Exact Solutions of Indirect Transverse Field Effects in Elongated Structures with Applications to CERN LHC and PS

The understanding of the electromagnetic interaction of the particle beam with the surrounding elements - so-called indirect space charge driven (ISCD) effects - in particle accelerators, is crucial for stable high-intensity performance. It is addressed and applied at the CERN accelerator complex. An appropriate quantitative explanation for the ISCD tune-shift which must be corrected during the operation of the Large Hadron Collider (LHC) was missing. This work developed an approach based on complex Green functions (solving the arbitrary Dirichlet and Neumann boundary problem) which matches measurements with unprecedented accuracy. As the primary origin of the ISCD tune-shift, the electric interaction with the beam-screen is identified. A closed-form model is obtained, that is also applicable to future accelerator projects as the High Luminosity (HL)-LHC, where these effects will be at least a factor two higher. During the Multi-Turn Extraction in the Proton Synchrotron (PS), the beam is split into the main beam and four islands so-called beamlets. In measurements, an intensity dependence in the beamlet position and tune was observed. ISCD effects are the cause as shown in calculations and numerical simulations based on closed-form expression acquired in this thesis. To obtain simple mathematical expressions, a novel Lorentz operator of the Green functions, on the Riemann-sphere (RS), and from it, the so-called image operators for arbitrary beam distributions are derived. These operators allow estimating the ISCD tune-shift of complex accelerator models. A novel method to approximate the fundamental electrostatic field (the Green function) of arbitrary simply connected domains, including an error bound, is proven. It is used to approximate the rect-elliptical LHC beam-screen. Additionally, a new integral representation of the Neumann function on the RS for smooth bounded simply-connected domains is derived. It allows for classifying domains concerning the solution of the Neumann function into bounded, unbounded star-like and exterior solutions. A novel method is presented, to obtain closed-forms in the case of unbounded star-like domains. Consequently, several novel closed-form solutions for the magnetic interaction of essential shapes as the $n$-poles or the combined-function magnets of the PS are obtained (so far only parallel plates were used). Finally, several new off-axis image tensors for standard geometries are provided.