SYMMETRY PRESERVING QUANTIZATION AND SELF-SIMILAR POTENTIALS

A particular one-dimensional model of classical mechanics obeying quad ratic Poisson symmetry algebra is considered. After a special quantiza tion the latter algebra is converted into the q-Weyl algebra describin g symmetries of a specific Schrodinger operator. In this way one recon structs the simplest self-similar potential having infinitely many bou nd states with eigenvalues forming one geometric series.