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The dissipation of ion-acoustic surface waves propagating in a semi-bounded and collisional plasma which has a boundary with vacuum is theoretically investigated and this result is used for the analysis of edge-relevant plasma simulated by Divertor Plasma Simulator-2 (DiPS-2). The collisional damping of the surface wave is investigated for weakly ionized plasmas by comparing the collisionless Landau damping with the collisional damping as follows: (1) the ratio of ion temperature $({T_i})$
to electron temperature $({T_e})$
should be very small for the weak collisionality $({T_i}/{T_e} \ll 1)$
; (2) the effect of collisionless Landau damping is dominant for the small parallel wavenumber, and the decay constant is given as $\gamma \approx{-} \sqrt {\mathrm{\pi }/2} {k_\parallel }{\lambda _{De}}\omega _{pi}^2/{\omega _{pe}}$
; and (3) the collisional damping dominates for the large parallel wavenumber, and the decay constant is given as $\gamma \approx{-} {\nu _{in}}/16$
, where ${\nu _{in}}$
is the ion–neutral collisional frequency. An experimental simulation of the above theoretical prediction has been done in the argon plasma of DiPS-2, which has the following parameters: plasma density ${n_e} = (\textrm{2--9)} \times \textrm{1}{\textrm{0}^{11}}\;\textrm{c}{\textrm{m}^{ - 3}}$
, ${T_e} = 3.7- 3.8\;\textrm{eV}$
, ${T_i} = 0.2- 0.3\;\textrm{eV}$
and collision frequency ${\nu _{in}} = 23- 127\;\textrm{kHz}$
. Although the wavelength should be specified with the given parameters of DiPS-2, the collisional damping is found to be $\gamma = ( - 0.9\;\textrm{to}\; - 5) \times {10^4}\;\textrm{rad}\;{\textrm{s}^{ - 1}}$
for ${k_\parallel }{\lambda _{De}} = 10$
, while the Landau damping is found to be $\gamma = ( - 4\;\textrm{to}\; - 9) \times {10^4}\;\textrm{rad}\;{\textrm{s}^{ - 1}}$
for ${k_\parallel }{\lambda _{De}} = 0.1$
.
Let 
$\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with 
$1\leq x\leq u\leq v\leq w$ and let 
${W}_{\alpha } $ be the associated weighted shift with weight sequence 
$\alpha $. The set of positive real numbers 
$x$ such that 
${W}_{\alpha } $ is quadratically hyponormal for some 
$u, v$ and 
$w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].
