Ultra-thin single wall carbon nanotubes (SWNTs) with a diameter of only 0.3 nm were synthesized in the nano-channels of AlPO4-11 porous single crystals. Raman spectra, with excitation wavelengths in the range from 457.9 to 647.1 nm, show excellent agreement with the density functional calculations of the Raman-active vibration modes of the armchair (2,2) SWCNTs. Calculated imaginary part of the dielectric function also displays qualitative agreement with the resonant Raman data. Interestingly, the (2,2) nanotube has two metastable ground state corresponding two slightly different lattice constants in axial direction, one state is metallic and the other is semiconducting. The polarizibility of the Raman modes agrees well with the calculated intensities for non-resonant Raman scattering, although the resonant Raman scattering plays a key role in the process. Both theory and experiment show the free-standing (2,2) SWNTs to be unstable. Confinement of the SWNTs in the nano-channels stablilizes the structure.