\Gamma, \qquad K_{\pm} = K_{\pm,{\rm B}} - \Gamma, \qquad Y = \frac{1}{2}
        \left(r \partial_r W + {b^{\dagger}}_x b_x + {b^{\dagger}}_y b_y - 1\right), \\
  K_{0,{\rm B}} & = D + \frac{1}{4} \omega r^2, \qquad K_{\pm,{\rm B}} = - D + \frac{1}{4} \omega r^2
        \mp \frac{1}{2} (r \partial_r + 1), \\
  D & = \frac{1}{4\omega} \left[- \partial_r^2 - \frac{1}{r} \partial_r - \frac{1}{r^2} \partial_{\varphi}^2 -
        \partial_r^2 W - \frac{1}{r} \partial_r W - \frac{1}{r^2} \partial_{\varphi}^2 W + (\partial_r W)^2
        + \frac{1}{r^2} (\partial_{\varphi} W)^2\right], \\
  \Gamma & = \frac{1}{2\omega} \biggl\{\biggl[\cos^2 \varphi \partial_r^2 W - \frac{2}{r} \sin \varphi
        \cos \varphi \partial_{r\varphi}^2 W + \frac{1}{r^2} \sin^2 \varphi \partial_{\varphi}^2 W + \frac{1}{r}
        \sin^2 \varphi \partial_r W \\
  & \quad + \frac{2}{r^2} \sin \varphi \cos \varphi \partial_{\varphi} W\biggr] {b^{\dagger}}_x b_x + \biggl[\sin \varphi
        \cos \varphi \partial_r^2 W + \frac{1}{r} (\cos^2 \varphi - \sin^2 \varphi) \partial_{r\varphi}^2 W \\
  & \quad - \frac{1}{r^2} \sin \varphi \cos \varphi \partial_{\varphi}^2 W - \frac{1}{r} \sin \varphi \cos \varphi
        \partial_r W - \frac{1}{r^2} (\cos^2 \varphi - \sin^2 \varphi) \partial_{\varphi} W\biggr] \\
  & \quad \times ({b^{\dagger}}_x b_y + {b^{\dagger}}_y b_x) + \biggl[\sin^2 \varphi \partial_r^2 W + \frac{2}{r} \sin \varphi
        \cos \varphi \partial_{r\varphi}^2 W + \frac{1}{r^2} \cos^2 \varphi \partial_{\varphi}^2 W \\
  & \quad + \frac{1}{r} \cos^2 