Modified Landau levels, damped harmonic oscillator and two-dimensional pseudo-bosons

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In a series of recent papers one of us has analyzed in some details a class of elementary excitations called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a,a†]=, which is replaced by [a,b]=, with b not necessarily equal to a†. Here, after a two-dimensional extension of the general framework, we apply the theory to a generalized version of the two-dimensional Hamiltonian describing Landau levels. Moreover, for this system, we discuss coherent states and we deduce a resolution of the identity. We also consider a different class of examples arising from a classical system, i.e. a damped harmonic oscillator.