We give an elementary introduction to the problems of quantization of dissipative systems considering the simplest but still not fully understood model of damped harmonic oscillator. We recall some old and introduce some new methods of quantization of the model starting from the classical damped harmonic oscillator equation. Our methods are based on Hamiltonian formulation of dynamical systems adopted to the case of dissipation and loss of energy. Interestingly, in our approach we can interpret the poles of the analytical continuation of the energy spectrum as resonant states, well-known in atomic physics.
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