Cosine and sine operators related to orthogonal polynomial sets on the interval [-1, 1]

The quantization of phase is still an open problem. In the approach of Susskind and Glogower, the so-called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related to the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way to arbitrary orthogonal polynomial sets on the interval [-1, 1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function, we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We also consider the arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states