A complete set of quantum-mechanical observables on a two-dimensional sphere

A study is made of the problem of diagonal operators on a two-dimensional sphere. A trigonometrie form of an elliptic system of coordinates on a sphere that is convenient for applications in physics is derived. Wave eigenfunctions of diagonal operators in the elliptic coordinate system – so-called spheroconieal functions – are constructed. Their main properties are derived. Conditions that determine the eigenvalues of the second diagonal operator in the elliptic coordinate system are found. Some matrix elements of spheroconicM functions are calculated. Possible applications in physics are discussed for the complete set of quantum-mechanical observables associated with the elliptic coordinate system on the twodimensional sphere.