Diffraction and Interference Phenomena with De Broglie Gravitational Waves

De Broglie gravitational waves form an axially symmetric class of vacuum solutions of the linearized Einstein equations. Unlike the well-known massless transverse perturbations in the weak-field regime, they possess an effective mass and exhibit longitudinal degrees of freedom, allowing them to act as guiding fields for elementary particles. In this paper, we show that the dynamics of the de Broglie gravitational wave, described in terms of the real part of a classical field ψ , is compatible with single particle Quantum Mechanics, and that the complex field ψ can be interpreted as the single particle wave function. In particular, we find that ψ satisfies the Schrödinger equation for a free particle and, using the principles of gravitational lensing, the same holds in the presence of a weak central potential. We show that when our gravitational wave passes through two closely separated slits, it diffracts and interferes, creating the characteristic pattern of bright and dark bands observed in quantum experiments. We propose two different interpretations: a wave-particle picture, in which wave and particle are two distinct entities, and an effective background field perspective, in the spirit of soliton theory. A variant of the double-slit experiment that could serve, in principle, as an experimental test of the proposed framework is also discussed.