For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and Laguerre polynomials. For these polynomials we obtain the explicit form of the hamiltonians, the energy levels and the explicit form of the impulse operators.KEY WORDS: classical orthogonal polynomials, generalized oscillator algebras, Poisson kernels, generalized Fourier transform. MSC (1991): 33C45, 33C80, 33D45, 33D80
The spectral measure of the position (momentum) operator X for q-deformed oscillator is calculated in the case of the indetermine Hamburger moment problem. The exposition is given for concrete choice of generators for q-oscillator algebra, although developed technique apply for every other cases with indetermine moment problem. The Stieltjes transformation m(z) of spectral measure is expressed in terms of the entries of Jacobi matrix X only. The direct connection between values of parameters labeling the spectral measures and related selfadjoint extensions of X is established.
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