left[\int d x_{a} F\left(x_{a}, x_{a} ; t_{b}-t_{a}\right)\right] e^{i A\left(x_{a}, x_{a} ; t_{b}-t_{a}\right) / \hbar} $$

$$ \frac{\partial}{\partial x_{b}} \frac{\partial}{\partial x_{a}} A\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=-\frac{1}{\dot{x}_{b} \dot{x}_{a}} \frac{\partial^{2} A}{\partial t_{b}^{2}} $$

$$ F\left(x_{b}, x_{a} ; t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}}\left[\frac{1}{\dot{x}\left(t_{b}\right) \dot{x}\left(t_{a}\right)} \frac{\partial^{2} A}{\partial t_{b}^{2}}\right]^{1 / 2} $$

$$ F\left(x_{a}, x_{a} ; t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}} \frac{1}{\dot{x}_{a}}\left[\frac{\partial^{2} A}{\partial t_{b}^{2}}\right]^{1 / 2} $$

$$ t_{b}-t_{a}=2 \int_{x_{-}}^{x_{+}} d x_{a} \frac{1}{\dot{x}_{a}}=2 \int_{x_{-}}^{x_{+}} d x \frac{M}{\sqrt{2 M[E-V(x)]}} $$

$$ Z_{\mathrm{QM}}^{\mathrm{sc}}\left(t_{b}-t_{a}\right)=\frac{t_{b}-t_{a}}{\sqrt{2 \pi i \hbar}}\left|\frac{\partial^{2} A}{\partial t_{b}^{2}}\right|^{1 / 2} e^{i A\left(t_{b}-t_{a}\right) / \hbar-i \pi} $$

$$ \begin{align*} \tilde{Z}_{\mathrm{QM}}^{\mathrm{sc}}(E) & =\int_{t_{a}}^{\infty} d t_{b} e^{i E\left(t_{b}-t_{a}\right) / \hbar} Z_{\mathrm{QM}}^{\mathrm{sc}}\left(t_{b}-t_{a}\right) \\ & =\frac{1}{\sqrt{2 \pi i \hbar}} \int_{t_{a}}^{\infty} d t_{b}\left(t_{b}-t_{a}\right)\left|\frac{\partial^{2} A}{\partial t_{b}^{2}}\right|^{1 / 2} e^{i\left[A\left(t_{b}-t_{a}\right)+\left(t_{b}-t_{a}\right) E\right] / \hbar-i \pi} \end{align*} $$

$$ A\left(t_{b}-t_{a}\right)+\left(t_{b}-t_{a}\right) E $$

$$ -\frac{\partial}{\partial t_{b}} A\left(t_{b}-t_{a}\right)=E $$

$$ \tilde{Z}_{\mathrm{QM}}^{\mathrm{sc}}(E)=t(E) e^{i[A(t)+t(E) E] / \hbar-i \pi} $$

$$ S(E)=A(t)-t \frac{\partial A(t)}{\partial t}, $$

$$ A(t)=S(E)-\frac{\partial S(E)}{\partial E} E $$

$$ S(E)=2 \int_{x_{-}}^{x_{+}} d x p(x)=2 \int_{x_{-}}^{x_{+}} d x \sqrt{2 M[E-V(x)]} $$

$$ \tilde{Z}_{\mathrm{QM}}^{\mathrm{sc}}(E)=\sum_{n=1}^{\infty} t(E) e^{i n[S(E) / \hbar-\pi]}=-t(E) \frac{e^{i S(E) / \hbar}}{1+e^{i S(E) / \hbar}} $$

$$ S\left(E_{n}\right)=2 \pi \hbar(n+1 / 2), \quad n=0, \pm 1, \pm 2, \ldots . $$

$$ \tilde{Z}_{\mathrm{QM}}^{\mathrm{sc}}(E) \approx t(E) \frac{i \hbar}{S^{\prime}\left(E_{n}\right)\left(E-E_{n}\right)} $$

$$ \tilde{Z}_{\mathrm{QM}}^{\mathrm{sc}}(E) \approx \frac{i \hbar}{E-E_{n}} $$

$$ \rho(E)=\frac{1}{2 \pi \hbar} \operatorname{disc} \tilde{Z}_{\mathrm{QM}}(E), $$

$$ \rho(E)=\frac{1}{\pi \hbar} \operatorname{Re} \tilde{Z}_{\mathrm{QM}}(E) . $$

$$ \bar{\rho}_{\mathrm{sc}}(E)=\frac{t(E)}{\pi \hbar} \sum_{n=1}^{\infty} \cos \{n[S(E) / \hbar-\pi]\} $$

$$ \Delta \rho_{\mathrm{sc}}(E)=\frac{t(E)}{2 \pi \hbar}\left(-1+\sum_{n=-\infty}^{\infty} e^{i n[S(E) / \hbar-\pi]}\right) $$

$$ \Delta \rho_{\mathrm{sc}}(E)=-\frac{t(E)}{2 \pi \hbar}+\frac{t(E)}{\hbar} \sum_{n=-\infty}^{\infty} \delta[S(E) / \hbar-2 \pi(n+1 / 2)] $$

$$ \Delta \rho_{\mathrm{sc}}(E)=-\frac{t(E)}{2 \pi \hbar}+\sum_{n=-\infty}^{\infty} \delta\left(E-E_{n}\right) . $$

$$ \Delta E_{n}=E_{n}-E_{n-1}=2 \pi \hbar \frac{\Delta E_{n}}{\Delta S_{n}} $$

$$ \rho_{\mathrm{av}}(E)=\frac{S^{\prime}(E)}{2 \pi \hbar}=\frac{t(E)}{2 \pi \hbar} . $$

$$ Z_{\mathrm{cl}}(E) \equiv \int d x \int \frac{d p}{2 \pi \hbar} \frac{i \hbar}{E-H(p, x)} $$

$$ \rho_{\mathrm{cl}}(E) \equiv \int d x \rho_{\mathrm{cl}}(E ; x) $$

$$ \rho_{\mathrm{cl}}(E ; x) \equiv \int \frac{d p}{2 \pi \hbar} \delta[E-H(p, x)] $$

$$ \delta(E-H(p, x))=\frac{M}{p(E ; x)}[\delta(p-p(E ; x))+\delta(p+p(E ; x))] $$

$$ p(E ; x)=\sqrt{2 M[E-V(x)]} $$

$$ \rho_{\mathrm{cl}}(E ; x)=\frac{1}{\pi \hbar} \frac{M}{p(E ; x)} $$

$$ \rho_{\mathrm{cl}}(E)=\int d x \frac{1}{\pi \hbar} \frac{M}{p(E ; x)}=\frac{1}{2 \pi \hbar} t(E)=\rho_{\mathrm{av}}(E) $$

$$ \rho_{\mathrm{sc}}(E)=\rho_{\mathrm{cl}}(E)+\Delta \rho_{\mathrm{sc}}(E) $$

$$ S(E)=2 \pi \hbar \int_{-\infty}^{E} d E \rho_{\mathrm{cl}}(E) $$

$$ S(E)=2 \pi \hbar N(E) $$

$$ Z_{\mathrm{cl}}(E) \equiv \int d^{D} x \int \frac{d^{D} p}{2 \pi \hbar} \frac{i \hbar}{E-H(\mathbf{p}, \mathbf{x})} $$

$$ \rho_{\mathrm{cl}}(E ; \mathbf{x}) \equiv \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \delta[E-H(\mathbf{p}, \mathbf{x})] $$

$$ \int \frac{d^{D} p}{(2 \pi \hbar)^{D}}=\int d p p^{D-1} \int d \hat{\mathbf{p}} $$

$$ \int d \hat{\mathbf{p}}=S_{D}=\frac{2 \pi^{D / 2}}{\Gamma(D / 2)} $$

$$ p(E ; \mathbf{x})=\sqrt{2 M[E-V(\mathbf{x})]} $$

$$ \rho_{\mathrm{cl}}(E) \equiv \int d^{D} x \rho_{\mathrm{cl}}(E ; \mathbf{x}) $$

$$ \rho_{\mathrm{cl}}(E ; \mathbf{x})=S_{D} \frac{M}{p^{2}(E ; \mathbf{x})} \frac{p^{D}(E ; \mathbf{x})}{(2 \pi \hbar)^{D}}=\frac{1}{\left(4 \pi \hbar^{2}\right)^{D / 2}} \frac{2 M}{\Gamma(D / 2)}\{2 M[E-V(\mathbf{x})]\}^{D / 2-1}, $$

$$ F\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar}^{D}}\left|\operatorname{det}_{D}\left[-\partial_{x_{b}^{i}} \partial_{x_{a}^{j}} A\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; t_{b}-t_{a}\right)\right]\right|^{1 / 2} e^{-i \pi \nu / 2} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\frac{1}{\sqrt{2 \pi i \hbar}^{D-1}} \sum_{p}\left|D_{S}\right|^{1 / 2} e^{i S\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; E\right) / \hbar-i \pi \nu^{\prime} / 2} $$

$$ D_{S}=(-1)^{D+1} \operatorname{det}\left(\begin{array}{cc} \frac{\partial^{2} S}{\partial \mathbf{x}_{b} \partial \mathbf{x}_{a}} & \frac{\partial^{2} S}{\partial E \partial \mathbf{x}_{a}} \\ \frac{\partial^{2} S}{\partial \mathbf{x}_{b} \partial E} & \frac{\partial^{2} S}{\partial E \partial E} \end{array}\right) $$

$$ \frac{\partial H}{\partial \mathbf{p}_{b}} \cdot \frac{\partial^{2} S}{\partial \mathbf{x}_{b} \partial \mathbf{x}_{a}}=\dot{\mathbf{x}}_{b} \cdot \frac{\partial^{2} S}{\partial \mathbf{x}_{b} \partial \mathbf{x}_{a}}=0 $$

$$ \dot{\mathbf{x}_{b}} \cdot \frac{\partial^{2} S}{\partial \mathbf{x}_{b} \partial E}=1, $$

$$ D_{S}=\frac{1}{\left|\dot{\mathbf{x}}_{b}\right|\left|\dot{\mathbf{x}}_{a}\right|} \operatorname{det}\left(-\frac{\partial^{2} S}{\partial \mathbf{x}_{b}^{\perp} \partial \mathbf{x}_{a}^{\perp}}\right) $$

$$ S\left(\mathbf{x}_{a}, \mathbf{x}_{b} ; E\right)=\sqrt{2 M E}\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right| $$

$$ D_{S}=\frac{M}{2 E} \frac{(2 M E)^{(D-1) / 2}}{\left|\mathbf{x}_{a}-\mathbf{x}_{b}\right|^{D-1}} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=\sqrt{\frac{M}{2 E}} \frac{1}{(2 \pi i \hbar)^{(D-1) / 2}} \frac{(2 M E)^{(D-1) / 4}}{\left|\mathbf{x}_{a}-\mathbf{x}_{b}\right|^{(D-1) / 2}} e^{i \sqrt{2 M E}\left|\mathbf{x}_{b}-\mathbf{x}_{a}\right| / \hbar} $$

$$ \frac{1}{2}\left(\mathbf{x}-\mathbf{x}^{\perp}\right)^{T} \frac{\partial^{2} S(\mathbf{x}, \mathbf{x} ; E)}{\partial \mathbf{x}^{\perp} \partial \mathbf{x}^{\perp}}\left(\mathbf{x}-\mathbf{x}^{\perp}\right) $$

$$ Z_{\mathrm{sc}}=t(E) \frac{\left|\frac{\partial^{2} S\left(\mathbf{x}_{b}, \mathbf{x}_{a} ; E\right)}{\partial \mathbf{x}_{b}^{\perp} \partial \mathbf{x}_{a}^{\perp}}\right|_{\mathbf{x}_{b}=\mathbf{x}_{a}=\mathbf{x}}^{1 / 2} \sum_{n=1}^{\infty} e^{i n[S(E) / \hbar-i \pi \nu / 2]}}{\left|\frac{\partial^{2} S(\mathbf{x}, \mathbf{x} ; E)}{\partial \mathbf{x}^{\perp} \partial \mathbf{x}^{\perp}}\right|^{1 / 2}} $$

$$ \binom{\delta \mathbf{x}_{b}^{\perp}}{\delta \mathbf{p}_{b}^{\perp}}=\left(\begin{array}{cc} A & B \\ C & D \end{array}\right)\binom{\delta \mathbf{x}_{a}^{\perp}}{\delta \mathbf{p}_{a}^{\perp}} \equiv M\binom{\delta \mathbf{x}_{a}^{\perp}}{\delta \mathbf{p}_{a}^{\perp}} $$

$$ \binom{\delta \mathbf{p}_{a}^{\perp}}{\delta \mathbf{p}_{b}^{\perp}}=\left(\begin{array}{rr} -a & -b \\ b^{T} & c \end{array}\right)\binom{\delta \mathbf{x}_{a}^{\perp}}{\delta \mathbf{x}_{b}^{\perp}} $$

$$ a=\frac{\partial^{2} S}{\partial \mathbf{x}_{a}^{\perp} \partial \mathbf{x}_{a}^{\perp}}, \quad b=\frac{\partial^{2} S}{\partial \mathbf{x}_{a}^{\perp} \partial \mathbf{x}_{b}^{\perp}}, \quad c=\frac{\partial^{2} S}{\partial \mathbf{x}_{b}^{\perp} \partial \mathbf{x}_{b}^{\perp}} $$

$$ A=-b^{-1} a, \quad B=-b^{-1}, \quad C=b^{T}-c b^{-1} a, \quad D=-c b^{-1} $$

$$ P(\lambda)=|M-\lambda|=\left|\begin{array}{ll} A-\lambda & B \\ C & D-\lambda \end{array}\right|=\left|\begin{array}{cl} -b^{-1} a-\lambda & -b^{-1} \\ b^{T}-c b^{-1} a & -c b^{-1}-\lambda \end{array}\right| . $$

$$ \begin{align*} P(\lambda) & =\left|\begin{array}{ll} -b^{-1} a-\lambda & -b^{-1} \\ b^{T}+\lambda & -\lambda \end{array}\right|=\frac{1}{|b|}\left|\begin{array}{lr} -a-\lambda b & -1 \\ b^{T}+(a+c) \lambda+\lambda^{2} b & 0 \end{array}\right| \\ & =\frac{1}{|b|}\left|b^{T}+(a+c) \lambda+\lambda^{2} b\right| \end{align*} $$

$$ \frac{\left|\frac{\partial^{2} S}{\partial \mathbf{x}_{b}^{\perp} \partial \mathbf{x}_{a}^{\perp}}\right|_{\mathbf{x}_{b} \approx \mathbf{x}_{a}=\mathbf{x}}^{1 / 2}}{\left|\frac{\partial^{2} S}{\partial \mathbf{x}_{b}^{\perp} \partial \mathbf{x}_{b}^{\perp}}+2 \frac{\partial S}{\partial \mathbf{x}_{b}^{\perp}} \frac{\partial S}{\partial \mathbf{x}_{a}^{\perp}}+\frac{\partial^{2} S}{\partial \mathbf{x}_{a}^{\perp} \partial \mathbf{x}_{a}^{\perp}}\right|_{\mathbf{x}_{b}=\mathbf{x}_{a}=\mathbf{x}}^{1 / 2}} $$

$$ Z_{\mathrm{sc}}(E)=t(E) \frac{1}{P(1)^{1 / 2}} \frac{e^{i S(E)-i \pi \nu / 2}}{1-e^{i S(E)-i \pi \nu / 2}} $$

$$ S\left(E_{n}\right)=2 \pi \hbar(n+\nu / 4) . $$

$$ P(1)=\prod_{i=1}^{2}\left(\lambda_{i}-1\right)\left(1 / \lambda_{i}-1\right) $$

$$ \left(\mathbf{p}_{b} \mid \mathbf{p}_{a}\right)_{E}=\frac{(2 \pi \hbar)^{D}}{\sqrt{2 \pi i \hbar}^{D-1}} \sum_{p}\left|\tilde{D}_{S}\right|^{1 / 2} e^{i S\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; E\right) / \hbar-i \pi \nu^{\prime} / 2} $$

$$ S\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; E\right)=S\left(\mathbf{p}_{b}, \mathbf{p}_{a} ; E\right)-\mathbf{p}_{b} \mathbf{x}_{b}+\mathbf{p}_{a} \mathbf{x}_{a}, $$

$$ D_{S}=\frac{1}{\left|\dot{\mathbf{p}}_{b}\right|\left|\dot{\mathbf{p}}_{a}\right|} \operatorname{det}\left(-\frac{\partial^{2} S}{\partial \mathbf{p}_{b}^{\perp} \partial \mathbf{p}_{a}^{\perp}}\right), $$

$$ \delta(E-\hat{H}) \equiv \sum_{n} \delta\left(E-E_{n}\right)|n\rangle\langle n| $$

$$ \delta(E-\hat{H})=\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} e^{-i(\hat{H}-E) t / \hbar} $$

$$ \rho(E ; \mathbf{x})=\langle\mathbf{x}| \delta(E-\hat{H})|\mathbf{x}\rangle=\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} e^{i E t / \hbar}\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}|\mathbf{x}\rangle $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}|\mathbf{x}\rangle=(\mathbf{x} t \mid \mathbf{x} 0) $$

$$ V(x)=\frac{\omega^{2}}{2}\left(x^{2}-2 a x\right)=\frac{\omega^{2}}{2}(x-a)^{2}-\frac{\omega^{2}}{2} a^{2} $$

$$ \left(x t_{b} \mid x t_{a}\right)=\frac{1}{\sqrt{2 \pi i \hbar / M}} \sqrt{\frac{\omega}{\sin \omega t}} \exp \left[i \frac{M \omega}{\hbar}\left(\tan \frac{\omega t}{2}(x-a)^{2}-\frac{\omega t a^{2}}{2}\right)\right] $$

$$ \begin{align*} \left(x t_{b} \mid x t_{a}\right)=\frac{e^{-i \frac{M}{2} \omega^{2}\left(x^{2}-2 a x\right) t / \hbar}}{\sqrt{2 \pi i \hbar t / M}}\{1 & +\frac{t^{2} \omega^{2}}{12}-\frac{i M}{\hbar} \frac{t^{3}}{24} \omega^{4}(x-a)^{2} \\ & \left.+\frac{t^{4} \omega^{4}}{160}-\frac{i M}{\hbar} \frac{11 t^{5} \omega^{2}}{1440} \omega^{4}(x-a)^{2}+\ldots\right\} \end{align*} $$

$$ e^{-i V(x) t / \hbar} $$

$$ \begin{align*} \omega^{2} & \rightarrow \frac{1}{M} V^{\prime \prime}(x), \\ \omega^{4}(x-a)^{2} & \rightarrow \frac{1}{M^{2}}\left[V^{\prime}(x)\right]^{2} . \end{align*} $$

$$ \begin{align*} \left(x t_{b} \mid x t_{a}\right)= & \frac{e^{-i V(x) t / \hbar}}{\sqrt{2 \pi i \hbar t / M}}\{1 \\ & +\frac{t^{2}}{12 M} V^{\prime \prime}(x)-\frac{i}{\hbar} \frac{t^{3}}{24 M}\left[V^{\prime}(x)\right]^{2} \\ & \left.+\frac{t^{4}}{160}\left[V^{\prime \prime}(x)\right]^{2}-\frac{i}{\hbar} \frac{11 t^{5}}{1440 M}\left[V^{\prime}(x)\right]^{2} V^{\prime \prime}(x)+\ldots\right\} . \end{align*} $$

$$ \rho(E ; x)=2 \int_{0}^{\infty} \frac{d \tau}{2 \pi \hbar} \frac{e^{-[E-V(x)] \tau / \hbar}}{\sqrt{2 \pi \hbar \tau / M}}\left\{1-\frac{\tau^{2}}{12 M} V^{\prime \prime}(x)-\frac{1}{\hbar} \frac{\tau^{3}}{24 M}\left[V^{\prime}(x)\right]^{2}+\ldots\right\} $$

$$ \rho_{\mathrm{cl}}(E ; x)=\frac{1}{\pi \hbar} \frac{M}{\sqrt{2 M[E-V(x)]}}=\frac{M}{\pi \hbar} \frac{1}{p(E ; x)} . $$

$$ \rho(E ; x)=\left\{1-\frac{\hbar^{2}}{12 M} V^{\prime \prime}(x) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}\left[V^{\prime}(x)\right]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \rho_{\mathrm{cl}}(E ; x) $$

$$ \begin{align*} \rho(E ; x)=\frac{1}{\pi \hbar} \sqrt{\frac{M}{2}}\left\{\frac{1}{[E-V(x)]^{1 / 2}}\right. & -\frac{\hbar^{2}}{12 M} V^{\prime \prime}(x) \frac{3}{4} \frac{1}{[E-V(x)]^{5 / 2}} \\ & \left.-\frac{\hbar^{2}}{24 M}\left[V^{\prime}(x)\right]^{2} \frac{15}{8} \frac{1}{[E-V(x)]^{7 / 2}}+\ldots\right\} . \end{align*} $$

$$ \rho(E)=\frac{1}{\pi \hbar} \sqrt{\frac{M}{2}} \int d x\left\{\frac{1}{[E-V(x)]^{1 / 2}}+\frac{\hbar^{2}}{24 M}\left[V^{\prime}(x)\right]^{2} \frac{15}{8} \frac{1}{[E-V(x)]^{7 / 2}}+\ldots\right\} . $$

$$ \left(\mathbf{x} t_{b} \mid \mathbf{x} t_{a}\right)=\frac{e^{-i V(\mathbf{x}) t / \hbar}}{\sqrt{2 \pi i \hbar t / M}^{D}}\left\{1+\frac{t^{2}}{12 M} \nabla^{2} V(\mathbf{x})-\frac{i}{\hbar} \frac{t^{3}}{24 M}[\nabla V(\mathbf{x})]^{2}+\ldots\right\} $$

$$ \int_{-\infty}^{\infty} \frac{d t}{2 \pi} \frac{1}{(i t+\eta)^{\nu}} e^{i t a}=\Theta(a) \frac{a^{\nu-1}}{\Gamma(\nu)} e^{-a \eta} $$

$$ \rho(E ; \mathbf{x})=\left\{1-\frac{\hbar^{2}}{12 M} \nabla^{2} V(\mathbf{x}) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \rho_{\mathrm{cl}}(E ; \mathbf{x}) $$

$$ \rho_{\mathrm{cl}}(E ; \mathbf{x})=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \delta[E-H(\mathbf{p}, \mathbf{x})]=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{M}{p(E ; \mathbf{x})} \delta[p-p(E ; \mathbf{x})] $$

$$ (\mathbf{x} t \mid \mathbf{x} 0)_{\mathrm{cl}}=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-i\left[\mathbf{p}^{2} / 2 M+V(\mathbf{x})\right] t / \hbar} $$

$$ \rho(E ; \mathbf{x})=\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{-i\left[p^{2}-p^{2}(E ; \mathbf{x})\right] t / 2 M \hbar} $$

$$ \begin{align*} \rho(E ; \mathbf{x})= & \left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2}\left\{\frac{1}{\Gamma(D / 2)}[E-V(\mathbf{x})]^{D / 2-1}\right. \\ & -\frac{\hbar^{2}}{12 M}\left[\nabla^{2} V(\mathbf{x})\right] \frac{1}{\Gamma(D / 2-2)}[E-V(\mathbf{x})]^{D / 2-3} \\ & \left.+\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{1}{\Gamma(D / 2-3)}[E-V(\mathbf{x})]^{D / 2-4}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \rho\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right) & =\left\langle\mathbf{x}_{b}\right| \delta(E-\hat{H})\left|\mathbf{x}_{a}\right\rangle=\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} e^{i E t / \hbar}\left\langle\mathbf{x}_{b}\right| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}_{a}\right\rangle \\ & =\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} e^{i E t / \hbar}\left(\mathbf{x}_{b} t \mid \mathbf{x}_{a} 0\right) \end{align*} $$

$$ \begin{align*} & \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)=\frac{e^{i M \Delta \mathbf{x}^{2} / 2 t \hbar}}{\sqrt{2 \pi i \hbar t / M}} e^{-i M \omega^{2} \overline{\mathbf{x}}^{2} t / 2 \hbar}\left\{1+\frac{t^{2} D}{12} \omega^{2}-\frac{i M}{\hbar} \frac{t^{3}}{24} \omega^{4} \overline{\mathbf{x}}^{2}-\frac{i M}{\hbar} \frac{t}{24} \Delta \mathbf{x}^{2} \omega^{2}\right. \\ & \left.-\frac{i M}{\hbar} \frac{t^{3}}{1440} \Delta \mathbf{x}^{2} \omega^{4}-\frac{i M}{\hbar} \frac{t^{3} D}{288} \Delta \mathbf{x}^{2} \omega^{4}+\frac{t^{4}\left(D+\frac{5}{4} D^{2}\right)}{360} \omega^{4}-\frac{1}{\hbar^{2}} \Delta \mathbf{x}^{4} \frac{t^{2}}{1152} \omega^{4} \overline{\mathbf{x}}^{2}+\ldots\right\} \end{align*} $$

$$ \begin{align*} \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)= & \frac{e^{i M \Delta \mathbf{x}^{2} / 2 t \hbar}}{\sqrt{2 \pi i \hbar t / M}^{D}} e^{-i V(\overline{\mathbf{x}}) t / \hbar}\left\{1+\frac{t^{2}}{12 M} \nabla^{2} V-\frac{i}{\hbar} \frac{t^{3}}{24 M}[\nabla V]^{2}-\frac{i}{\hbar} \frac{t}{24}(\Delta \mathbf{x} \nabla)^{2} V\right. \\ \left.-\frac{i}{\hbar} \frac{t^{3}}{1440 M}\left[(\Delta \mathbf{x} \nabla) \Delta x_{i} \nabla_{j} V\right] \nabla_{i} \nabla_{j} V\right] & -\frac{i M}{\hbar} \frac{t^{3}}{288}(\Delta \mathbf{x})^{2}\left[\nabla^{2} V\right]^{2}+\frac{t^{4}}{360 M^{2}}\left(\left[\nabla_{i} \nabla_{j} V\right]^{2}+\frac{5}{4}[\nabla V]^{2}\right) \\ & \left.-\frac{t^{2}}{1152 \hbar^{2}}\left[\Delta x_{i} \Delta x_{j} \nabla_{k} \nabla_{l} V\right]\left[\Delta x_{k} \Delta x_{l} \nabla_{i} \nabla_{j} V\right]+\ldots\right\} .(4.267) \end{align*} $$

$$ \Delta \mathbf{x}^{2} M \omega^{2} \rightarrow(\Delta \mathbf{x} \nabla)^{2} V(\overline{\mathbf{x}}) $$

$$ \left(\mathbf{x} t \mid \mathbf{x}^{\prime} 0\right)=\frac{e^{i M \Delta \mathbf{x}^{2} / 2 t \hbar}}{\sqrt{2 \pi i \hbar t / M}^{D}} A(\mathbf{x}, \Delta \mathbf{x}, t) $$

$$ \hbar i \partial_{t} A\left(\mathbf{x}, \Delta \mathbf{x}^{\prime} ; t\right)=\left[-\frac{\hbar^{2}}{2 M} \nabla^{2}-V(\mathbf{x})\right] A-\frac{i \hbar}{t} \Delta \mathbf{x} \nabla A(\mathbf{x}, \Delta \mathbf{x} ; t) $$

$$ A(\mathbf{x}, \Delta \mathbf{x} ; t)=1+\sum_{n=1}^{\infty} t^{n} \sum_{p=0}^{\infty} \Delta x_{i_{1}} \cdots \Delta x_{i_{p}} a_{i_{1}, \ldots, i_{p}}^{(n)}(\mathbf{x}) $$

$$ \begin{align*} \rho\left(E ; \mathbf{x}_{b},\right. & \left.\mathbf{x}_{a}\right)=\int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} \frac{e^{i M \Delta \mathbf{x}^{2} / 2 t \hbar}}{\sqrt{2 \pi i \hbar t / M}^{D}} e^{-i[V(\overline{\mathbf{x}})-E] t / \hbar} \\ \times & \left\{1+\frac{t^{2}}{12 M} \nabla^{2} V(\overline{\mathbf{x}})-\frac{i}{\hbar} \frac{t^{3}}{24 M}[\nabla V(\overline{\mathbf{x}})]^{2}-\frac{i}{\hbar} \frac{t}{24}(\Delta \mathbf{x} \nabla)^{2} V(\overline{\mathbf{x}})+\ldots\right\} . \end{align*} $$

$$ \left(\mathbf{x}_{b} t_{b} \mid \mathbf{x}_{a} t_{a}\right)_{\mathrm{cl}}=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} e^{i \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar} e^{-i H(\mathbf{p}, \overline{\mathbf{x}}) t / \hbar} $$

$$ \rho_{\mathrm{cl}}\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right)=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \delta(E-H(\mathbf{p}, \overline{\mathbf{x}})) e^{i \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar} $$

$$ \int d \hat{\mathbf{p}} e^{i \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar}=S_{D}(p R / \hbar) $$

$$ S_{D}(z)=(2 \pi)^{D / 2} J_{D / 2-1}(z) / z^{D / 2-1} $$

$$ J_{\nu}(z) \approx \frac{(z / 2)^{\nu}}{\Gamma(\nu+1)} $$

$$ \rho_{\mathrm{cl}}\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right)=\left(\frac{1}{2 \pi \hbar^{2}}\right)^{D / 2} M \frac{J_{D / 2-1}(p(E ; \overline{\mathbf{x}}) R / \hbar)}{(R / \hbar)^{D / 2-1}} . $$

$$ J_{1 / 2}(z)=\sqrt{\frac{2}{\pi z}} \sin z $$

$$ \rho_{\mathrm{cl}}\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right)=\left(\frac{1}{2 \pi \hbar^{2}}\right)^{3 / 2} \frac{1}{\Gamma(3 / 2)} \frac{M}{\sqrt{2}} \frac{\sin [p(E ; \overline{\mathbf{x}}) R / \hbar]}{R / \hbar} . $$

$$ \begin{align*} & \rho\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right)=\left\{1-\frac{\hbar^{2}}{12 M}\left[\nabla^{2} V(\overline{\mathbf{x}})\right] \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\overline{\mathbf{x}})]^{2} \frac{d^{3}}{d V^{3}}\right. \\ &\left.+\frac{1}{24}\left[\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) \nabla\right]^{2} V(\overline{\mathbf{x}}) \frac{d}{d V}+\ldots\right\} \rho_{\mathrm{cl}}\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right) . \end{align*} $$

$$ \operatorname{Tr} \log \hat{H}=\int d^{D} x \int_{-\infty}^{\infty} d E \rho(E ; \mathbf{x}) \log E $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=\int d^{D} x \int_{E_{0}}^{\infty} d E \rho_{\mathrm{cl}}(E ; \mathbf{x}) \log E=\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \int d^{D} x I_{D / 2}(V(\mathbf{x})) $$

$$ I_{\alpha}(V) \equiv \frac{1}{\Gamma(\alpha)} \int_{V}^{\infty} d E(E-V)^{\alpha-1} \log E $$

$$ I_{\alpha}^{\eta}(V) \equiv \frac{1}{\Gamma(\alpha)} \int_{V}^{\infty} d E(E-V)^{\alpha-1} E^{-\eta}=V^{\alpha-\eta} \frac{\Gamma(-\alpha+\eta)}{\Gamma(\eta)} $$

$$ I_{\alpha}(V)=-\Gamma(-\alpha) V^{\alpha} $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=-\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \Gamma(-D / 2) \int d^{D} x[V(\mathbf{x})]^{D / 2} $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=-\int d^{D} x \int_{-\infty}^{\infty} d E \int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} \frac{e^{i t[E-V(\mathrm{x})] / \hbar}}{(2 \pi i \hbar t / M)^{D / 2}} \int_{0}^{\infty} \frac{d t^{\prime}}{t^{\prime}} e^{-i E t^{\prime} / \hbar} $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=-\int d^{D} x \int_{0}^{\infty} \frac{d t}{t} \frac{1}{(2 \pi i \hbar t / M)^{D / 2}} e^{-i t V(\mathrm{x}) / \hbar} $$

$$ \operatorname{Tr} \log \hat{H}=-\int d^{D} x \int_{0}^{\infty} \frac{d t}{t}\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}|\mathbf{x}\rangle $$

$$ \begin{align*} \operatorname{Tr} \log \hat{H} & =-\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \Gamma(-D / 2) \\ & \times \int d^{D} x\left\{1-\frac{\hbar^{2}}{12 M} \nabla^{2} V(\mathbf{x}) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\}[V(\mathbf{x})]^{D / 2} \end{align*} $$

$$ \operatorname{Tr} \log \hat{H}=-\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \Gamma(-D / 2) \int d^{D} x\left\{1+\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\}[V(\mathbf{x})]^{D / 2} . $$

$$ \left\{1-\frac{\hbar^{2}}{24 M} \frac{\Gamma(3-D / 2)}{\Gamma(-D / 2)} \frac{[\nabla V(\mathbf{x})]^{2}}{[V(\mathbf{x})]^{3}}+\ldots\right\} . $$

$$ \operatorname{Tr} \log \left[-\hbar^{2} \partial_{x}^{2}+V(x)\right]=\frac{1}{\hbar} \int d x \sqrt{V(x)}\left\{1+\frac{\hbar^{2}}{32} \frac{\left[V^{\prime}(x)\right]^{2}}{V^{3}(x)}+\ldots\right\} $$

$$ \begin{align*} \operatorname{Tr} \log \left[-\frac{\hbar^{2}}{2 M} \boldsymbol{\nabla}^{2}+V(\mathbf{x})\right] & =\operatorname{Tr} \log \left[-\frac{\hbar^{2}}{2 M} \boldsymbol{\nabla}^{2}+V+\delta V(\mathbf{x})\right] \\ & =\operatorname{Tr} \log \left(-\frac{\hbar^{2}}{2 M} \boldsymbol{\nabla}^{2}+V\right)+\operatorname{Tr} \log \left(1+\Delta_{V} \delta V\right) \end{align*} $$

$$ \Delta_{V}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left(-\frac{\hbar^{2}}{2 M} \nabla^{2}+V\right)^{-1}=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{e^{i \mathbf{p}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) / \hbar}}{\mathbf{p}^{2} / 2 M+V} \equiv \Delta_{V}\left(\mathbf{x}-\mathbf{x}^{\prime}\right) $$

$$ \operatorname{Tr} \log \hat{H}=\left.[\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}\right|_{V(\mathbf{x}) \rightarrow V}+\operatorname{Tr} \log \left(1+\Delta_{V} \delta V\right) $$

$$ \operatorname{Tr} \log \left(1+\Delta_{V} \delta V\right)=\operatorname{Tr} \Delta_{V} \delta V-\frac{1}{2} \operatorname{Tr}\left(\Delta_{V} \delta V\right)^{2}+\ldots, $$

$$ \operatorname{Tr} \Delta_{V} \delta V=\int d^{D} x \Delta_{V}(\mathbf{x}, \mathbf{x}) \delta V(\mathbf{x})=\Delta_{V}(\mathbf{0}) \int d^{D} x \delta V(\mathbf{x}) $$

$$ \Delta_{V}(\mathbf{0})=\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{1}{\mathbf{p}^{2} / 2 M+V}=\left.\partial_{V}[\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}\right|_{V(\mathbf{x}) \rightarrow V^{\prime}} $$

$$ -\frac{1}{2} \operatorname{Tr}\left(\Delta_{V} \delta V\right)^{2}=-\frac{1}{2} \int d^{D} x \int d^{D} x^{\prime} \Delta_{V}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \delta V\left(\mathbf{x}^{\prime}\right) \Delta_{V}\left(\mathbf{x}^{\prime}, \mathbf{x}\right) \delta V(\mathbf{x}) $$

$$ [f(\hat{A}), \hat{B}]=f^{\prime}(\hat{A})[\hat{A}, \hat{B}]-\frac{1}{2} f^{\prime \prime}(\hat{A})[\hat{A},[\hat{A}, \hat{B}]]+\ldots, $$

$$ \delta V \Delta_{V}=\Delta_{V} \delta V+\Delta_{V}^{2}[\hat{T}, \delta V]+\Delta_{V}^{3}[\hat{T},[\hat{T}, \delta V]]+\ldots $$

$$ \begin{align*} {[\hat{T}, f] } & =-\frac{\hbar^{2}}{2 M}\left[\left(\boldsymbol{\nabla}^{2} f\right)+2(\boldsymbol{\nabla} f) \cdot \boldsymbol{\nabla}\right], \\ {[\hat{T},[\hat{T}, f]] } & =\frac{\hbar^{4}}{4 M^{2}}\left\{\left[\left(\boldsymbol{\nabla}^{2}\right)^{2} f\right]+4\left[\boldsymbol{\nabla} \boldsymbol{\nabla}^{2} f\right] \cdot \boldsymbol{\nabla}+4\left[\nabla_{i} \nabla_{i} f\right] \nabla_{i} \nabla_{j}\right\}, \\ & \vdots \end{align*} $$

$$ -\frac{1}{2} \int d^{D} x \int d^{D} x^{\prime} \Delta_{V}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \Delta_{V}\left(\mathbf{x}^{\prime}, \mathbf{x}\right)[\delta V(\mathbf{x})]^{2} $$

$$ \begin{gather*} \int d^{D} x \int d^{D} x_{1} \cdots \int d^{D} x_{n} \Delta_{V}\left(\mathbf{x}, \mathbf{x}_{1}\right) \Delta_{V}\left(\mathbf{x}_{1}, \mathbf{x}_{2}\right) \cdots \Delta_{V}\left(\mathbf{x}_{n-1}, \mathbf{x}_{n}\right) \Delta_{V}\left(\mathbf{x}_{n}, \mathbf{x}\right) \\ =\int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{1}{\left(\mathbf{p}^{2} / 2 M+V\right)^{n+1}}=\frac{(-1)^{n}}{n!} \partial_{V}^{n} \Delta_{V}(\mathbf{0}) \end{gather*} $$

$$ \frac{1}{2} \partial_{V} \Delta_{V}(\mathbf{0}) \int d^{D} x[\delta V(\mathbf{x})]^{2} $$

$$ \begin{array}{rl} \frac{\hbar^{2}}{4 M} \int d^{D} & x \int d^{D} x_{1} \int d^{D} x_{2} \Delta_{V}\left(\mathbf{x}, \mathbf{x}_{1}\right) \Delta_{V}\left(\mathbf{x}_{1}, \mathbf{x}_{2}\right) \Delta_{V}\left(\mathbf{x}_{2}, \mathbf{x}\right) \\ & \times\left\{\left[\nabla^{2} \delta V(\mathbf{x})\right] \delta V(\mathbf{x})+2[\nabla \delta V(\mathbf{x})]^{2}+2[\nabla \delta V(\mathbf{x})] \delta V(\mathbf{x}) \nabla\right\} \end{array} $$

$$ \frac{\hbar^{2}}{8 M} \int d^{D} x\left\{\left[\boldsymbol{\nabla}^{2} \delta V(\mathbf{x})\right][\delta V(\mathbf{x})]+2[\boldsymbol{\nabla} \delta V(\mathbf{x})]^{2}\right\} \partial_{V}^{2} \Delta_{V}(\mathbf{0}) $$

$$ \begin{gather*} -\frac{\hbar^{4}}{8 M^{2}} \int d^{D} x \int d^{D} x_{1} \int d^{D} x_{2} \int d^{D} x_{3} \Delta_{V}\left(\mathbf{x}, \mathbf{x}_{1}\right) \Delta_{V}\left(\mathbf{x}_{1}, \mathbf{x}_{2}\right) \Delta_{V}\left(\mathbf{x}_{2}, \mathbf{x}_{3}\right) \Delta_{V}\left(\mathbf{x}_{3}, \mathbf{x}\right) \\ \times 4\left[\nabla_{i} \nabla_{j} \delta V(\mathbf{x})\right] \delta V(\mathbf{x}) \nabla_{i} \nabla_{j} \end{gather*} $$

$$ \begin{align*} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{-4 p_{i} p_{j} / \hbar^{2}}{\left(\mathbf{p}^{2} / 2 M+V\right)^{4}} & =-\frac{8 M}{\hbar^{2}} \frac{\delta_{i j}}{D} \int \frac{d^{D} p}{(2 \pi \hbar)^{D}}\left[\frac{1}{\left(\mathbf{p}^{2} / 2 M+V\right)^{3}}-\frac{V}{\left(\mathbf{p}^{2} / 2 M+V\right)^{4}}\right] \\ & =-\frac{8 M}{\hbar^{2}} \frac{\delta_{i j}}{D}\left(\frac{1}{2} \partial_{V}^{2}+\frac{1}{6} V \partial_{V}^{3}\right) \Delta_{V}(\mathbf{0}) \end{align*} $$

$$ -\frac{\hbar^{2}}{M} \int d^{D} x[\boldsymbol{\nabla} \delta V(\mathbf{x})]^{2} \frac{\delta_{i j}}{D}\left(\frac{1}{2} \partial_{V}^{2}+\frac{1}{6} V \partial_{V}^{3}\right) \Delta_{V}(\mathbf{0}) $$

$$ \operatorname{Tr} \log \left[-\hbar^{2} \partial_{x}^{2}+V(x)\right]=\frac{1}{\hbar} \int d x \sqrt{V(x)}\left\{1+\frac{\hbar^{2}}{32} \frac{\left[V^{\prime}(x)\right]^{2}}{V^{3}(x)}+\ldots\right\} $$

$$ \begin{gather*} \frac{1}{\hbar} \int d x \sqrt{V(x)}\left\{1-\hbar \frac{V^{\prime}}{4 V^{3 / 2}}-\hbar^{2}\left(\frac{5 V^{\prime 2}}{32 V^{3}}-\frac{V^{\prime \prime}}{8 V^{2}}\right)-\hbar^{3}\left(\frac{15 V^{\prime 3}}{64 V^{9 / 2}}-\frac{9 V^{\prime} V^{\prime \prime}}{32 V^{7 / 2}}+\frac{V^{(3)}}{16 V^{5 / 2}}\right)\right. \\ \left.-\hbar^{4}\left(\frac{1105 V^{\prime 4}}{2048 V^{6}}-\frac{221 V^{\prime 2} V^{\prime \prime}}{256 V^{5}}+\frac{19 V^{\prime \prime 2}}{128 V^{4}}+\frac{7 V^{\prime} V^{(3)}}{32 V^{4}}-\frac{V^{(4)}}{32 V^{3}}\right)\right\} \end{gather*} $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=e^{-i t\left[\hat{\mathbf{p}}^{2} / 2 M+V(\mathbf{x})\right] / \hbar}\left\langle\mathbf{x} \mid \mathbf{x}^{\prime}\right\rangle=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{-i t\left[-\hbar^{2} \nabla^{2} / 2 M+V(\mathbf{x})\right] / \hbar} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} $$

$$ e^{-i \mathbf{k x}} \boldsymbol{\nabla} e^{i \mathbf{k x}}=\boldsymbol{\nabla}+i \mathbf{k} $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)} e^{-i t\left[-\hbar^{2}(\boldsymbol{\nabla}+i \mathbf{k})^{2} / 2 M+V(\mathbf{x})\right] / \hbar} $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-i t \hbar \mathbf{k}^{2} / 2 M} e^{-i t V(\mathbf{x}) / \hbar} C(\mathbf{x} ; \mathbf{k}, t) $$

$$ C(\mathbf{x} ; \mathbf{k}, t) \equiv e^{i t\left[\hbar^{2} k^{2}+V(\mathbf{x})\right] / \hbar} e^{-i t\left[-\hbar^{2}(\boldsymbol{\nabla}+i \mathbf{k})^{2} / 2 M+V(\mathbf{x})\right] / \hbar}=e^{i t V(\mathbf{x}) / \hbar} e^{-i t\left[\hat{H}-i \hbar^{2} \mathbf{k} \boldsymbol{\nabla} / M\right] / \hbar} $$

$$ C(\mathbf{x} ; \mathbf{k}, t)=\sum_{n=0}^{\infty} t^{n} C^{(n)}(\mathbf{x} ; \mathbf{k}) $$

$$ \int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-i t \hbar \mathbf{k}^{2} / 2 M}=\frac{1}{(2 \pi i \hbar / M)^{D / 2}} e^{i M\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2 \hbar t} $$

$$ \langle f(\mathbf{k})\rangle_{k} \equiv \int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-i t \hbar \mathbf{k}^{2} / 2 M} f(\mathbf{k}) / \int \frac{d^{D} k}{(2 \pi)^{D}} e^{i \mathbf{k}\left(\mathbf{x}-\mathbf{x}^{\prime}\right)-i t \hbar \mathbf{k}^{2} / 2 M} $$

$$ \langle f(\mathbf{k})\rangle_{k}=f\left(-i \nabla_{\mathbf{x}}\right) e^{i M\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2 \hbar t} $$

$$ \begin{align*} \left\langle k_{i}\right\rangle_{k} & =-i \nabla_{i} e^{i M\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2 \hbar t}=\kappa_{i} e^{i M(\Delta \mathbf{x})^{2} / 2 \hbar t}, \quad \kappa_{i} \equiv \frac{M}{\hbar} \frac{\left(x-x^{\prime}\right)_{i}}{t}, \\ \left\langle k_{i} k_{j}\right\rangle_{k} & =i \nabla_{i} \nabla_{j} e^{i M\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2 \hbar t}=\left(\kappa_{i} \kappa_{j}+\frac{M}{i \hbar t} \delta_{i j}\right) e^{i M(\Delta \mathbf{x})^{2} / 2 \hbar t}, \\ \left\langle k_{i} k_{j} k_{k}\right\rangle_{k} & =-\nabla_{i} \nabla_{j} \nabla_{k} e^{i M \Delta \mathbf{x}^{2} / 2 \hbar t}=\left[\kappa_{i} \kappa_{j} \kappa_{k}+\frac{M}{i \hbar t}\left(\delta_{i j} \kappa_{k}+\delta_{j k} \kappa_{i}+\delta_{k i} \kappa_{j}\right)\right] e^{i M(\Delta \mathbf{x})^{2} / 2 \hbar t} . \end{align*} $$

$$ \begin{align*} \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle & =\frac{1}{(2 \pi i \hbar / M)^{D / 2}}\langle C(\mathbf{x}, \mathbf{k}, t)\rangle_{q} \\ & =\frac{1}{(2 \pi i \hbar / M)^{D / 2}} C\left(\mathbf{x} ;-i \nabla_{\mathbf{x}}, t\right) e^{i M\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2 \hbar t} \end{align*} $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\frac{e^{i M(\Delta \mathbf{x})^{2} / 2 \hbar t}}{(2 \pi i \hbar / M)^{D / 2}} \sum_{n=0}^{\infty} t^{n} a_{n}(\mathbf{x}, \Delta \mathbf{x}) $$

$$ \left.C(\mathbf{x} ; \mathbf{q} \sqrt{M / t \hbar}, t)=\left[e^{-i t\left[-\hbar^{2} \nabla^{2} / 2 M+V(\mathbf{x})-i \mathbf{q} \sqrt{i \hbar / M t}\right.} \hbar \nabla\right] / \hbar\right]_{Y^{n}}, $$

$$ \langle\mathbf{x}| e^{-i \hat{H} t / \hbar}\left|\mathbf{x}^{\prime}\right\rangle=\frac{e^{i M(\Delta \mathbf{x})^{2} / 2 \hbar t}}{(2 \pi i \hbar / M)^{D / 2}}\left[e^{-\Delta \mathbf{x}^{2} / 2}\langle C(\mathbf{x}, \mathbf{q} \sqrt{M / t \hbar}, t)\rangle_{q}\right]_{Y^{n}}, $$

$$ \langle f(\mathbf{q})\rangle_{q} \equiv \int \frac{d^{D} q}{(2 \pi)^{D}} f(\mathbf{q}) e^{i \mathbf{q} \Delta \mathbf{x}-q^{2} / 2} / \int \frac{d^{D} q}{(2 \pi)^{D}} e^{i \mathbf{q} \Delta \mathbf{x}-q^{2} / 2} $$

$$ \begin{align*} \int \frac{d^{D} q}{(2 \pi)^{D}} e^{-q^{2} / 2}\left\{1, q_{i} q_{j}, q_{i_{1}} q_{i_{2}} q_{i_{3}} q_{4}, \ldots \ldots,\right. & \left.q_{i_{1}} q_{i_{2}} \cdots q_{i_{2 n-1}} q_{i_{2 n}}, \ldots\right\} \\ = & \frac{1}{(2 \pi)^{D / 2}}\left\{1, \delta_{i j}, \delta_{i_{1} i_{2} i_{3} i_{4}}, \ldots, \delta_{i_{1} i_{2} \ldots i_{2 n-1} i_{2 n}}, \ldots\right\} \end{align*} $$

$$ \begin{align*} \left\langle q_{i}\right\rangle_{q} & =-i \nabla_{i} e^{-(\Delta \mathbf{x})^{2} / 2}=\Delta x_{i} e^{-(\Delta \mathbf{x})^{2} / 2} \\ \left\langle q_{i} q_{j}\right\rangle_{q} & =i \nabla_{i} \nabla_{j} e^{-\left(\mathbf{x}-\mathbf{x}^{\prime}\right)^{2} / 2}=\left(\Delta x_{i} \Delta x_{j}+\delta_{i j}\right) e^{-(\Delta \mathbf{x})^{2} / 2} \\ \left\langle q_{i} q_{j} q_{q}\right\rangle_{q} & =-\nabla_{i} \nabla_{j} \nabla_{k} e^{i M \Delta \mathbf{x}^{2} / 2}=\left[\Delta x_{i} \Delta x_{j} \Delta x_{k}+\left(\delta_{i j} \Delta x_{k}+\delta_{j k} \Delta x_{i}+\delta_{k i} \Delta x_{j}\right)\right] e^{-(\Delta \mathbf{x})^{2} / 2} \end{align*} $$

$$ \delta_{i_{1} \ldots i_{2 n}}=\delta_{i_{1} i_{2}} \delta_{i_{3} i_{4} \ldots i_{2 n}}+\delta_{i_{1} i_{3}} \delta_{i_{2} i_{4} \ldots i_{2 n}}+\ldots+\delta_{i_{1} i_{2 n}} \delta_{i_{2} i_{3} \ldots i_{2 n-1}} . $$

$$ \left[D_{i}, D_{j}\right]=-i \epsilon_{i j k}(e / c \hbar) B_{k} $$

$$ \begin{align*} & a_{0}=1, \quad a_{1}=0 \\ & a_{2}=\left\langle\frac{1}{2}\left(-\hat{H}^{2}-V^{2}\right)-\frac{\hbar^{2}}{3!M}\left[\hat{H}(\mathbf{q D})^{2}+(\mathbf{q D}) \hat{H}(\mathbf{q D})+(\mathbf{q D})^{2} \hat{H}\right]-\frac{\hbar^{4}}{4!M^{2}}(\mathbf{q D})^{4}\right\rangle_{q} \end{align*} $$

$$ a_{2}=-\frac{1}{2}\left(\hat{H}^{2}-V^{2}\right)-\frac{\hbar^{2}}{3!M}\left[\hat{H} \mathbf{D}^{2}+\mathbf{D} \hat{H} \mathbf{D}+\mathbf{D}^{2} \hat{H}\right]-\frac{\hbar^{4}}{8 M^{2}}\left(\mathbf{D}^{4}+D_{i} D_{j} D_{i} D_{j}+D_{i} \mathbf{D}^{2} D_{i}\right) $$

$$ \begin{align*} a_{2}=-\frac{1}{2} & {\left[\left(\frac{\hbar^{2} \mathbf{D}^{2}}{2 M}\right)^{2}-\frac{\hbar^{2}}{2 M}\left(\mathbf{D}^{2} V+V \mathbf{D}^{2}\right)\right]-\frac{\hbar^{2}}{3!M}\left[\hat{H} \mathbf{D}^{2}+\mathbf{D} \hat{H} \mathbf{D}+\mathbf{D}^{2} \hat{H}\right] } \\ & -\frac{\hbar^{4}}{8 M^{2}}\left(\mathbf{D}^{4}+D_{i} D_{j} D_{i} D_{j}+D_{i} \mathbf{D}^{2} D_{i}\right) . \end{align*} $$

$$ a_{2}=\frac{\hbar^{2}}{12 M}\left[\mathbf{D}^{2} V(x)+V(x) \mathbf{D}^{2}\right]-\frac{\hbar^{2}}{6 M} D_{i} V(x) D_{i}-\frac{\hbar^{4}}{48 M^{2}}\left[D_{i}, D_{j}\right]^{2} $$

$$ a_{2}=\frac{\hbar^{2}}{12 M} \nabla^{2} V(x) $$

$$ \Delta a_{2}^{B}=-\frac{\hbar^{4}}{48 M^{2}}\left(\left[D_{1}, D_{2}\right]^{2}+\left[D_{2}, D_{1}\right]^{2}\right)=\frac{\hbar^{2}}{24 M^{2}}\left(\frac{e}{c}\right)^{2} B^{2} $$

$$ F_{\omega}=-\frac{\hbar}{\sqrt{\pi}} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{-1 / 2}\left[1+2 \sum_{n=1}^{\infty} e^{-(n \hbar \beta)^{2} / 4 \tau}\right] e^{-\tau \omega^{2}} $$

$$ \frac{1}{b} \sum_{m} e^{-\tau \hbar k_{m}^{2} / 2 M}=\sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{-\tau \hbar k^{2} / 2 M+i b k n}=\frac{1}{\sqrt{2 \pi \hbar \tau / M}} \sum_{n=-\infty}^{\infty} e^{-n^{2} M b^{2} / 2 \hbar \tau} $$

$$ \rho_{\mathrm{cl}}(b, E ; x)=2 \int_{0}^{\infty} \frac{d \tau}{2 \pi \hbar} \sum_{n=-\infty}^{\infty} \frac{e^{-n^{2} M b^{2} / 2 \hbar \tau-\tau[E-V(x)] / \hbar}}{(2 \pi \hbar \tau / M)^{1 / 2}} $$

$$ \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{\nu} e^{-n^{2} M b^{2} / 2 \hbar \tau-[E-V(x)] \tau / \hbar}=2\left[\frac{n M b}{p(E ; x)}\right]^{\nu} K_{\nu}(n p(E ; x) b / \hbar) $$

$$ \rho_{\mathrm{cl}}(b, E ; x)=\frac{1}{\pi \hbar} \frac{1}{\sqrt{2 \pi \hbar / M}} 2 \sum_{n=0}^{\infty}\left[\frac{n M b}{p(E ; x)}\right]^{1 / 2} K_{1 / 2}(n p(E ; x) b / \hbar) $$

$$ \rho_{\mathrm{cl}}(b, E ; x)=\frac{M}{\pi \hbar} \frac{1}{p(E ; x)}\left(1+2 \sum_{n=1}^{\infty} e^{-n p(E ; x) b / \hbar}\right) $$

$$ \rho_{\mathrm{cl}}(b, E ; x)=\frac{M}{\pi \hbar} \frac{\operatorname{coth}[p(E ; x) b / 2 \hbar]}{p(E ; x)}=\frac{M}{\pi \hbar} \frac{\operatorname{coth} \sqrt{2 M[E-V(x)]} b / 2 \hbar}{\sqrt{2 M[E-V(x)]}} . $$

$$ \rho(b, E ; x)=\left\{1-\frac{\hbar^{2}}{12 M} V^{\prime \prime}(x) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}\left[V^{\prime}(x)\right]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \rho_{\mathrm{cl}}(b, E ; x) $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=-\int_{0}^{b} d x \int_{-\infty}^{\infty} d E \int_{-\infty}^{\infty} \frac{d t}{2 \pi \hbar} \sum_{n=-\infty}^{\infty} \frac{e^{-i n^{2} M b^{2} / 2 \hbar t+i t[E-V(x)] / \hbar}}{(2 \pi i \hbar t / M)^{1 / 2}} \int_{0}^{\infty} \frac{d t^{\prime}}{t^{\prime}} e^{-i E t^{\prime} / \hbar} $$

$$ [\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}}=-\sum_{n=-\infty}^{\infty} \int_{0}^{b} d x \int_{0}^{\infty} \frac{d t}{t} \frac{1}{(2 \pi i \hbar t / M)^{1 / 2}} e^{-i n^{2} M b^{2} / 2 \hbar t-i t V(x) / \hbar} $$

$$ \begin{align*} {[\operatorname{Tr} \log \hat{H}]_{\mathrm{cl}} } & =-\left.\pi \int_{0}^{b} d x \int_{V(x)}^{\infty} d V \rho_{\mathrm{cl}}(b, E ; x)\right|_{E-V(x) \rightarrow V} \\ & =\pi \int_{0}^{b} d x \int_{V(x)}^{\infty} \frac{d V}{E_{b}} \frac{1}{4 \pi b} \frac{\operatorname{coth} \frac{1}{2} \sqrt{V / E_{b}}}{\frac{1}{2} \sqrt{V / E_{b}}} \end{align*} $$

$$ \operatorname{Tr} \log \hat{H}=\int_{0}^{b} d x\left\{1+\frac{\hbar^{2}}{24 M}\left[V^{\prime}(x)\right]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \frac{2}{b} \log \left\{2 \sinh \left[\frac{1}{2} \sqrt{\frac{V(x)}{E_{b}}}\right]\right\} $$

$$ \operatorname{Tr} \log \left[-\partial_{\tau}^{2}+\omega^{2}(\tau)\right]=\int_{0}^{\hbar \beta} d \tau\left\{1+\frac{\left[\partial_{\tau} \omega^{2}(\tau)\right]^{2}}{12} \partial_{\omega^{2}}^{3}+\ldots\right\} \frac{2}{\hbar \beta} \log \left[2 \sinh \frac{\hbar \beta \omega(\tau)}{2}\right] $$

$$ \rho(E)=-\frac{1}{\pi} \partial_{E} \operatorname{Im} \operatorname{Tr} \log \left\{-\partial_{x}^{2}+[V(x)-E]\right\} $$

$$ -\operatorname{Im} \operatorname{Tr} \log \left\{-\partial_{x}^{2}+[V(x)-E]\right\}=\pi(n+1 / 2) $$

$$ S_{\mathrm{qc}}(E)=-2 \hbar \operatorname{Im} \operatorname{Tr} \log \left\{-\partial_{x}^{2}+[V(x)-E]\right\} $$

$$ \begin{align*} & S_{\mathrm{qc}}(E)=2 \int d x \sqrt{E-V(x)}\left\{1+\hbar^{2}\left[\frac{5 V^{\prime 2}}{32(E-V)^{3}}+\frac{V^{\prime \prime}}{8(E-V)^{2}}\right]\right. \\ & \left.\quad-\hbar^{4}\left[\frac{1105 V^{\prime 4}}{2048(E-V)^{6}}+\frac{221 V^{\prime 2} V^{\prime \prime}}{256(E-V)^{5}}+\frac{19 V^{\prime \prime 2}}{128(E-V)^{4}}+\frac{7 V^{\prime} V^{(3)}}{32(E-V)^{4}}+\frac{V^{(4)}}{32(E-V)^{3}}+\ldots\right]\right\} \end{align*} $$

$$ \begin{align*} & \rho(E)=\frac{1}{2 \pi \hbar} S_{\mathrm{qc}}(E)=\frac{1}{2 \pi \hbar} \int d x \frac{1}{\sqrt{E-V}}\left\{1-\hbar^{2}\left[\frac{25 V^{\prime 2}}{32(E-V)^{3}}+\frac{3 V^{\prime \prime}}{8(E-V)^{2}}\right]\right. \\ & \left.\quad+\hbar^{3}\left[\frac{12155 V^{\prime 4}}{2048(E-V)^{6}}+\frac{1989 V^{\prime 2} V^{\prime \prime}}{256(E-V)^{5}}+\frac{133 V^{\prime \prime 2}}{128(E-V)^{4}}+\frac{49 V^{\prime} V^{(3)}}{32(E-V)^{4}}+\frac{5 V^{(4)}}{32(E-V)^{3}}\right]\right\} . \end{align*} $$

$$ \begin{align*} N(\nu) & =\nu-\frac{1}{12 \pi \nu}+\frac{11 \pi^{2}}{10368 \Gamma^{8}\left(\frac{3}{4}\right) \nu^{3}}+\frac{4697 \pi}{1866240 \Gamma^{8}\left(\frac{3}{4}\right) \nu^{5}}-\frac{390065 \pi^{4}}{501645312 \Gamma^{16}\left(\frac{3}{4}\right) \nu^{7}} \\ & -\frac{53352893 \pi^{3}}{7739670528 \Gamma^{16}\left(\frac{3}{4}\right) \nu^{9}}+\ldots=n+1 / 2 \end{align*} $$

$$ \kappa^{(n)}=\left[\nu^{(n)} 3 \Gamma(3 / 2)^{2} / 2 \sqrt{\pi}\right]^{4 / 3} . $$

$$ \begin{align*} \nu^{(n)}=\left(n+\frac{1}{2}\right)\left[1+\frac{0.026525823}{\left(n+\frac{1}{2}\right)^{2}}\right. & -\frac{0.002762954}{\left(n+\frac{1}{2}\right)^{4}}-\frac{0.001299177}{\left(n+\frac{1}{2}\right)^{6}} \\ & \left.+\frac{0.003140091}{\left(n+\frac{1}{2}\right)^{8}}+\frac{0.007594497}{\left(n+\frac{1}{2}\right)^{10}}+\ldots\right] . \end{align*} $$

$$ Z_{\mathrm{qc}}(E)=t_{\mathrm{qc}}(E) \frac{1}{P_{\mathrm{qc}}(1)^{1 / 2}} \frac{e^{i S_{\mathrm{qc}}(E)-i \pi \nu / 2}}{1-e^{i S_{\mathrm{qc}}(E)-i \pi \nu / 2}} . $$

$$ \rho_{\mathrm{cl}}^{(-)}(\mathbf{x})=\int_{V(\mathbf{x})}^{0} d E \rho_{\mathrm{cl}}(E ; \mathbf{x})=\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \frac{1}{\Gamma(D / 2+1)}[-V(\mathbf{x})]^{D / 2} . $$

$$ E_{F}=\frac{p_{F}(\mathbf{x})^{2}}{2 M}+V(\mathbf{x}) . $$

$$ p_{F}(\mathbf{x})=p\left(E_{F} ; \mathbf{x}\right)=\sqrt{2 M\left[E_{F}-V(\mathbf{x})\right]} . $$

$$ \rho_{\mathrm{cl}}^{(-)}(\mathbf{x})=\int_{|\mathbf{p}| \leq p_{F}(\mathbf{x})} \frac{d^{D} p}{(2 \pi \hbar)^{D}}=\frac{1}{(2 \pi \hbar)^{D}} S_{D} \int_{0}^{p_{F}(\mathbf{x})} d p p^{D-1}=\frac{1}{(2 \pi \hbar)^{D}} \frac{2 \pi^{D / 2}}{\Gamma(D / 2)} \frac{p_{F}^{D}(\mathbf{x})}{D} $$

$$ n(\mathbf{x})=2 \rho_{\mathrm{cl}}^{(-)}(\mathbf{x}) $$

$$ E_{\mathrm{potTF}}^{(-)}(\mathbf{x})=V(\mathbf{x}) \rho_{\mathrm{cl}}^{(-)}(\mathbf{x})=-\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \frac{1}{\Gamma(D / 2+1)}[-V(\mathbf{x})]^{D / 2+1} $$

$$ \begin{align*} E_{\mathrm{kinTF}}^{(-)}(\mathbf{x}) & =\int_{V(\mathbf{x})}^{0} d E[E-V(\mathbf{x})] \rho_{\mathrm{cl}}(E ; \mathbf{x}) \\ & =\frac{D / 2}{D / 2+1}\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \frac{1}{\Gamma(D / 2+1)}[-V(\mathbf{x})]^{D / 2+1} \end{align*} $$

$$ E_{\mathrm{kinTF}}^{(-)}(\mathbf{x})=\int_{|\mathbf{p}| \leq p_{F}(\mathbf{x})} \frac{d^{D} p}{(2 \pi \hbar)^{D}} \frac{p^{2}}{2 M} $$

$$ E_{\mathrm{kinTF}}^{(-)}(\mathbf{x})=\frac{1}{(2 \pi \hbar)^{D}} S_{D} \frac{1}{2 M} \int_{0}^{p_{F}(\mathbf{x})} d p p^{D+1}=\frac{1}{(2 \pi \hbar)^{D}} \frac{S_{D}}{D+2} \frac{p_{F}^{D+2}(\mathbf{x})}{2 M} $$

$$ \begin{align*} E_{\mathrm{TF}}^{(-)}(\mathbf{x}) & =\int_{V(\mathbf{x})}^{0} d E E \rho_{\mathrm{cl}}(E ; \mathbf{x}) \\ & =-\frac{1}{D / 2+1}\left(\frac{M}{2 \pi \hbar^{2}}\right)^{D / 2} \frac{1}{\Gamma(D / 2+1)}[-V(\mathbf{x})]^{D / 2+1} \end{align*} $$

$$ E_{\mathrm{TF}}^{(-)}(\mathbf{x})=-\frac{1}{D / 2} E_{\mathrm{kin} \mathrm{TF}}^{(-)}(\mathbf{x})=\frac{1}{D / 2+1} E_{\mathrm{pot} \mathrm{TF}}^{(-)}(\mathbf{x}) $$

$$ \nabla^{2} V(\mathbf{x})=4 \pi e^{2}\left[Z \delta^{(3)}(\mathbf{x})-n(\mathbf{x})\right] \equiv 4 \pi e^{2}\left[n_{\mathrm{C}}(\mathbf{x})-n(\mathbf{x})\right] . $$

$$ V_{\mathrm{C}}(\mathbf{x})=-\frac{Z e^{2}}{r} $$

$$ a_{H}=\frac{\hbar^{2}}{M e^{2}}=\frac{1}{\alpha} \lambda_{M}^{\mathrm{C}} $$

$$ \lambda_{M}^{\mathrm{C}} \equiv \hbar / M c \approx 3.86159323 \times 10^{-13} \mathrm{~cm} . $$

$$ V(\mathbf{x})=-\frac{Z e^{2}}{r} f(r) $$

$$ f(0)=1, $$

$$ a_{\mathrm{TF}}=\frac{1}{e^{2} Z^{1 / 3}} \frac{2 \pi \hbar^{2}}{M}\left[\frac{\Gamma(5 / 2)}{2 \cdot 4 \pi}\right]^{2 / 3}=\frac{1}{2}\left(\frac{3 \pi}{4}\right)^{2 / 3} \frac{a_{H}}{Z^{1 / 3}} \approx 0.8853 \frac{a_{H}}{Z^{1 / 3}}, $$

$$ r=a_{\mathrm{TF}} \xi . $$

$$ n(\mathbf{x})=-\frac{\left(2 Z e^{2} M\right)^{3 / 2}}{3 \pi^{2} \hbar^{3}}\left[\frac{f(\xi)}{a_{\mathrm{TF}} \xi}\right]^{3 / 2}=\frac{Z}{4 \pi a_{\mathrm{TF}}^{3}}\left[\frac{f(\xi)}{\xi}\right]^{3 / 2} . $$

$$ \nabla^{2} V(\mathbf{x})=\frac{1}{r} \frac{d}{d r^{2}} r V(\mathbf{x})=-\frac{Z e^{2}}{a_{\mathrm{TF}}^{3}} \frac{1}{\xi} f^{\prime \prime}(\xi), $$

$$ f^{\prime \prime}(\xi)=\frac{1}{\sqrt{\xi}} f^{3 / 2}(\xi), \quad \xi>0 . $$

$$ f(\xi)=1-s \xi+\ldots, $$

$$ s \approx 1.58807 . $$

$$ f(\xi) \approx \frac{144}{\xi^{3}} $$

$$ E_{\mathrm{kin}}^{(-)}=2 \frac{3}{5}\left(\frac{M}{2 \pi \hbar^{2}}\right)^{3 / 2} \frac{1}{\Gamma(5 / 2)} \int d^{3} x[-V(\mathbf{x})]^{5 / 2} $$

$$ E_{\mathrm{kin}}^{(-)}=\frac{3}{5} \kappa \int d^{3} x n^{5 / 3}(\mathbf{x}) $$

$$ \kappa \equiv \frac{\hbar}{2 M}\left(3 \pi^{2}\right)^{2 / 3} $$

$$ E_{\mathrm{pot}}^{(-)}=\int d^{3} x V(\mathbf{x}) n(\mathbf{x}) $$

$$ E_{\mathrm{pot}}^{(-)}=-\frac{5}{3} E_{\mathrm{kin}}^{(-)} $$

$$ E_{\mathrm{e}}^{(-)}=E_{\mathrm{kin}}^{(-)}+E_{\mathrm{pot}}^{(-)}=\frac{2}{5} E_{\mathrm{pot}}^{(-)} $$

$$ E_{\mathrm{e}}^{(-)}=\mathcal{E}_{\mathrm{e}}[n] \equiv \frac{3}{5} \kappa \int d^{3} x n^{5 / 3}(\mathbf{x})+\int d^{3} x V(\mathbf{x}) n(\mathbf{x}) $$

$$ E_{\mathrm{pot}}^{(-)}=E_{\mathrm{C}}^{(-)} \equiv \int d^{3} x V_{\mathrm{C}}(\mathbf{x}) n(\mathbf{x}) $$

$$ E_{\mathrm{ee}}^{(-)}=\mathcal{E}_{\mathrm{ee}}[n]=\frac{e^{2}}{2} \int d^{3} x d^{3} x^{\prime} n(\mathbf{x}) \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} n\left(\mathbf{x}^{\prime}\right) $$

$$ \mathcal{E}_{\text {tot }}[n]=\frac{3}{5} \kappa \int d^{3} x n^{5 / 3}(\mathbf{x})+\int d^{3} x V_{\mathrm{C}}(\mathbf{x}) n(\mathbf{x})+\frac{e^{2}}{2} \int d^{3} x d^{3} x^{\prime} n(\mathbf{x}) \frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} n\left(\mathbf{x}^{\prime}\right) $$

$$ \nabla^{2} \varphi(\mathbf{x})=-4 \pi e^{2} n(\mathbf{x}) $$

$$ \kappa n^{2 / 3}(\mathbf{x})=-V(\mathbf{x}) $$

$$ V(\mathbf{x})=V_{\mathrm{C}}(\mathbf{x})+\varphi(\mathbf{x}) $$

$$ \mathcal{E}_{\text {tot }}[\varphi]=-\frac{2}{5} \int d^{3} x V(\mathbf{x}) n(\mathbf{x})+\frac{1}{8 \pi e^{2}} \int d^{3} x \varphi(\mathbf{x}) \nabla^{2} \varphi(\mathbf{x}) $$

$$ 2 \frac{M^{3 / 2}}{(2 \pi \hbar)^{3 / 2}} \frac{4 \pi}{\Gamma(5 / 2)} e^{3}=\frac{Z^{-1 / 2}}{a_{\mathrm{TF}}^{3 / 2}} $$

$$ E_{\mathrm{pot}}^{(-)}=-\frac{Z^{2} e^{2}}{a} \int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi) $$

$$ E_{\mathrm{ee}}^{(-)}=\frac{1}{2} \int d^{3} x n(\mathbf{x}) \varphi(\mathbf{x}) $$

$$ E_{\mathrm{ee}}^{(-)}=\frac{1}{2} \int d^{3} x n(\mathbf{x})\left[V(\mathbf{x})-V_{\mathrm{C}}(\mathbf{x})\right]=\frac{1}{2} E_{\mathrm{pot}}^{(-)}-\frac{1}{2} E_{\mathrm{C}}^{(-)} $$

$$ E_{\mathrm{tot}}^{(-)}=\frac{2}{5} E_{\mathrm{pot}}^{(-)}-E_{\mathrm{ee}}^{(-)}=-\frac{1}{10} E_{\mathrm{pot}}^{(-)}+\frac{1}{2} E_{\mathrm{C}}^{(-)} $$

$$ E_{\mathrm{C}}^{(-)}=-\frac{1}{4 \pi e^{2}} \int d^{3} x \varphi(\mathbf{x}) \nabla^{2} V_{\mathrm{C}}(\mathbf{x})=-\int d^{3} x \varphi(\mathbf{x}) n_{\mathrm{C}}(\mathbf{x}) $$

$$ \varphi(\mathbf{x})=V(\mathbf{x})-V_{\mathrm{C}}(\mathbf{x})=-\frac{Z e^{2}}{r}[f(\xi)-1], \quad n_{\mathrm{C}}=Z \delta^{(3)}(\mathbf{x}) $$

$$ \varphi(\mathbf{0})=\frac{Z e^{2}}{a} s $$

$$ E_{\mathrm{C}}^{(-)}=-\frac{Z^{2} e^{2}}{a} s $$

$$ I[f]=\int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi) $$

$$ \mathcal{E}_{\mathrm{tot}}[\varphi]=-\frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} \varepsilon[f] $$

$$ \varepsilon[f] \equiv \frac{2}{5} I[f]+\frac{1}{2} J[f]=\int_{0}^{\infty} d \xi\left\{\frac{2}{5} \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi)-\frac{1}{2}[f(\xi)-1] f^{\prime \prime}(\xi)\right\} $$

$$ J[f]=\int_{0}^{\infty} d \xi\left[f^{\prime}(\xi)\right]^{2} $$

$$ J[f]=-\int_{0}^{\infty} d \xi[f(\xi)-1] f^{\prime \prime}(\xi)=\int_{0}^{\infty} d \xi\left[f^{\prime}(\xi)\right]^{2}-\left.[f(\xi)-1] f^{\prime}(\xi)\right|_{0} ^{\infty} $$

$$ f(\xi) \rightarrow \bar{f}(\xi)=f(\lambda \xi) $$

$$ \varepsilon_{\lambda}[f]=\frac{2}{5} \lambda^{-1 / 2} I[f]+\frac{1}{2} \lambda J[f] $$

$$ I[f]=\frac{5}{2} J[f] $$

$$ J=-\int_{0}^{\infty} d \xi f(\xi) f^{\prime \prime}(\xi)+\int_{0}^{\infty} d \xi f^{\prime \prime}(\xi)=-\int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi)-f^{\prime}(0)=-I+s $$

$$ I=\frac{5}{7} s, \quad J=\frac{2}{7} s $$

$$ \begin{array}{rlrl} E_{\mathrm{kin}}^{(-)} & =\frac{3}{5} \frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} \frac{5}{7} s, & E_{\mathrm{pot}}^{(-)} & =-\frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} \frac{5}{7} s, \\ E_{\mathrm{C}}^{(-)} & =-\frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} s, & E_{\mathrm{ee}}^{(-)}=\frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} \frac{1}{7} s \end{array} $$

$$ E_{\mathrm{tot}}^{(-)}=\frac{2}{5} E_{\mathrm{pot}}^{(-)}-E_{\mathrm{ee}}^{(-)}=-\frac{1}{10} E_{\mathrm{pot}}^{(-)}+\frac{1}{2} E_{\mathrm{C}}^{(-)}=-\frac{3}{7} \frac{Z^{2} e^{2}}{a_{\mathrm{TF}}} s \approx-0.7687 Z^{7 / 3} \frac{e^{2}}{a_{H}} . $$

$$ \bar{\varepsilon}[f]=-\frac{1}{10} \int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi)-\frac{1}{2} f(0) f^{\prime}(0) $$

$$ \bar{\varepsilon}[f] \equiv-\frac{1}{10} I[f]+\frac{1}{2} J_{\mathrm{C}}[f]=-\frac{1}{10} \int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{5 / 2}(\xi)+\frac{1}{2} \int_{0}^{\infty} d \xi \frac{1}{\sqrt{\xi}} f^{3 / 2}(\xi) $$

$$ E_{\mathrm{tot}}=E_{\mathrm{kin}}+E_{\mathrm{pot}} $$

$$ \lambda^{2} E_{\mathrm{kin}}+\lambda E_{\mathrm{pot}} . $$

$$ 2 E_{\mathrm{kin}}+E_{\mathrm{pot}}=0 $$

$$ E_{\mathrm{tot}}=-E_{\mathrm{kin}} . $$

$$ \rho_{\mathrm{cl}}^{(-)}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right)=\int_{V(\overline{\mathbf{x}})}^{0} d E \rho_{\mathrm{cl}}^{(-)}\left(E ; \mathbf{x}_{b}, \mathbf{x}_{a}\right) $$

$$ \int_{V(\overline{\mathbf{x}})}^{0} d E=\frac{1}{M} \int_{0}^{p_{F}(\overline{\mathbf{x}})} d p p $$

$$ \rho_{\mathrm{cl}}^{(-)}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right)=\frac{p_{F}^{3}(\overline{\mathbf{x}})}{2 \pi^{2} \hbar^{3}} \frac{1}{z^{3}}(\sin z-z \cos z) $$

$$ z \equiv p_{F}(\overline{\mathbf{x}}) R / \hbar $$

$$ \rho_{\mathrm{cl}}^{(-)}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right)=\int_{|\mathbf{p}| \leq p_{F}(\overline{\mathbf{x}})} \frac{d^{3} p}{(2 \pi \hbar)^{D}} e^{i \mathbf{p}\left(\mathbf{x}_{b}-\mathbf{x}_{a}\right) / \hbar} $$

$$ \rho^{(-)}(\mathbf{x})=\rho^{(-)}\left(\mathbf{x}_{b}, \mathbf{x}_{a}\right) $$

$$ E_{\mathrm{ee}}^{(-)}=4 \times \frac{e^{2}}{2} \int d^{3} x d^{3} x^{\prime} \rho^{(-)}(\mathbf{x}, \mathbf{x}) \frac{1}{4 \pi\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} \rho^{(-)}\left(\mathbf{x}^{\prime}, \mathbf{x}^{\prime}\right) $$

$$ E_{\mathrm{exch}}^{(-)}=-2 \times \frac{e^{2}}{2} \int d^{3} x d^{3} x^{\prime} \rho^{(-)}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \frac{1}{4 \pi\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} \rho^{(-)}\left(\mathbf{x}^{\prime}, \mathbf{x}\right) $$

$$ \int_{0}^{\infty} d z z^{2} \frac{1}{z}\left[\frac{1}{z^{3}}(\sin z-z \cos z)\right]^{2}=\frac{1}{4} $$

$$ E_{\mathrm{exch}}^{(-)}=-\frac{e^{2}}{4 \pi^{3}} \int d^{3} \overline{\mathbf{x}}\left[\frac{p_{F}(\overline{\mathbf{x}})}{\hbar}\right]^{4} $$

$$ p_{F}(\mathbf{x})=\sqrt{\frac{2 Z}{a_{H} a_{\mathrm{TF}}} \frac{f(\xi)}{\xi}} $$

$$ E_{\mathrm{exch}}^{(-)}=-\frac{4}{\pi^{2}} \frac{a_{\mathrm{TF}}}{a_{H}} I_{2} \frac{Z^{2}}{a_{H}} \approx-0.3588 Z^{5 / 3} \frac{e^{2}}{a_{H}} I_{2} $$

$$ I_{2} \equiv \int_{0}^{\infty} d \xi f^{2}(\xi) \approx 0.6154 $$

$$ E_{\mathrm{exch}}^{(-)} \approx-0.2208 Z^{5 / 3} \frac{e^{2}}{a_{H}} $$

$$ C_{\mathrm{exch}}(Z)=1+0.2872 Z^{-2 / 3} $$

$$ \frac{Z^{2} e^{2}}{a_{H}} \frac{1}{Z^{2 / 3}} \ll-\varepsilon \ll \frac{Z^{2} e^{2}}{a_{H}} $$

$$ -\varepsilon \equiv \frac{Z^{2}}{2 a_{H} \nu^{2}} $$

$$ 1 \ll \nu^{2} \ll Z^{2 / 3} $$

$$ \frac{p^{2}}{2 M}-\frac{Z e^{2}}{r}<-\varepsilon $$

$$ f(\xi) \rightarrow[-\varepsilon-V(r)] \frac{r}{Z e^{2}}=f(\xi)-\xi / \xi_{\mathrm{m}} $$

$$ \xi_{\mathrm{m}}=\frac{2 \nu^{2}}{Z} \frac{a_{H}}{a} $$

$$ \Delta E_{\mathrm{tot}}^{(-)}=-\frac{3}{5} \frac{Z^{2} e^{2}}{a} \int_{0}^{\xi_{\max }} d \xi \frac{1}{\sqrt{\xi}}\left[f(\xi)-\xi / \xi_{\mathrm{m}}\right]^{5 / 2} $$

$$ \xi_{\max }=Z e^{2} \varepsilon f\left(\xi_{\max }\right) / a $$

$$ -\varepsilon-V\left(r_{\max }\right)=0 $$

$$ \xi_{\max } \approx \xi_{\mathrm{m}} $$

$$ \int_{0}^{1 / c} d \lambda \frac{1}{\sqrt{\lambda}}(1-c \lambda)^{5 / 2}=\frac{1}{\sqrt{c}} B(1 / 2,7 / 2)=\frac{1}{\sqrt{c}} \frac{5}{8} \pi $$

$$ \Delta E_{\mathrm{tot}}^{(-)}=-\frac{3}{5} \frac{Z^{2} e^{2}}{a} \frac{5}{8} \frac{\pi}{M} \sqrt{\frac{a_{H}}{2 a}} \frac{\nu}{Z^{1 / 3}} $$

$$ \Delta E_{\mathrm{tot}}^{(-)}=-\frac{Z^{2} e^{2}}{a_{H}} \nu $$

$$ E_{n}=-\frac{Z^{2} e^{2}}{a_{H}} \frac{1}{2 n^{2}} $$

$$ \begin{align*} \Delta_{\mathrm{QM}} E_{\mathrm{tot}}^{(-)} & =-2 \frac{Z^{2} e^{2}}{a_{H}} \frac{1}{2} \sum_{n=0}^{\nu} 1 \\ & =-\frac{Z^{2} e^{2}}{a_{H}}[\nu] \end{align*} $$

$$ \Delta E_{\mathrm{corr}}^{(-)}=-\frac{Z^{2} e^{2}}{a_{H}}([\nu]-\nu) $$

$$ \langle[\nu]\rangle=\nu-\frac{1}{2}, $$

$$ \Delta E_{\mathrm{corr}}^{(-)}=\frac{Z^{2} e^{2}}{a_{H}} \frac{1}{2} $$

$$ C_{\text {sing }}(Z)=1-\frac{7 a_{\mathrm{TF}}}{6 a_{H} s} \approx 1-0.6504 Z^{-1 / 3} $$

$$ \rho^{(-)}(\mathbf{x})=\langle\mathbf{x}| \Theta(-\hat{H})|\mathbf{x}\rangle $$

$$ \Theta(-\hat{H})=\int_{-\infty}^{\infty} \frac{d t}{2 \pi i(t-i \eta)} e^{-i \hat{H} t / \hbar} $$

$$ \rho^{(-)}(\mathbf{x})=\int_{-\infty}^{\infty} \frac{d t}{2 \pi i(t-i \eta)}(\mathbf{x} t \mid \mathbf{x} 0) $$

$$ \rho(\mathbf{x})=\left\{1-\frac{\hbar^{2}}{12 M} \nabla^{2} V(\mathbf{x}) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \rho_{\mathrm{TF}}(\mathbf{x}) $$

$$ E_{\mathrm{pot}}^{(-)}(\mathbf{x})=\langle\mathbf{x}| V(\mathbf{x}) \Theta(-\hat{H})|\mathbf{x}\rangle=\int d^{3} x V(\mathbf{x}) \rho^{(-)}(\mathbf{x}) $$

$$ E_{\mathrm{pot}}^{(-)}(\mathbf{x})=V(\mathbf{x})\left\{1-\frac{\hbar^{2}}{12 M} \nabla^{2} V(\mathbf{x}) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} \rho_{\mathrm{TF}}(\mathbf{x}) $$

$$ E^{(-)}(\mathbf{x})=\langle\mathbf{x}| \hat{H} \Theta(-\hat{H})|\mathbf{x}\rangle . $$

$$ \frac{\partial}{\partial V(\mathbf{x})} E^{(-)}(\mathbf{x})=\rho^{(-)}(\mathbf{x}) $$

$$ E^{(-)}(\mathbf{x})=i \hbar \int_{-\infty}^{\infty} \frac{d t}{2 \pi i(t-i \eta)^{2}}(\mathbf{x} t \mid \mathbf{x} 0) $$

$$ E_{\mathrm{TF}}^{(-)}(\mathbf{x})=-\left(\frac{M}{2 \pi \hbar}\right)^{D / 2} \frac{1}{\Gamma(D / 2+2)}[-V(\mathbf{x})]^{D / 2+1} $$

$$ E^{(-)}(\mathbf{x})=\left\{1-\frac{\hbar^{2}}{12 M} \nabla^{2} V(\mathbf{x}) \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}[\nabla V(\mathbf{x})]^{2} \frac{d^{3}}{d V^{3}}+\ldots\right\} E_{\mathrm{TF}}^{(-)}(\mathbf{x}) $$

$$ E^{(-)}(\mathbf{x})=\frac{1}{2 M}\langle\mathbf{x}| \hat{\mathbf{p}}^{2} \Theta(-\hat{H})|\mathbf{x}\rangle $$

$$ \begin{align*} E_{\mathrm{kin}}^{(-)}(\mathbf{x}) & =-M \frac{\partial}{\partial M} i \hbar \int_{-\infty}^{\infty} \frac{d t}{2 \pi i(t-i \eta)^{2}}\langle\mathbf{x}| e^{-i \hat{H} t / \hbar}|\mathbf{x}\rangle \\ & =-M \frac{\partial}{\partial M} E^{(-)}(\mathbf{x}) \end{align*} $$

$$ \begin{align*} \Delta E_{\mathrm{e}}^{(-)}= & -\int d^{3} x\left[-\frac{\hbar^{2}}{12 M} \nabla^{2} V \frac{d^{2}}{d V^{2}}-\frac{\hbar^{2}}{24 M}(\nabla V)^{2} \frac{d^{3}}{d V^{3}}\right] \\ & \times 2 \frac{2}{5}\left(\frac{M}{2 \pi \hbar}\right)^{3 / 2} \frac{1}{\Gamma(5 / 2)}(-V)^{5 / 2} \\ = & \frac{\sqrt{2 M}}{12 \hbar \pi^{2}} \int d^{3} x\left[(-V)^{1 / 2} \nabla^{2} V-\frac{1}{4}(-V)^{-3 / 2}(\nabla V)^{2}\right] \end{align*} $$

$$ \nabla^{2} V^{3 / 2}=\frac{3}{2} \nabla\left[V^{1 / 2} \nabla V\right]=\frac{3}{4}\left[V^{-1 / 2}(\nabla V)^{2}+2 V^{1 / 2} \nabla^{2} V\right] $$

$$ \Delta E_{\mathrm{e}}^{(-)}=\frac{\sqrt{2 M}}{24 \hbar \pi^{2}} \int d^{3} x\left[(-V)^{1 / 2} \nabla^{2} V-\frac{2}{3} \nabla^{2}(-V)^{3 / 2}\right] $$

$$ E_{\mathrm{tot}}^{(-)}=E_{\mathrm{e}}^{(-)}+\Delta E_{\mathrm{e}}^{(-)}-\mathcal{E}_{\varphi \varphi}[\varphi] $$

$$ \frac{\delta}{\delta V(\mathbf{x})}\left[E_{\mathrm{e}}^{(-)}+\Delta E_{\mathrm{e}}^{(-)}\right]+\frac{1}{e^{2}} \nabla^{2}[\varphi(\mathbf{x})+\Delta \varphi(\mathbf{x})]=0 $$

$$ \frac{1}{e^{2}} \nabla^{2} \Delta \varphi(\mathbf{x})=-\frac{\delta \Delta E_{\mathrm{e}}^{(-)}}{\delta V(\mathbf{x})} $$

$$ E_{\mathrm{tot}}^{(-)}=E_{\mathrm{e}}^{(-)}+\Delta E_{\mathrm{e}}^{(-)}+\int d^{3} x \frac{\delta E_{\mathrm{e}}^{(-)}}{\delta V(\mathbf{x})} \Delta \varphi(\mathbf{x})-\mathcal{E}_{\varphi \varphi}[\varphi]-\frac{1}{4 \pi e^{2}} \int d^{3} x \boldsymbol{\nabla} \varphi(\mathbf{x}) \boldsymbol{\nabla} \Delta \varphi(\mathbf{x}) $$

$$ E<-\varepsilon \text {. } $$

$$ r