Kleinert · 제14장 Duru–Kleinert 변환

$$ \hat{H}=\hat{T}+\hat{V}, $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|\mathbf{x}_{a}\right\rangle \approx \frac{r_{b}^{\lambda} r_{a}^{1-\lambda}}{{\sqrt{2 \pi \epsilon_{s} \hbar r^{1-\lambda} r^{\lambda} / M}}^{D_{\mathcal{C}} / 2}} \prod_{n=1}^{N}\left[\int \frac{d^{D_{\mathcal{C}}} \Delta x_{n}}{{\sqrt{2 \pi \epsilon_{s} \hbar r_{n} / M}}^{D_{\mathcal{C}}}}\right] e^{-\mathcal{A}_{E}^{N} / \hbar} $$

$$ \begin{gather*} e^{i}{ }_{\mu, \xi}=e^{i}{ }_{\mu}, \\ e^{i}{ }_{\mu, \beta}=\left(\begin{array}{llll} -e^{\xi \frac{\sinh \beta}{\cosh ^{2} \beta} \cos \phi} & -e^{\xi \frac{1-\sinh ^{2} \beta}{\cosh ^{3} \beta} \cos \phi} & e^{\xi \frac{\sinh \beta}{\cosh ^{2} \beta} \sin \phi} & 0 \\ -e^{\xi \frac{\sinh ^{2}}{\cosh ^{2} \beta} \sin \phi} & -e^{\xi \frac{\xi \sinh ^{2} \beta}{\cosh ^{3} \beta} \sin \phi} & -e^{\xi \frac{\sinh \beta}{\cosh ^{2} \beta} \cos \phi} & 0 \\ -e^{\xi} \cosh ^{-2} \beta & 2 e^{\xi \frac{\sinh \beta}{\cosh ^{3} \beta}} & 0 & 0 \\ 0 & 0 & -e^{\xi} \cosh ^{-2} \beta & 0 \end{array}\right), \\ e^{i}{ }_{\mu, \phi}=\left(\begin{array}{llll} -e^{\xi} \cosh ^{-1} \beta \sin \phi & e^{\xi} \frac{\sinh \beta}{\cosh ^{2} \beta} \sin \phi & -e^{\xi} \cosh ^{-1} \beta \cos \phi & 0 \\ e^{\xi} \cosh ^{-1} \beta \cos \phi & -e^{\xi \frac{\sinh \beta}{\cosh ^{2} \beta} \cos \phi} & -e^{\xi} \cosh ^{-1} \beta \sin \phi & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right), \\ e^{i}{ }_{\mu, \alpha}=0 . \end{gather*} $$

$$ \begin{align*} \hat{L}_{i j} & =-\frac{i}{2}\left(x_{i} \partial_{x^{j}}-x_{j} \partial_{x^{i}}\right)+q \frac{r}{\mathbf{x}_{\perp}^{2}} x_{\perp}^{k} \\ \hat{L}_{i 4} & =\frac{1}{2}\left[-x^{i} \partial_{\mathbf{x}}^{2}-x^{i}+2 \partial_{x^{i}} \mathbf{x} \partial_{\mathbf{x}}+2 i q \frac{r}{\mathbf{x}_{\perp}^{2}}\left(\mathbf{x}_{\perp} \times \boldsymbol{\nabla}\right)_{i}-(-)^{\delta_{i 3}} q^{2} \frac{x_{i}}{\mathbf{x}_{\perp}^{2}}\right] \\ \hat{L}_{i 5} & =\frac{1}{2}\left[-x^{i} \partial_{\mathbf{x}}^{2}+x^{i}+2 \partial_{x^{i}} \mathbf{x} \partial_{\mathbf{x}}+2 i q \frac{r}{\mathbf{x}_{\perp}^{2}}\left(\mathbf{x}_{\perp} \times \boldsymbol{\nabla}\right)_{i}-(-)^{\delta_{i 3}} q^{2} \frac{x_{i}}{\mathbf{x}_{\perp}^{2}}\right] \\ \hat{L}_{i 6} & =-i r \partial_{x^{i}}-\frac{q}{\mathbf{x}_{\perp}^{2}}\left(\mathbf{x} \times \mathbf{x}_{\perp}\right)_{i} \\ \hat{L}_{45} & =-i\left(\mathbf{x} \partial_{\mathbf{x}}+1\right) \\ \hat{L}_{46} & =\frac{1}{2}\left[-r \partial_{\mathbf{x}}^{2}-r+2 i q \frac{z}{\mathbf{x}_{\perp}^{2}}(\mathbf{x} \times \boldsymbol{\nabla})_{3}+q^{2} \frac{r}{\mathbf{x}_{\perp}^{2}}\right] \\ \hat{L}_{56} & =\frac{1}{2}\left[-r \partial_{\mathbf{x}}^{2}+r+2 i q \frac{z}{\mathbf{x}_{\perp}^{2}}(\mathbf{x} \times \boldsymbol{\nabla})_{3}+q^{2} \frac{r}{\mathbf{x}_{\perp}^{2}}\right] \end{align*} $$

$$ \hat{H}_{E}=\hat{H}-E, $$

$$ \mathcal{A}_{E}^{N}=-(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s}} \frac{\left(x_{n}-x_{n-1}\right)^{2}}{r_{n}^{1-\lambda} r_{n-1}^{\lambda}}+\epsilon_{s} E r_{n}^{1-\lambda} r_{n-1}^{\lambda}\right] $$

$$ \left\langle x_{b}\right| \hat{U}_{E}(t)\left|x_{a}\right\rangle \equiv\left\langle x_{b}\right| e^{-i t \hat{H}_{E} / \hbar}\left|x_{a}\right\rangle . $$

$$ \left\langle\mathbf{x}_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|\mathbf{x}_{a}\right\rangle=\frac{1}{\left(r_{b} r_{a}\right)^{D_{\mathcal{C}}-1 / 2}} \sum\left\langle r_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|r_{a}\right\rangle_{l} Y_{l m}\left(\hat{\mathbf{x}}_{b}\right) Y_{l m}^{*}\left(\hat{\mathbf{x}}_{a}\right) $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\int_{t_{a}}^{\infty} d t_{b}\left\langle x_{b}\right| \hat{U}_{E}\left(t_{b}-t_{a}\right)\left|x_{a}\right\rangle $$

$$ \mathcal{A}_{E}^{N}=-(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s} r_{n}^{1-\lambda} r_{n-1}^{\lambda}}\left(r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1} \cos \vartheta_{n}\right)+\epsilon_{s} E r_{n}\right], $$

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

$$ \exp \left(\frac{M}{\epsilon_{s}} r_{n}^{\lambda} r_{n-1}^{1-\lambda} \cos \vartheta_{n}\right)=e^{h} \sum_{l_{\mathcal{C}}=0}^{\infty} \tilde{a}_{l_{\mathcal{C}}}(h) \sum_{\mathbf{m}} Y_{l_{\mathcal{C}}} \mathbf{m}\left(\hat{\mathbf{x}}_{b}\right) Y_{l_{\mathcal{C}}}^{*} \mathbf{m}\left(\hat{\mathbf{x}}_{a}\right) $$

$$ \begin{gather*} \Gamma_{\xi \mu \nu}=g_{\mu \nu}, \\ \Gamma_{\beta \mu \nu}=\left(\begin{array}{llll} 0 & e^{2 \xi} \cosh ^{-2} \beta & 0 & 0 \\ -e^{2 \xi} \cosh ^{-2} & -e^{2 \xi \frac{\sinh \beta}{\cosh ^{3} \beta}} & 0 & 0 \\ 0 & 0 & 0 & -e^{2 \xi} \cosh ^{-2} \beta \\ 0 & 0 & 0 & 0 \end{array}\right), \\ \Gamma_{\phi \mu \nu}=\left(\begin{array}{llll} 0 & 0 & e^{2 \xi} \cosh ^{-2} \beta & 0 \\ 0 & 0 & -e^{2 \xi \frac{\sinh \beta}{\cosh ^{3} \beta}} & 0 \\ -e^{2 \xi} \cosh ^{-2} & e^{2 \xi \frac{\sinh \beta}{\cosh ^{3} \beta}} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right), \\ \Gamma_{\alpha \mu \nu}=0 . \end{gather*} $$

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

$$ a_{l_{\mathcal{C}}}(h)=\left(\frac{2 \pi}{h}\right)^{\left(D_{\mathcal{C}}-1\right) / 2} \tilde{I}_{D_{\mathcal{C}} / 2-1+l_{\mathcal{C}}}(h), \quad h=\frac{M}{\hbar \epsilon_{s}} r_{n}^{\lambda} r_{n-1}^{1-\lambda} $$

$$ \hat{\mathcal{H}}_{E}=f_{l}(x)(\hat{H}-E) f_{r}(x) . $$

$$ \left\langle r_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|r_{a}\right\rangle_{l_{C}} \approx \frac{r_{b}{ }^{\lambda} r_{a}{ }^{1-\lambda}}{\sqrt{2 \pi \hbar \epsilon_{s} r_{b}{ }^{1-\lambda} r_{a}{ }^{\lambda} / M}} \prod_{n=2}^{N+1}\left[\int \frac{d \Delta r_{n} r_{n-1}^{-1 / 2}}{\sqrt{2 \pi \hbar \epsilon_{s} / M}}\right] e^{-\mathcal{A}_{E}^{N} / \hbar} $$

$$ \left\langle x_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|x_{a}\right\rangle \equiv f_{r}\left(x_{b}\right) f_{l}\left(x_{a}\right)\left\langle x_{b}\right| e^{-i S \hat{\mathcal{H}}_{E} / \hbar}\left|x_{a}\right\rangle $$

$$ \mathcal{A}_{E}^{N}=-(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s}} \frac{\left(r_{n}-r_{n-1}\right)^{2}}{r_{n}^{1-\lambda} r_{n-1}^{\lambda}}-\hbar \log \tilde{I}_{D_{\mathcal{C} / 2-1+l_{\mathcal{C}}}}\left(\frac{M}{\hbar \epsilon_{s}} r_{n}^{\lambda} r_{n-1}^{1-\lambda}\right)-\epsilon_{s} E r_{n}\right] $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\int_{0}^{\infty} d S\left\langle x_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|x_{a}\right\rangle $$

$$ u_{n}=\sqrt{r_{n}} $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\int_{0}^{\infty} d S\left[f_{r}\left(x_{b}\right) f_{l}\left(x_{a}\right) \int \mathcal{D} x(s) e^{i \mathcal{A}_{E}^{f}[x] / \hbar}\right] $$

$$ \begin{align*} \Delta r_{n} & =\left(u_{n}+u_{n-1}\right) \Delta u_{n} \\ & =2 u_{n}\left(1-\Delta u_{n} / 2 u_{n}\right) \Delta u_{n} \\ \frac{\partial \Delta r_{n}}{\partial \Delta u_{n}} & =2 u_{n}\left(1-\Delta u_{n} / u_{n}\right) \\ r_{n-1}^{-1 / 2} & =u_{n}^{-1}\left(1-\Delta u_{n} / u_{n}\right)^{-1} \end{align*} $$

$$ \mathcal{A}_{E}^{f}[x]=\int_{0}^{S} d s\left\{\frac{M}{2 f(x(s))} x^{\prime 2}(s)-f(x(s))[V(x(s))-E]\right\} $$

$$ \frac{\sqrt{u_{b} u_{a}}}{\sqrt{2 \pi \hbar \epsilon_{s} / M}} \prod_{n=1}^{N} \int \frac{d \Delta u_{n}}{\sqrt{2 \pi \hbar \epsilon_{s} / M}} $$

$$ d t=d s f(x(s)) $$

$$ \frac{4 \bar{u}_{n}^{2}\left(\Delta u_{n}\right)^{2}}{2 \epsilon_{s} u_{n} u_{n-1}}=\frac{4}{2 \epsilon_{s}}\left[\left(\Delta u_{n}\right)^{2}+\frac{1}{4} \frac{\left(\Delta u_{n}\right)^{4}}{u_{n}^{2}}+\ldots\right] . $$

$$ x=h(q) $$

$$ V_{\mathrm{eff}}\left(u_{n}^{2}\right)=\epsilon_{s} \hbar^{2} \frac{1}{2 \cdot 4 M} \frac{3}{4 u_{n}^{2}} $$

$$ d x=h^{\prime}(q) d q $$

$$ \left\langle r_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|r_{a}\right\rangle_{l_{\mathcal{C}}} \approx \frac{\sqrt{u_{b} u_{a}}}{\sqrt{2 \pi \hbar \epsilon_{s} / M}} \prod_{n=2}^{N+1}\left[\int_{0}^{\infty} \frac{2 d u_{n}}{\sqrt{2 \pi \hbar \epsilon_{s} / M}}\right] e^{-\mathcal{A}_{E}^{N} / \hbar} $$

$$ h^{\prime 2}(q)=f(h(q)) . $$

$$ \mathcal{A}_{E}^{N}=-(N+1) \epsilon_{s}+\sum_{n=1}^{N+1}\left[\frac{4 M}{2} \frac{\left(\Delta u_{n}\right)^{2}}{2 \epsilon_{s}}+V_{\mathrm{eff}}\left(u_{n}^{2}\right)-\hbar \log \tilde{I}_{D_{\mathcal{C} / 2-1+l_{\mathcal{C}}}}\left(\frac{M}{\hbar \epsilon_{s}} u_{n} u_{n-1}\right)\right] . $$

$$ \mathcal{A}_{E}^{f, q}=\int_{0}^{S} d s\left\{\frac{M}{2} q^{\prime 2}(s)-f(q(s))[V(q(s))-E]\right\} $$

$$ M_{\mathcal{O}}=4 M $$

$$ f(q) \equiv f(h(q)), \quad V(q) \equiv V(h(q)) $$

$$ -\hbar \log \tilde{I}_{D_{\mathcal{C} / 2-1+l_{\mathcal{C}}}}\left(\frac{M_{\mathcal{O}} / 4}{\hbar \epsilon_{s}} u_{n} u_{n-1}\right)+\epsilon_{s} \hbar^{2} \frac{3}{8 M_{\mathcal{O}} u_{n}^{2}}+\ldots $$

$$ V_{\mathrm{eff}}(q)=-\frac{\hbar^{2}}{4 M}\left[\frac{h^{\prime \prime \prime}}{h^{\prime}}-\frac{3}{2}\left(\frac{h^{\prime \prime}}{h^{\prime}}\right)^{2}\right] $$

$$ \epsilon_{s} \frac{\hbar^{2}}{2 M_{\mathcal{O}}} \frac{4}{u_{n} u_{n-1}}\left[\left(\frac{D_{\mathcal{C}}}{2}-1+l_{\mathcal{C}}\right)^{2}-\frac{1}{4}\right]+\epsilon_{s} \hbar^{2} \frac{3}{8 M_{\mathcal{O}} u_{n}^{2}}+\ldots $$

$$ \mathcal{A}_{E, \mathcal{E}}^{\mathrm{DK}}[q]=\int_{0}^{S} d s\left\{\frac{M}{2} q^{\prime 2}(s)-f(q(s))[V(q(s))-E]-V_{\mathrm{eff}}(q(s))+\mathcal{E}\right\} $$

$$ \epsilon_{s} \frac{\hbar}{2 M_{\mathcal{O}}} \frac{1}{u_{n} u_{n-1}}\left[\left(D_{\mathcal{C}}-2+2 l_{\mathcal{C}}\right)^{2}-\frac{1}{4}\right] $$

$$ \left(q_{b} \mid q_{a}\right)_{\mathcal{E}}=\int_{0}^{\infty} d S \int \mathcal{D} q(s) e^{i \mathcal{A}_{E, \mathcal{E}}^{\mathrm{DK}}[q]} $$

$$ \mu_{\mathcal{O}}=2 \mu_{\mathcal{C}} $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\left[f\left(x_{b}\right) f\left(x_{a}\right)\right]^{1 / 4}\left(q_{b} \mid q_{a}\right)_{\mathcal{E}=0} $$

$$ -\hbar \log \tilde{I}_{D_{\mathcal{C}}-2+2 l_{\mathcal{C}}}\left(\frac{M_{\mathcal{O}}}{\hbar \epsilon_{s}} u_{n} u_{n-1}\right) $$

$$ \int d x|x\rangle\langle x|=1 $$

$$ \left\langle\mathbf{x}_{E}\right| \hat{\mathcal{U}}_{e}(s)\left|\mathbf{x}_{a}\right\rangle \approx \frac{r_{b}{ }^{2} r_{a}{ }^{2-2 \lambda}}{{\sqrt{2 \pi \epsilon_{s} \hbar r_{b}{ }^{2-2 \lambda} r_{a}{ }^{2 \lambda} / M}}^{D_{\mathcal{C}} / 2}} \prod_{n=1}^{N}\left[\int \frac{d^{D_{\mathcal{C}}} \Delta x_{n}}{{\sqrt{2 \pi \epsilon_{s} \hbar r_{n}^{2} / M}}^{D_{\mathcal{C}}}}\right] e^{-\mathcal{A}_{E}^{f N} / \hbar} $$

$$ \int d q \sqrt{f(q)}|h(q)\rangle\langle h(q)|=1 $$

$$ \mathcal{A}_{E}^{f N}=-(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s} r_{n}^{2-2 \lambda} r_{n-1}^{2 \lambda}}\left(r_{n}^{2}+r_{n-1}^{2}-2 r_{n} r_{n-1} \cos \vartheta_{n}\right)-\epsilon_{s} E r_{n}^{2}\right] $$

$$ \int d q|q\rangle\langle q| \equiv 1 $$

$$ \mathcal{A}_{E}^{f N}=-(N+1) \epsilon_{s} e^{2}+\sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s}} \frac{\left(r_{n}-r_{n-1}\right)^{2}}{r_{n} r_{n-1}}-\hbar \log \tilde{I}_{D_{\mathcal{C}} / 2-1+l_{\mathcal{C}}}\left(\frac{M}{\hbar \epsilon_{s}}\right)-\epsilon_{s} E r_{n}^{2}\right] $$

$$ |x\rangle=f(q)^{-1 / 4}|q\rangle $$

$$ \epsilon_{s} \frac{\hbar}{2 M_{\mathcal{C}}}\left[\left(D_{\mathcal{C}} / 2-1+l_{\mathcal{C}}\right)^{2}-1 / 4\right] $$

$$ \left(x_{b} \mid x_{a}\right)_{E}=\left[h^{\prime}\left(q_{b}\right) h^{\prime}\left(q_{a}\right)\right]^{1 / 2}\left(q_{b} \mid q_{a}\right)_{\mathcal{E}=0} $$

$$ r=h(x)=e^{x} $$

$$ \begin{align*} & \left\langle x_{b}\right| \hat{\mathcal{U}}_{E}(S)\left|x_{a}\right\rangle \\ & \quad \approx \frac{f_{r}\left(x_{b}\right) f_{l}\left(x_{a}\right)}{\sqrt{2 \pi i \epsilon_{s} \hbar f_{l}\left(x_{b}\right) f_{r}\left(x_{a}\right) / M}} \prod_{n=1}^{N}\left[\int \frac{d x_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar f_{n} / M}}\right] \exp \left(\frac{i}{\hbar} \mathcal{A}^{N}\right) \end{align*} $$

$$ \frac{\sqrt{r_{b} r_{a}}}{{\sqrt{2 \pi \hbar \epsilon_{s} / M}}^{N+1}} \prod_{n=2}^{N+1} \int \frac{d \Delta r_{n}}{r_{n-1}} $$

$$ \mathcal{A}^{N}=\sum_{n=1}^{N+1}\left\{\frac{M}{2 \epsilon_{s}} \frac{\left(\Delta x_{n}\right)^{2}}{f_{l}\left(x_{n}\right) f_{r}\left(x_{n-1}\right)}+\epsilon_{s}\left[E-V\left(x_{n}\right)\right] f_{l}\left(x_{n}\right) f_{r}\left(x_{n-1}\right)\right\} $$

$$ \frac{1}{r_{n-1}}=\frac{1}{r_{n}}\left(\frac{r_{n}}{r_{n-1}}\right)=\frac{e^{\Delta x_{n}}}{e^{x_{n}}} $$

$$ \frac{\left[f\left(x_{b}\right) f\left(x_{a}\right)\right]^{1 / 4}}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}}\left[\frac{f_{r}\left(x_{a}\right)}{f_{r}\left(x_{b}\right)}\right]^{-5 / 4}\left[\frac{f_{l}\left(x_{a}\right)}{f_{l}\left(x_{b}\right)}\right]^{1 / 4} \prod_{n=2}^{N+1} \int \frac{d \Delta x_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar f_{n} / M}} $$

$$ \Delta r=e^{x}-e^{x-\Delta x}=e^{x}\left(1-e^{-\Delta x}\right) $$

$$ \mathcal{A}_{0}^{N}=\sum_{n=1}^{N+1} \frac{M}{2 \epsilon_{s}}\left(\Delta q_{n}\right)^{2} $$

$$ \frac{\partial \Delta r}{\partial \Delta x}=e^{x} e^{-\Delta x} $$

$$ \Delta x=x(q)-x(q-\Delta q)=e_{1} \Delta q-\frac{1}{2} e_{2}(\Delta q)^{2}+\frac{1}{6} e_{3}(\Delta q)^{3}+\ldots, $$

$$ \frac{e^{\left(x_{b}+x_{a}\right) / 2}}{{\sqrt{2 \pi \hbar \epsilon_{s} / M}}^{N+1}} \prod_{n=2}^{N+1} \int d \Delta x_{n} $$

$$ e_{1} \equiv h^{\prime}=f^{1 / 2}, e_{2} \equiv h^{\prime \prime}, e_{3} \equiv h^{\prime \prime \prime}, \ldots $$

$$ \mathcal{A}_{E}^{N}=\sum_{n=1}^{N+1} \frac{M}{\epsilon_{s}}\left(1-\cos \Delta x_{n}\right) $$

$$ \bar{e} \equiv 1 / e_{1}=1 / h^{\prime}=1 / f^{1 / 2} $$

$$ \mathcal{A}_{E}^{N}=\sum_{n=1}^{N+1} \frac{M}{2 \epsilon_{s}}\left[(\Delta x)^{2}-\frac{1}{12}\left(\Delta x_{n}\right)^{4}+\ldots\right] . $$

$$ \begin{align*} \frac{\left(\Delta x_{n}\right)^{2}}{2 \epsilon_{s} f_{l}\left(x_{n}\right) f_{r}\left(x_{n-1}\right)}= & \frac{(\Delta q)^{2}}{2 \epsilon_{s}}\left\{1-\bar{e} e_{2} \Delta q+\left[\frac{1}{3} \bar{e} e_{3}+\frac{1}{4}\left(\bar{e} e_{2}\right)^{2}\right](\Delta q)^{2}+\ldots\right\} \\ & \times\left\{1+\frac{f_{r}^{\prime}}{f_{r}} \Delta q+\left[\left(\frac{f_{r}^{\prime}}{f_{r}}\right)^{2}-\frac{1}{2} \frac{f_{r}^{\prime \prime}}{f_{r}}\right](\Delta q)^{2}+\ldots\right\} \end{align*} $$

$$ J=\frac{\partial \Delta x}{\partial \Delta q}=f^{1 / 2}\left[1-\bar{e} e_{2} \Delta q+\frac{1}{2} \bar{e} e_{3}(\Delta q)^{2}+\ldots\right] $$

$$ \frac{f_{b}^{3 \lambda / 2} f_{a}^{(1-3 \lambda) / 2}}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}} \prod_{n=2}^{N+1} \int \frac{d \Delta x_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar f_{n} / M}} $$

$$ f_{b}^{3 \lambda / 2} f_{a}^{(1-3 \lambda) / 2}=f_{b}^{1 / 4} f_{a}^{1 / 4} \prod_{n=1}^{N+1}\left(\frac{f_{n-1}}{f_{n}}\right)^{1 / 4-3 \lambda / 2} $$

$$ \begin{align*} & \left\langle x_{b}\right| \hat{\mathcal{U}}_{E}(s)\left|x_{a}\right\rangle \approx \frac{f_{b}^{1 / 4} f_{a}^{1 / 4}}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}} \prod_{n=1}^{N}\left[\int \frac{d \Delta q_{n}}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}}\right] \\ & \quad \times \exp \left\{\frac{i}{\hbar}\left[\sum_{n=1}^{N+1} \frac{M}{2 \epsilon_{s}}\left(\Delta q_{n}\right)^{2}+\epsilon_{s} f\left(q_{n}\right)\left[E-V\left(q_{n}\right)\right]+\ldots\right]\right\}\left[1+C\left(q_{n}, \Delta q_{n}\right)\right] \end{align*} $$

$$ 1+C \equiv\left(1+C_{\mathrm{meas}}\right)\left(1+C_{f}\right)\left(1+C_{\mathrm{act}}\right) $$

$$ C_{\mathrm{meas}}=-\bar{e} e_{2} \Delta q+\frac{1}{2} \bar{e} e_{3}(\Delta q)^{2}+\ldots $$

$$ \begin{align*} C_{f}= & \left(\frac{1}{4}-\frac{3 \lambda}{2}\right)\left[-\frac{f^{\prime}}{f} \Delta q+\frac{1}{2} \frac{f^{\prime \prime}}{f}(\Delta q)^{2}\right] \\ & -\frac{1}{2}\left(\frac{3}{4}+\frac{3 \lambda}{2}\right)\left(\frac{1}{4}-\frac{3 \lambda}{2}\right)\left(\frac{f^{\prime}}{f}\right)^{2}(\Delta q)^{2}+\ldots \end{align*} $$

$$ \begin{align*} C_{\mathrm{act}}= & \frac{i}{\hbar} M \frac{(\Delta q)^{2}}{2 \epsilon_{s}}\left\{-\left(\bar{e} e_{2}-\lambda \frac{f^{\prime}}{f}\right) \Delta q\right. \\ + & {\left.\left[\frac{1}{3} \bar{e} e_{3}+\frac{1}{4}\left(\bar{e} e_{2}\right)^{2}+\frac{1}{2}\left(-\lambda \frac{f^{\prime \prime}}{f}+\lambda(\lambda+1)\left(\frac{f^{\prime}}{f}\right)^{2}\right)-\lambda \bar{e} e_{2} \frac{f^{\prime}}{f}\right](\Delta q)^{2}\right\} } \\ & -\frac{M^{2}}{2 \hbar^{2}} \frac{(\Delta q)^{4}}{4 \epsilon_{s}^{2}}\left(\bar{e} e_{2}-\lambda \frac{f^{\prime}}{f}\right)^{2}(\Delta q)^{2}+\ldots \end{align*} $$

$$ \left\langle(\Delta q)^{2 n}\right\rangle_{0}=\left(\frac{i \hbar}{M}\right)^{n}(2 n-1)!!. $$

$$ \langle C \Delta q\rangle_{0}=i \hbar \epsilon_{s}\left[-\bar{e} e_{2}-\left(\frac{1}{4}-\frac{3 \lambda}{2}\right) \frac{f^{\prime}}{f}+\frac{3}{2}\left(\bar{e} e_{2}-\lambda \frac{f^{\prime}}{f}\right)\right] . $$

$$ e_{1}^{2}=f $$

$$ 2 e_{1} e_{2}=f^{\prime}, \quad 2 \bar{e} e_{2}=f^{\prime} / f $$

$$ f^{\prime \prime} / f=2\left[\left(\bar{e} e_{2}\right)^{2}+\bar{e} e_{3}\right] $$

$$ \begin{align*} C_{f}= & \left(\frac{1}{4}-\frac{3 \lambda}{2}\right)\left\{-2 \bar{e} e_{2} \Delta q+\left[\left(\bar{e} e_{2}\right)^{2}+\bar{e} e_{3}\right](\Delta q)^{2}\right\} \\ & -2\left(\frac{3}{4}+\frac{3 \lambda}{2}\right)\left(\frac{1}{4}-\frac{3 \lambda}{2}\right)\left(\bar{e} e_{2}\right)^{2}(\Delta q)^{2}+\ldots \end{align*} $$

$$ \begin{gather*} C_{\mathrm{act}}=i \frac{M}{\hbar} \frac{(\Delta q)^{2}}{2 \epsilon_{s}}\left\{-(1-2 \lambda) \bar{e} e_{2} \Delta q\right. \\ \left.+\left[\frac{1}{3} \bar{e} e_{3}+\frac{1}{4}\left(\bar{e} e_{2}\right)^{2}-\lambda\left(\bar{e} e_{2}+\bar{e} e_{3}\right)+2 \lambda(\lambda+1)\left(\bar{e} e_{2}\right)^{2}-2 \lambda\left(\bar{e} e_{2}\right)^{2}\right](\Delta q)^{2}\right\} \\ -\frac{M^{2}}{2 \hbar^{2}} \frac{(\Delta q)^{4}}{4 \epsilon_{s}^{2}}(1-2 \lambda)^{2}\left(\bar{e} e_{2}\right)^{2}(\Delta q)^{2}+\ldots \end{gather*} $$

$$ \begin{align*} C= & \bar{e} e_{2}\left(\frac{1}{2}-\lambda\right) \Delta q\left[-\frac{i M}{\hbar \epsilon_{s}}(\Delta q)^{2}-3\right] \\ + & \left(\bar{e} e_{2}\right)^{2}\left[\frac{9}{2}\left(\lambda-\frac{1}{6}\right)\left(\lambda-\frac{1}{2}\right)(\Delta q)^{2}+i\left(4 \lambda^{2}-\frac{7}{2} \lambda+\frac{7}{8}\right) \frac{M}{\hbar \epsilon_{s}}(\Delta q)^{4}\right. \\ & \left.\quad-\frac{1}{2}\left(\lambda-\frac{1}{2}\right)^{2} \frac{M^{2}}{\hbar^{2} \epsilon_{s}^{2}}(\Delta q)^{6}\right] \\ & +\bar{e} e_{3}\left[-\frac{3}{2}\left(\lambda-\frac{1}{2}\right)(\Delta q)^{2}-\frac{1}{2}\left(\lambda-\frac{1}{3}\right) i \frac{M}{\hbar \epsilon_{s}}(\Delta q)^{4}\right]+\ldots \end{align*} $$

$$ \langle C\rangle_{0}=-\frac{\epsilon_{s} \hbar}{M}\left[\frac{1}{4} \bar{e} e_{3}-\frac{3}{8}\left(\bar{e} e_{2}\right)^{2}\right] $$

$$ V_{\mathrm{eff}}=-\frac{i \hbar^{2}}{M}\left[\frac{1}{4} \bar{e} e_{3}-\frac{3}{8}\left(\bar{e} e_{2}\right)^{2}\right] $$

$$ K^{\epsilon_{s}}(\Delta q)=\frac{1}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}} \exp \left\{\frac{i}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s}}\left(\Delta q_{n}\right)^{2}+\epsilon_{s} E f\left(q_{n}\right)\right]\right\}[1+C] $$

$$ K^{\epsilon_{s}}(\Delta q)=\frac{1}{\sqrt{2 \pi i \epsilon_{s} \hbar / M}} \exp \left\{\frac{i}{\hbar} \sum_{n=1}^{N+1}\left[\frac{M}{2 \epsilon_{s}}\left(\Delta q_{n}\right)^{2}+\epsilon_{s} E f\left(q_{n}\right)-\epsilon_{s} V_{\mathrm{eff}}\right]\right\} $$

$$ \left[-\frac{\hbar^{2}}{2 M} \partial_{x}^{2}-E\right] \psi(x, t)=i \hbar \partial_{t} \psi(x, t) $$

$$ \left[-\frac{\hbar^{2}}{2 M} f_{l}(x) \partial_{x}^{2} f_{r}(x)-E f(x)\right] \psi_{f}(x, t)=f(x) i \hbar \partial_{t} \psi_{f}(x, t) $$

$$ \left[-\frac{\hbar}{2 M} f_{l}(q)\left(\frac{1}{h^{\prime}(q)} \partial_{q}\right)^{2} f_{r}(q)-E f(q)\right] \psi_{f}(q, t)=f(q) i \hbar \partial_{t} \psi_{f}(x, t) $$

$$ \left[-\frac{\hbar^{2}}{2 M} f_{r}^{-1}(q)\left(\partial_{q}^{2}-\frac{h^{\prime \prime}}{h^{\prime}} \partial_{q}\right) f_{r}(q)-E f(q)\right] \psi_{f}(q, t)=f(q) i \hbar \partial_{t} \psi_{f}(q, t) $$

$$ \begin{align*} & {\left[-\frac{\hbar^{2}}{2 M} h^{\prime}(q)^{-1 / 2}\left(\partial_{q}^{2}-\frac{h^{\prime \prime}}{h^{\prime}} \partial_{q}\right) h^{\prime}(q)^{1 / 2}-E f(q)\right] \phi(q, t)} \\ & \quad=\left[-\frac{1}{2 M} \partial_{q}^{2}+V_{\text {eff }}-E f(q)\right] \phi(q, t)=f(q) i \hbar \partial_{t} \phi(q, t) \end{align*} $$

$$ V_{\mathrm{eff}}=-\frac{\hbar^{2}}{4 M}\left[\frac{h^{\prime \prime \prime}}{h^{\prime}}-\frac{3}{2}\left(\frac{h^{\prime \prime}}{h^{\prime}}\right)^{2}\right]=\frac{\hbar^{2}}{2 M} \frac{1}{4} $$

$$ \mathcal{A}_{E}=\int d t\left[\frac{M}{2} \dot{x}^{2}(t)-V(x)+E\right] $$

$$ \mathcal{A}_{\mathcal{O}}=\int d t\left[\frac{M}{2} \dot{r}^{2}-\hbar^{2} \frac{\mu_{\mathcal{O}}^{2}-1 / 4}{2 M r^{2}}-\frac{M}{2} \omega^{2} r^{2}+E_{\mathcal{O}}\right] $$

$$ \mu_{\mathcal{O}}=D_{\mathcal{O}} / 2-1+l_{\mathcal{O}} $$

$$ \epsilon \hbar^{2} \frac{\mu_{\mathcal{O}}^{2}-1 / 4}{2 M r_{n}^{2}} \longrightarrow i \hbar \log \tilde{I}_{\mu_{\mathcal{O}}}\left(\frac{M}{i \hbar \epsilon} r_{n} r_{n-1}\right) $$

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

$$ \nu=\nu_{\mathcal{O}} \equiv \frac{E_{\mathcal{O}}}{2 \omega \hbar}, \quad \mu=\mu_{\mathcal{O}} $$

$$ f(r)=r^{2} . $$

$$ \mathcal{H}_{\mathcal{O}}=r^{2} \frac{p^{2}}{M}+\hbar^{2} \frac{\mu_{\mathcal{O}}^{2}-1 / 4}{2 M}+\frac{M}{2} \omega^{2} r^{4}-E_{\mathcal{O}} r^{2} $$

$$ \mathcal{A}_{\mathcal{O}}^{f=r^{2}}=\int_{0}^{S} d s\left(\frac{M}{2} \frac{r^{\prime 2}}{r^{2}}-\frac{\mu_{\mathcal{O}}^{2}-1 / 4}{2 M}-\frac{M}{2} \omega^{2} r^{4}+E_{\mathcal{O}} r^{2}\right) $$

$$ r=h(x) \equiv e^{x}, $$

$$ h^{\prime 2}=e^{2 x}=f(r)=r^{2} $$

$$ V_{\mathrm{eff}}=-\frac{\hbar^{2}}{M}\left[\frac{1}{4} \frac{h^{\prime \prime \prime}}{h^{\prime}}-\frac{3}{8}\left(\frac{h^{\prime \prime}}{h^{\prime}}\right)^{2}\right]=\frac{\hbar^{2}}{8 M} $$

$$ \mathcal{A}_{\mathcal{O}}^{\mathrm{DK}}=\int_{0}^{S} d s\left[\frac{M}{2} x^{\prime 2}-\frac{\mu_{\mathcal{O}}^{2}}{2 M}-\frac{M \omega^{2}}{2} e^{4 x}+E_{\mathcal{O}} e^{2 x}\right] $$

$$ \begin{align*} A & =\frac{M}{2} \omega^{2}, \\ B & =E_{\mathcal{O}}, \\ C & =\frac{\hbar^{2} \mu_{\mathcal{O}}^{2}}{2 M}+E_{\mathcal{M}}, \end{align*} $$

$$ \mathcal{A}_{\mathcal{M}}=\int_{0}^{S} d s\left[\frac{M}{2} x^{\prime 2}-\left(V_{\mathcal{M}}-E_{\mathcal{M}}\right)\right] $$

$$ V_{\mathcal{M}}(x)=A e^{4 x}-B e^{2 x}+C . $$

$$ \left(x_{b} \mid x_{a}\right)_{E_{\mathcal{M}}}=\int_{0}^{\infty} d S \int \mathcal{D} x(s) e^{i \mathcal{A}_{\mathcal{M}} / \hbar} $$

$$ \left(r_{b} \mid r_{a}\right)_{E_{\mathcal{O}}, l}=e^{\left(x_{b}+x_{a}\right) / 2}\left(x_{b} \mid x_{a}\right)_{E_{\mathcal{M}}}, $$

$$ B=\hbar \omega\left(\mu_{\mathcal{O}}+n_{r}+1 / 2\right), \quad n_{r}=0,1,2,3, \ldots, $$

$$ B=\hbar \sqrt{2 A / M}\left(\mu_{\mathcal{O}}+n_{r}+1 / 2\right) $$

$$ E_{\mathcal{M}}=C-\frac{B^{2}}{4 A}\left[1-\frac{\hbar}{B} \sqrt{\frac{2 A}{M}}\left(n_{r}+1 / 2\right)\right]^{2}, \quad\left(0 \leq n_{r} \leq \sqrt{M B^{2} / 2 A \hbar^{2}}-1 / 2 .\right. $$

$$ E_{\mathcal{M}}=-V_{0}\left[1-\frac{\hbar}{\sqrt{M V_{0} / 2}}\left(n_{r}+1 / 2\right)\right]^{2}, \quad\left(0 \leq n_{r} \leq \sqrt{M V_{0} / 2 \hbar^{2}}-1 / 2\right. $$

$$ \mathcal{A}_{\mathcal{C}}=\int d t\left[\frac{M}{2} \dot{r}^{2}-\hbar^{2} \frac{\mu_{\mathcal{C}}^{2}-1 / 4}{2 M r^{2}}+\frac{e^{2}}{r}+E_{\mathcal{C}}\right] $$

$$ \mu_{\mathcal{C}}=D_{\mathcal{C}} / 2-1+l_{\mathcal{C}} $$

$$ f(r)=r^{2} $$

$$ \mathcal{A}_{\mathcal{C}}^{f=r^{2}}=\int_{0}^{S} d s\left[\frac{M}{2} \frac{r^{\prime 2}}{r^{2}}-\hbar^{2} \frac{\mu_{\mathcal{C}}^{2}-1 / 4}{2 M}+e^{2} r+E_{\mathcal{C}} r^{2}\right] $$

$$ r=e^{x} . $$

$$ V_{\mathrm{eff}}=\frac{\hbar^{2}}{2 M} \frac{1}{4} $$

$$ \mathcal{A}_{\mathcal{C}}^{\mathrm{DK}}=\int_{0}^{S} d s\left[\frac{M}{2} x^{\prime 2}-\hbar^{2} \frac{\mu_{\mathcal{C}}^{2}}{2 M}+e^{2} e^{x}+E_{\mathcal{C}} e^{2 x}\right] $$

$$ \begin{align*} x & =2 \bar{x}, \\ M & =\bar{M} / 4, \\ \mu_{\mathcal{C}} & =2 \bar{\mu}, \end{align*} $$

$$ \mathcal{A}_{\mathcal{C}}^{\mathrm{DK}}=\int_{0}^{S} d s\left[\frac{\bar{M}}{2} \bar{x}^{\prime 2}-\hbar^{2} \frac{\bar{\mu}^{2}}{2 \bar{M}}+e^{2} e^{2 \bar{x}}+E_{\mathcal{C}} e^{4 \bar{x}}\right] $$

$$ \left(r_{b} \mid r_{a}\right)_{E_{\mathcal{C}}, l_{\mathcal{C}}}=\frac{1}{2} e^{\left(x_{b}+x_{a}\right)}\left(x_{b} \mid x_{a}\right)_{E_{\mathcal{M}}} $$

$$ \begin{align*} A & =-E_{\mathcal{C}} \\ B & =e^{2} \\ C & =\hbar^{2} \frac{\mu_{\mathcal{C}}^{2}}{2 M}+E_{\mathcal{M}} \end{align*} $$

$$ E_{\mathcal{C}}=-\frac{M e^{4}}{\hbar^{2}} \frac{1}{2\left(\mu_{\mathcal{C}}+n_{r}+\frac{1}{2}\right)^{2}}=-M c^{2} \frac{\alpha^{2}}{2 n^{2}} $$

$$ \begin{align*} M_{\mathcal{O}} & =4 M_{\mathcal{C}} \\ \mu_{\mathcal{O}} & =2 \mu_{\mathcal{C}} \\ E_{\mathcal{O}} & =e^{2} \\ -\frac{M_{\mathcal{O}}}{2} \omega^{2} & =E_{\mathcal{C}} \\ r_{\mathcal{O}} & =\sqrt{r_{\mathcal{C}}} \end{align*} $$

$$ D_{\mathcal{O}} / 2-1+l_{\mathcal{O}}=2\left(D_{\mathcal{C}} / 2-1+l_{\mathcal{C}}\right) $$

$$ l_{\mathcal{O}}=2 l_{\mathcal{C}} $$

$$ D_{\mathcal{O}}=2 D_{\mathcal{C}}-2 $$

$$ \left(r_{b} \mid r_{a}\right)_{E_{\mathcal{C}}, \mu_{\mathcal{C}}}=\frac{1}{2} \sqrt{u_{b} u_{a}}\left(u_{b} \mid u_{a}\right)_{E_{\mathcal{O}}, \mu_{\mathcal{O}}} $$

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

$$ \int_{0}^{\infty} d u 2 u|r\rangle\langle r|=\int d u|u\rangle\langle u|=1 $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E_{\mathcal{C}}}=\sum_{l_{\mathcal{C}}=0}^{\infty} \sum_{m=-l_{\mathcal{C}}}^{l_{\mathcal{C}}} \frac{1}{r_{b} r_{a}}\left(r_{b} \mid r_{a}\right)_{E_{\mathcal{C}}, l_{\mathcal{C}}} Y_{l_{\mathcal{C}} m}\left(\theta_{b}, \varphi_{b}\right) Y_{l_{\mathcal{C}} m}^{*}\left(\theta_{a}, \varphi_{a}\right) $$

$$ \begin{align*} \left(\vec{u}_{b} \mid \vec{u}_{a}\right)_{E_{\mathcal{O}}} & =\sum_{l_{\mathcal{O}}=0}^{\infty}\left(u_{b} \mid u_{a}\right)_{E_{\mathcal{O}}, l_{\mathcal{O}}} \\ & \times \frac{l_{\mathcal{O}}+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l_{\mathcal{O}} / 2}^{l_{\mathcal{O} / 2}} \mathcal{D}_{m_{1} m_{2}}^{l_{\mathcal{O}} / 2}\left(\varphi_{n}, \theta_{n}, \gamma_{n}\right) \mathcal{D}_{m_{1} m_{2}}^{l_{\mathcal{O}} / 2{ }^{*}}\left(\varphi_{n-1}, \theta_{n-1}, \gamma_{n-1}\right) \end{align*} $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E_{\mathcal{C}}}=\int_{0}^{\infty} d S e^{i e^{2} S / \hbar} \frac{1}{16} \int_{0}^{4 \pi} d \gamma_{a}\left(\vec{u}_{b} S \mid \vec{u}_{a} 0\right) $$

$$ \frac{l_{\mathcal{O}}+1}{2 \pi^{2}} \sum_{m_{1}, m_{2}=-l_{\mathcal{O} / 2}}^{l_{\mathcal{O} / 2}} d_{m_{1} m_{2}}^{l_{\mathcal{O}} / 2}\left(\theta_{b}\right) d_{m_{1} m_{2}}^{l_{\mathcal{O}} / 2}\left(\theta_{a}\right) e^{i m_{1}\left(\varphi_{b}-\varphi_{a}\right)+i m_{2}\left(\gamma_{b}-\gamma_{a}\right)} $$

$$ 8 \sum_{m}^{l_{\mathcal{O}} / 2} Y_{l_{\mathcal{O} / 2}, m, 0}\left(\theta_{b}, \phi_{b}\right) Y_{l_{\mathcal{O}} / 2, m, 0}^{*}\left(\theta_{a}, \phi_{a}\right) $$

$$ Y_{l_{\mathcal{O} / 2, m}}(\theta, \phi)=\sqrt{\frac{l_{\mathcal{O}}+1}{4 \pi}} e^{i m \phi} d_{m, 0}^{l_{\mathcal{O}} / 2}(\theta) $$

$$ \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E_{\mathcal{C}}, l_{\mathcal{C}}}=\frac{1}{\left(r_{b} r_{a}\right)^{\left(D_{\mathcal{C}}-1\right) / 2}} \sum_{l_{\mathcal{C}}=0}^{\infty}\left(r_{b} \mid r_{a}\right)_{E_{\mathcal{C}}, l_{\mathcal{C}}} \frac{2 l_{\mathcal{C}}+D_{\mathcal{C}}-2}{D_{\mathcal{C}}-2} \frac{1}{S_{D_{\mathcal{C}}}} C_{l_{\mathcal{C}}}^{\left(D_{\mathcal{C}} / 2-1\right)}\left(\cos \Delta \vartheta_{n}\right) . $$

$$ \left(u_{b} \mid u_{a}\right)_{E_{\mathcal{O}}, l_{\mathcal{O}}}=-i \frac{M_{\mathcal{O}}}{\hbar} \sqrt{u_{b} u_{a}} \int_{0}^{1} \frac{d \varrho}{2 \varrho} \varrho^{-\nu} e^{-\kappa\left(u_{b}^{2}+u_{a}^{2}\right) \frac{1+\varrho}{1-\varrho}} I_{l_{\mathcal{O}}+D_{\mathcal{O}} / 2-1}\left(2 \kappa u_{b} u_{a} \frac{2 \sqrt{\varrho}}{1-\varrho}\right), $$

$$ \kappa \equiv \frac{M_{\mathcal{O}} \omega}{2 \hbar}, \quad \nu \equiv E_{\mathcal{O}} / 2 \hbar \omega $$

$$ E_{\mathcal{O} n}=2 \hbar \omega\left(\frac{n}{2}+\frac{D_{\mathcal{O}}}{4}\right)=\hbar \omega\left(2 n_{r}+l+\frac{D_{\mathcal{O}}}{2}\right) $$

$$ \begin{gather*} \left(\frac{1}{2} k z\right)^{D_{\mathcal{C} / 2-1 / 2}} I_{D_{\mathcal{C} / 2-3 / 2}}(k z)=k^{D_{\mathcal{C}}-2} \sum_{l=0}^{\infty} \frac{1}{l!} \frac{\Gamma\left(l+D_{\mathcal{C}}-2\right)}{\Gamma\left(D_{\mathcal{C}} / 2-1 / 2\right)}\left(2 l+D_{\mathcal{C}}-2\right) \\ \times F\left(-l, l+D_{\mathcal{C}}-2 ; D_{\mathcal{C}} / 2-1 / 2 ;\left(1+k^{2}\right) / 2\right)(-)^{l} I_{2 l+D_{\mathcal{C}}-2}(z) \end{gather*} $$

$$ \begin{align*} & \frac{1}{2} \frac{1}{(2 \pi)^{D_{\mathcal{C}} / 2-1 / 2}}\left(\frac{z}{2}\right)^{D_{\mathcal{C}}-2} I_{D_{\mathcal{C}} / 2-3 / 2}(k z) /(k z)^{D_{\mathcal{C}} / 2-3 / 2} \\ & \quad=\sum_{l_{\mathcal{C}}=0}^{\infty} \frac{2 l_{\mathcal{C}}+D_{\mathcal{C}}-2}{D_{\mathcal{C}}-2} \frac{1}{S_{D_{\mathcal{C}}}} C_{l_{\mathcal{C}}}^{\left(D_{\mathcal{C}} / 2-1\right)}\left(\left(1+k^{2}\right) / 2\right) I_{2 l_{\mathcal{C}}+D_{\mathcal{C}}-2}(z) \end{align*} $$

$$ z \equiv 2 \kappa u_{b} u_{a} \frac{2 \sqrt{\varrho}}{1-\varrho}, \quad k \equiv \cos (\vartheta / 2) $$

$$ \begin{align*} \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E} & =-i \frac{M}{\hbar} \frac{\kappa^{D_{\mathcal{C}}-2}}{(2 \pi)^{\left(D_{\mathcal{C}}-1\right) / 2}} \int_{0}^{1} \frac{d \varrho}{(1-\varrho)^{2}} \varrho^{-\nu} \\ & \times\left(\frac{2 \sqrt{\varrho}}{1-\varrho}\right)^{\left(D_{\mathcal{C}}-3\right) / 2} e^{-\kappa \frac{1+\varrho}{1-\varrho}\left(r_{b}+r_{a}\right)} I_{D_{\mathcal{C}} / 2-3 / 2}(k z) /(k z)^{D_{\mathcal{C}} / 2-3 / 2} \end{align*} $$

$$ k z=2 \kappa \frac{2 \sqrt{\varrho}}{1-\varrho} \sqrt{\left(r_{b} r_{a}+\mathbf{x}_{b} \mathbf{x}_{a}\right) / 2} $$

$$ \begin{align*} & \left(\mathbf{x}_{b} \mid \mathbf{x}_{a}\right)_{E}=-i \frac{M}{2 \hbar} \frac{\kappa^{D_{\mathcal{C}}-2}}{(2 \pi)^{\left(D_{\mathcal{C}}-1\right) / 2}} \frac{\pi e^{i \pi\left(\nu-D_{\mathcal{C}} / 2+3 / 2\right)}}{\sin \left[\pi\left(\nu-D_{\mathcal{C}} / 2+3 / 2\right)\right]} \\ & \quad \times \int_{C} \frac{d \zeta}{2 \pi i}(\zeta-1)^{-\nu+D_{\mathcal{C}} / 2-3 / 2}(\zeta+1)^{\nu+D_{\mathcal{C}} / 2-3 / 2} e^{-\kappa \zeta\left(r_{b}+r_{a}\right)} I_{D_{\mathcal{C}} / 2-3 / 2}(z) / z^{D_{\mathcal{C}} / 2-3 / 2} \end{align*} $$

$$ E_{\mathcal{C}}=-\frac{M e^{4}}{\hbar^{2}} \frac{1}{2 n^{2}}, \quad \text { with } \quad n=n_{r}+l_{\mathcal{C}}+\left(D_{\mathcal{C}}-1\right) / 2, \quad n_{r}=0,1,2, \ldots, $$

$$ \mathcal{A}_{\mathcal{P} \mathcal{T}}=\int d t\left[\frac{\mu}{2} \dot{\theta}^{2}+\frac{\hbar^{2}}{8 \mu}-\frac{\hbar^{2}}{2 \mu} \frac{m^{2}-1 / 4 "}{\sin ^{2} \theta}+E_{\mathcal{P} \mathcal{T}}\right] $$

$$ \begin{align*} \left(\theta_{b} \mid \theta_{a}\right)_{m, E_{\mathcal{P} \mathcal{T}}}= & \sqrt{\sin \theta_{b} \sin \theta_{a}} \sum_{n=0}^{\infty} \frac{i \hbar}{E_{\mathcal{P} \mathcal{T}}-\hbar^{2} L_{2} / 2 \mu} \\ & \quad \times \frac{2 n+2 m+1}{2} \frac{(n+2 m)!}{n!} P_{n+m}^{-m}\left(\cos \theta_{b}\right) P_{n+m}^{-m}\left(\cos \theta_{a}\right) \end{align*} $$

$$ n+m=l=l\left(E_{\mathcal{P} \mathcal{T}}\right) \equiv-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{2 \mu E_{\mathcal{P} \mathcal{T}}}{\hbar^{2}}} $$

$$ \begin{align*} \left(\theta_{b} \mid \theta_{a}\right)_{m, E_{\mathcal{P} \mathcal{T}}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \frac{-i \mu}{\hbar} \Gamma & \left(m-l\left(E_{\mathcal{P} \mathcal{T}}\right)\right) \Gamma\left(l\left(E_{\mathcal{P} \mathcal{T}}\right)+m+1\right) \\ & \times P_{l\left(E_{\mathcal{P} \mathcal{T}}\right)}^{-m}\left(-\cos \theta_{b}\right) P_{l\left(E_{\mathcal{P} \mathcal{T}}\right)}^{-m}\left(\cos \theta_{a}\right) \end{align*} $$

$$ V_{\mathcal{P} \mathcal{T}}(\theta)=\frac{\hbar^{2}}{2 \mu} \frac{m^{2}}{\sin ^{2} \theta} $$

$$ f(\theta)=\sin ^{2} \theta, $$

$$ \mathcal{A}_{\mathcal{P} \mathcal{T}}^{f=\sin ^{2} \theta}=\int_{0}^{S} d s\left[\frac{\mu}{2 \sin ^{2} \theta} \theta^{\prime 2}+\frac{\hbar^{2}}{8 \mu} \sin ^{2} \theta-\frac{\hbar^{2}}{2 \mu}\left(m^{2}-1 / 4\right)+E_{\mathcal{P} \mathcal{T}} \sin ^{2} \theta\right] $$

$$ \sin \theta=\frac{1}{\cosh x}, \quad \cos \theta=-\tanh x $$

$$ h^{\prime}(x)=\sin \theta=\frac{1}{\cosh x} $$

$$ h^{\prime \prime}(x)=-\frac{\tanh x}{\cosh x}, \quad h^{\prime \prime \prime}(x)=-\frac{1}{\cosh x}\left(1-2 \tanh ^{2} x\right) $$

$$ V_{\mathrm{eff}}=\frac{\hbar^{2}}{8 \mu}\left(1+\frac{1}{\cosh ^{2} x}\right) $$

$$ \mathcal{A}_{\mathcal{P} \mathcal{T}}^{\mathrm{DK}}=\int_{0}^{S} d s\left[\frac{\mu}{2} x^{\prime 2}-\frac{\hbar^{2} m^{2}}{2 \mu}+E_{\mathcal{P} \mathcal{T}} \frac{1}{\cosh ^{2} x}\right] $$

$$ V_{\mathcal{R} \mathcal{M}}(x)=-\frac{\hbar^{2}}{2 \mu} \frac{s(s+1)}{\cosh ^{2} x} $$

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

$$ \left(\theta_{b} \mid \theta_{a}\right)_{m, E_{\mathcal{P} \mathcal{T}}}=\sqrt{\sin \theta_{b} \sin \theta_{a}}\left(x_{b} \mid x_{a}\right)_{m, E_{\mathcal{R} \mathcal{M}}} $$

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

$$ m\left(E_{\mathcal{R} \mathcal{M}}\right)=s-n, \quad n=0,1,2, \ldots,[s] $$

$$ \psi_{n}(x)=\sqrt{\Gamma(2 s-n+1)(s-n) / n} P_{s}^{n-s}(\tanh x) $$

$$ F(a, b ; c ; z)=(1-z)^{c-a-b} F(c-a, c-b ; c ; z) $$

$$ P_{s}^{n-s}(\tanh x)=\frac{2^{n-s}}{\Gamma(s-n+1)} \frac{1}{\cosh ^{s-n} x} F(-n, 1+2 s-n ; s-n+1 ;(1-\tanh x) / 2) . $$

$$ \mathcal{A}_{\mathcal{P} \mathcal{T}^{\prime}}=\int d t\left[\frac{\mu}{2} \dot{\theta}^{2}+\frac{\hbar^{2}}{32 \mu}-\frac{\hbar^{2}}{2 \mu} \frac{m_{1}^{2}+m_{2}^{2}-2 m_{1} m_{2} \cos \theta-1 / 4 "}{\sin ^{2} \theta}+E_{\mathcal{P} \mathcal{T}^{\prime}}\right] $$

$$ V_{\mathcal{P} \mathcal{T}^{\prime}}(\theta)=\frac{\hbar^{2}}{2 \mu}\left[\frac{s_{1}\left(s_{1}+1\right)}{\sin ^{2}(\theta / 2)}+\frac{s_{2}\left(s_{2}+1\right)}{\cos ^{2}(\theta / 2)}\right] $$

$$ \begin{align*} \left(\theta_{b} \mid \theta_{a}\right)_{m_{1}, m_{2}, E_{\mathcal{P} \mathcal{T}^{\prime}}} & =\sqrt{\sin \theta_{b} \sin \theta_{a}} \\ & \times \sum_{n=0}^{\infty} \frac{i \hbar}{E_{\mathcal{P} \mathcal{T}^{\prime}}-\hbar^{2} L_{2} / 8 \mu} \frac{2 n+2 m_{1}+1}{2} d_{m_{1}, m_{2}}^{n+m_{1}}\left(\theta_{b}\right) d_{m_{1}, m_{2}}^{n+m_{1}}\left(\theta_{a}\right), \end{align*} $$

$$ 2 n+2 m_{1}=l=l\left(E_{\mathcal{P} \mathcal{T}^{\prime}}\right) \equiv-1+2 \sqrt{\frac{1}{16}+\frac{2 \mu E_{\mathcal{P} \mathcal{T}^{\prime}}}{\hbar^{2}}} $$

$$ \begin{align*} & \left(\theta_{b} \mid \theta_{a}\right)_{m_{1}, m_{2}, E_{\mathcal{P} \mathcal{T}^{\prime}}}=\sqrt{\sin \theta_{b} \sin \theta_{a}} \frac{-2 i \mu}{\hbar} \\ & \quad \times \Gamma\left(m_{1}-l\left(E_{\mathcal{P} \mathcal{T}^{\prime}}\right) / 2\right) \Gamma\left(l\left(E_{\mathcal{P} \mathcal{T}^{\prime}}\right) / 2-m_{1}+1\right) \frac{1}{2} d_{m_{1},-m_{2}}^{l\left(E_{\mathcal{P} \mathcal{T}^{\prime}}\right) / 2}\left(\theta_{b}-\pi\right) d_{m_{1}, m_{2}}^{l\left(E_{\mathcal{P} \mathcal{T}^{\prime}}\right) / 2}\left(\theta_{a}\right) \end{align*} $$

$$ E_{\mathcal{P} \mathcal{T}^{\prime}}=\frac{\hbar^{2}}{2 \mu}\left[\left(m_{1}+n+\frac{1}{2}\right)^{2}-\frac{1}{16}\right], \quad n=0,1,2, \ldots $$

$$ f(\theta)=\sin ^{2} \theta $$

$$ \begin{align*} \mathcal{A}_{\mathcal{P} \mathcal{T}^{\prime}}^{f=\sin ^{2} \theta}=\int_{0}^{S} d s[ & \frac{\mu}{2 \sin ^{2} \theta} \theta^{\prime 2}+\frac{\hbar^{2}}{32 \mu} \sin ^{2} \theta \\ & \left.\quad-\frac{\hbar^{2}}{2 \mu}\left(m_{1}^{2}+m_{2}^{2}-1 / 4-2 m_{1} m_{2} \cos \theta\right)+E_{\mathcal{P} \mathcal{T}^{\prime}} \sin ^{2} \theta\right] \end{align*} $$

$$ \sin \theta= \pm \frac{1}{\cosh x}, \quad \cos \theta=-\tanh x $$

$$ V_{\mathrm{eff}}=\frac{\hbar^{2}}{8 \mu}\left(1+\frac{1}{\cosh ^{2} x}\right) $$

$$ \mathcal{A}_{\mathcal{P} \mathcal{T}^{\prime}}^{\mathrm{DK}}=\int_{0}^{S} d s\left[\frac{\mu}{2} x^{\prime 2}-\frac{\hbar^{2}}{2 \mu}\left(m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \tanh x\right)+\left(E_{\mathcal{P} \mathcal{T}^{\prime}}-\frac{3 \hbar^{2}}{32 \mu}\right) \frac{1}{\cosh ^{2} x}\right] $$

$$ V_{\mathcal{R} \mathcal{M}^{\prime}}(x)=-\left(E_{\mathcal{P} \mathcal{T}^{\prime}}-\frac{3 \hbar^{2}}{32 \mu}\right) \frac{1}{\cosh ^{2} x}+\frac{\hbar^{2}}{2 \mu} 2 m_{1} m_{2} \tanh x $$

$$ V_{\mathcal{R} \mathcal{M}^{\prime}}(x)=\frac{\hbar^{2}}{2 \mu}\left[-\frac{s(s+1)}{\cosh ^{2} x}+2 c \tanh x\right] $$

$$ E_{\mathcal{P} \mathcal{T}^{\prime}}=\frac{\hbar^{2}}{2 \mu}\left[s(s+1)+\frac{3}{16}\right], \quad m_{1} m_{2}=c $$

$$ \mathcal{A}_{\mathcal{R} \mathcal{M}^{\prime}}=\int_{0}^{S} d s\left[\frac{\mu}{2} x^{\prime 2}-\left(V_{\mathcal{R} \mathcal{M}^{\prime}}-E_{\mathcal{R} \mathcal{M}^{\prime}}\right)\right] $$

$$ E_{\mathcal{R} \mathcal{M}^{\prime}}=-\frac{\hbar^{2}}{2 \mu}\left(m_{1}^{2}+m_{2}^{2}\right)=-\frac{\hbar^{2}}{2 \mu}\left(m_{1}^{2}+c^{2} / m_{1}^{2}\right) $$

$$ \left(\theta_{b} \mid \theta_{a}\right)_{m_{1}, m_{2}, E_{\mathcal{P} \mathcal{T}^{\prime}}}=\sqrt{\sin \theta_{b} \sin \theta_{a}}\left(x_{b} \mid x_{a}\right)_{m_{1}, m_{2}, E_{\mathcal{R} \mathcal{M}^{\prime}}} $$

$$ \begin{align*} \left(x_{b} \mid x_{a}\right)_{m_{1}, m_{2}, E_{\mathcal{R} \mathcal{M}^{\prime}}} & =\frac{-2 i \mu}{\hbar} \Gamma\left(m_{1}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right)-s\right) \Gamma\left(s-m_{1}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right)+1\right) \\ & \times \frac{1}{2} d_{m_{1}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right),-m_{2}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right)}\left(\theta_{b}\left(x_{b}\right)-\pi\right) d_{m_{1}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right), m_{2}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right)}\left(\theta_{a}\left(x_{a}\right)\right) \end{align*} $$

$$ m_{1}\left(E_{\mathcal{R} \mathcal{M}^{\prime}}\right)=s-n, \quad n=0,1, \ldots,[s] $$

$$ \begin{align*} \Psi_{n}(x) & =\sqrt{\frac{m_{1}^{2}-m_{2}^{2}}{m_{1}} \frac{\Gamma\left(s+1-m_{1}\right) n!}{\Gamma\left(s+1-m_{2}\right) \Gamma\left(s+1+m_{2}\right)}} \\ & \times\left[\frac{1}{2}(1+\tanh x)\right]^{\left(m_{1}-m_{2}\right) / 2}\left[\frac{1}{2}(1-\tanh x)\right]^{\left(m_{1}+m_{2}\right) / 2} P_{n}^{\left(m_{1}-m_{2}, m_{1}+m_{2}\right)}(-\tanh x) \end{align*} $$

$$ \begin{align*} \Psi_{n}(x) & =\sqrt{\frac{m_{1}^{2}-m_{2}^{2}}{m_{1}} \frac{\Gamma\left(s+1+m_{1}\right) \Gamma\left(s+1-m_{2}\right)}{n!\Gamma\left(1+m_{1}-m_{2}\right)^{2} \Gamma\left(s+1+m_{2}\right)}} \\ & \times\left[\frac{1}{2}(1+\tanh x)\right]^{\left(m_{1}-m_{2}\right) / 2}\left[\frac{1}{2}(1-\tanh x)\right]^{\left(m_{1}+m_{2}\right) / 2} \\ & \times F\left(2 s-n+1,-n ; 1+m_{1}-m_{2} ; \frac{1}{2}(1+\tanh x)\right) \end{align*} $$

$$ V_{\mathcal{H}}(r)=g \frac{1}{e^{r / a}-1} $$

$$ \mathcal{A}_{\mathcal{H}}=\int d t\left[\frac{M}{2} \dot{r}^{2}-V_{\mathcal{H}}(r)+E_{\mathcal{H}}\right] $$

$$ f(r)=4\left(1-e^{-r / a}\right)^{2} $$

$$ \mathcal{A}_{\mathcal{H}}^{f}=\int_{0}^{\infty} d s\left[\frac{M}{2} \frac{r^{\prime 2}}{4\left(1-e^{-r / a}\right)^{2}}-g 4 e^{-r / a}\left(1-e^{-r / a}\right)+E_{\mathcal{H}} 4\left(1-e^{-r / a}\right)^{2}\right] $$

$$ \frac{d r}{d x}=h^{\prime}(x), $$

$$ h^{\prime}=\sqrt{f}=2\left(1-e^{-r / a}\right) $$

$$ \frac{r}{a}=x+a \log [2 \cosh (x / a)]=\log \left(e^{2 x / a}+1\right) $$

$$ h^{\prime}(x)=2 \frac{e^{2 x / a}}{e^{2 x / a}+1}=\frac{e^{x / a}}{\cosh (x / a)} $$

$$ \begin{align*} h^{\prime \prime}(x) & =\frac{1}{a} \frac{1}{\cosh ^{2}(x / a)}=\frac{1}{a} \frac{e^{x / a}}{\cosh (x / a)}[1-\tanh (x / a)] \\ h^{\prime \prime \prime}(x) & =-\frac{2}{a^{2}} \frac{\sinh x}{\cosh ^{3} x}=-\frac{2}{a^{2}} \frac{e^{x / a}}{\cosh (x / a)}\left[\tanh (x / a)-\tan ^{2}(x / a)\right] \end{align*} $$

$$ \begin{align*} \frac{h^{\prime \prime}}{h^{\prime}} & =\frac{1}{a} \frac{e^{-x / a}}{\cosh (x / a)}=\frac{1}{a}[1-\tanh (x / a)] \\ \frac{h^{\prime \prime \prime}}{h^{\prime}} & =-\frac{2}{a^{2}} \frac{e^{-x / a} \sinh (x / a)}{\cosh ^{2}(x / a)}=-\frac{2}{a^{2}} \tanh (x / a)[1-\tanh (x / a)] \end{align*} $$

$$ V_{\mathrm{eff}}=\frac{\hbar^{2}}{8 M a^{2}}\left[2-2 \tanh (x / a)-\frac{4}{\cosh ^{2}(x / a)}\right] $$

$$ \begin{align*} \mathcal{A}_{\mathcal{H}}^{\mathrm{DK}}=\int_{0}^{S} d s & \left\{\frac{M}{2} x^{\prime 2}-\left(g+E_{\mathcal{H}}-\frac{\hbar^{2}}{2 M a^{2}}\right) \frac{1}{\cosh ^{2}(x / a)}\right. \\ & \left.+\left(2 E_{\mathcal{H}}+\frac{\hbar^{2}}{4 M a^{2}}\right) \tanh (x / a)+\left(2 E_{\mathcal{H}}-\frac{\hbar^{2}}{4 M a^{2}}\right)\right\} \end{align*} $$

$$ V_{\mathcal{R} \mathcal{M}^{\prime}}(x / a)=\frac{\hbar^{2}}{2 M a^{2}}\left[-\frac{s(s+1)}{\cosh ^{2}(x / a)}+c \tanh (x / a)\right] $$

$$ \left(r_{b} \mid r_{a}\right)_{E_{\mathcal{H}}}=e^{\left(x_{b}+x_{a}\right) / 2 a}\left[\cosh \left(x_{b} / a\right) \cosh \left(x_{a} / a\right)\right]^{-1 / 2}\left(x_{b} \mid x_{a}\right)_{E_{\mathcal{R} \mathcal{M}^{\prime}}}, $$

$$ E_{\mathcal{H} n}=g\left(\frac{n_{g}^{2}-n^{2}}{2 n_{g} n}\right)^{2}=-\frac{\hbar^{2}}{2 M a^{2}} \frac{1}{4 n^{2}}\left(n_{g}^{2}-n^{2}\right)^{2}, \quad 1 \leq n