the low-lying dipole resonance in neutron-rich Ne isotopes, and we discuss the isotopic dependence with special attention to the deformation effects. To this end, we have developed a deformed QRPA code in the matrix formulation based on the coordinate-space Skyrme-Hartree-Fock-Bogoliubov (HFB) theory.

The paper is organized as follows: In Sec. \[model\], we explain our method. In Sec. \[check\], we check the results of our new calculation scheme by comparing the existing QRPA results. In Sec. \[results\], we present the results of the deformed QRPA and we discuss the microscopic structure of the low-lying dipole state in $^{26,28,30}$Ne. Finally, we summarize the paper in Sec. \[summary\].

\[model\]Model
==============

We briefly summarize here our approach (see Ref. [@yos06] for details). In order to discuss simultaneously effects of nuclear deformation and pairing correlations including the continuum, we solve the HFB equations [@dob84; @bul80] $$\begin{gathered}
\begin{pmatrix}
h^{\tau}(\boldsymbol{r}\sigma)-\lambda^{\tau} & \tilde{h}^{\tau}(\boldsymbol{r}\sigma) \\
\tilde{h}^{\tau}(\boldsymbol{r}\sigma) & -(h^{\tau}(\boldsymbol{r}\sigma)-\lambda^{\tau}) \end{pmatrix}
\begin{pmatrix}
\varphi^{\tau}_{1,\alpha}(\boldsymbol{r}\sigma) \\ 
\varphi^{\tau}_{2,\alpha}(\boldsymbol{r}\sigma)
\end{pmatrix}
\\
= E_{\alpha}
\begin{pmatrix}
\varphi^{\tau}_{1,\alpha}(\boldsymbol{r}\sigma) \\ 
\varphi^{\tau}_{2,\alpha}(\boldsymbol{r}\sigma)
\end{pmatrix} \label{eq:HFB1}\end{gathered}$$ directly in the cylindrical coordinates assuming axial and reflection symmetries. Here, $\tau=\nu$ (neutron) and $\pi$ (proton), and $\boldsymbol{r}=(\rho,z,\phi)$. For the mean-field Hamiltonian $h$, we employ the SkM\* interaction [@bar82]. Details for expressing the densities and currents in the cylindrical coordinate representation can be found in Refs. [@ter03; @sto05]. The pairing field is treated by using the density-dependent contact interaction [@ber91; @ter95], $$v_{pp}(\boldsymbol{r},\boldsymbol{r}^{\prime})=V_{0}\dfrac{1