Non-invasive beam diagnostics are essential for operating modern high-intensity accelerators such as the Large Hadron Collider (LHC). Among these diagnostics, Schottky monitors stand out for their ability to measure tune, chromaticity, momentum spread, and emittance based on the intrinsic noise (Schottky noise) of the circulating beam. Yet, the accuracy of Schottky analysis can be compromised by collective effects - particularly beam coupling impedance - and by non-linear fields, such as those introduced by Landau octupoles.

This thesis presents a comprehensive theoretical and computational framework for Schottky diagnostics, addressing the effects of beam coupling impedance and octupolar non-linearities. It provides a rigorous analysis of the statistical properties of Schottky spectra, demonstrating that individual "instantaneous" spectra represent only a single realization of an underlying random process. We develop and validate a novel simulation method that computes the beam's spectral content efficiently for the LHC, overcoming the limitations of standard discrete Fourier transform approaches.

Building on this foundation, the influence of Landau octupoles is explored in detail. We derive new expressions that reveal how octupole-induced amplitude detuning modifies the betatron sidebands of the Schottky spectrum, and we confirm these findings through macro-particle simulations and experiments with LHC ion beams. Despite introducing distortions that complicate classical parameter-extraction formulas, the presence of octupoles can also mitigate coherent components, sometimes improving overall diagnostic capabilities.

The study concludes with an investigation into beam coupling impedance, demonstrated to significantly influence synchrotron and betatron tunes in high-intensity proton beams. Broad-band resonator models are integrated into macro-particle simulations and analytical formulas, highlighting alterations in the Schottky spectrum and enabling comparisons with measured spectra across varying bunch intensities at the LHC. Preliminary evidence suggests that parts of the LHC's longitudinal impedance model may be underestimated.