Abstract: This paper considers some self-adjoint operators N_u in Bernoulli noise functionals that essentially belong to the category of potential operators. By the Full-Wiener integral transform, the unitarily equivalent multiplication operators are explicitly obtained, and thus the explicit expressions of their spectrums are further obtained. Under some mild conditions, the fact that these operators have only pure point spectra is proved. Finally, in application, the proof that the quantum system with such an operator as the Hamiltonian is stable is proved.

Key words: Bernoulli noise functional, potential operator, spectral property, weight function