Oscillatons in Scalar-Field Dark Matter from a Full Fourier Expansion of an Exponential Potential

Real, time-dependent scalar fields can form oscillating, self-gravitating configurations-oscillatonsthat are viable candidates for scalar-field dark matter (SFDM). We revisit oscillatons with an exponential self-interaction and develop a full Fourier (Jacobi{Anger) treatment that resums the time dependence of both the metric and the potential, thereby unifying quadratic, quartic, and higher-order interactions within a single framework. After fixing the small-amplitude normalization V0 = m2 Φ=(λ2k0), we derive a closed, dimensionless boundary-value problem for the radial profiles and solve it numerically via Bessel-series truncation with controlled convergence. We compute time-resolved and time-averaged observables energy density, radial energy flux, radial/tangential pressures, and total mass and map their dependence on the coupling λ and central amplitude. The geometry exhibits only even harmonics of the fundamental frequency, while composite observables inherit a DC part plus even harmonics; the radial flux oscillates predominantly at 2!. Apparent negative instantaneous pressures arise from coherent oscillations and are assessed consistently through classical energy-condition diagnostics (WEC/NEC/SEC). Our formulation provides a reproducible and extensible baseline for stability analyses and observational constraints on SFDM oscillatons