  The excited hyperon $\Lambda(1405)$ is the important building block for
kaonic nuclei which are nuclear many-body system with anti-kaons. We have been
investigating the $\Lambda(1405)$ resonance with the coupled-channel Complex
Scaling Method (ccCSM) in which the $\Lambda(1405)$ is treated as a
hadron-molecular state of a $\bar{K}N$-$\pi\Sigma$ coupled system. We use a
$\bar{K}N$(-$\pi Y$) potential based on the chiral SU(3) theory. In this
article, we report the double-pole nature of the $\Lambda(1405)$, which is a
characteristic property predicted by many studies with chiral SU(3)-based
models. With the help of the complex-range Gaussian basis in ccCSM, we have
found successfully another pole besides a pole near the $\bar{K}N$ threshold
(called higher pole) which was found in our previous work with the real-range
Gaussian basis. The new pole (called lower pole) is found far below $\bar{K}N$
threshold: $(M, -\Gamma/2)=(1395, -138)$ MeV when $f_\pi=110$ MeV. In spite of
so broad width of the lower-pole state, the state is clearly identified with
good separation from continuum states, since the oscillatory behavior of the
continuum states is improved owing to the complex-range Gaussian basis.
Analyzing the ccCSM wave function of the lower pole, we have revealed
explicitly that the lower-pole state is dominated by the $\pi\Sigma$ component
rather than the $\bar{K}N$ component. We have confirmed that the ccCSM wave
function is correctly connected to the asymptotic form of the resonance wave
function. We have estimated the meson-baryon mean distance for the lower-pole
state as well as for the higher-pole state. In addition, we have investigated
the origin of the appearance of the lower pole. The lower pole is confirmed to
be generated by the energy dependence attributed to the chiral dynamics, by
comparing the result of an energy-independent potential.
