Spectral problems for the Weyl-ordered form of operators $\left(\frac{1}{\hat{p}}\right)^{n} \hat{q}^{n}$ 1 p ̂ n q ̂ n

In this paper, we consider quantization of powers of the ratio between the Hamiltonian coordinates for position and momentum in one-dimensional systems. The domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and self-adjointness. The spectral problems for the operators that result from using Weyl-ordering are discussed by introducing Fredholm integral operator forms in position representation, and the symmetry of the actions of the parity and time reversal operators on the kernels is discussed. Finally, the general structures and properties of the eigenfunctions and eigenvalues are also derived and analyzed.