[en] In this thesis, the discovery potential of the ATLAS experiment at the Large Hadron Collider (LHC) at CERN for the heavy neutral Higgs bosons H/A of theMinimal Supersymmetric extension of the Standard Model of particle physics (MSSM) in the decay channels H/A→τ⁺τ^-→e/μ+X and H/A→μ⁺μ^- has been studied. The ATLAS detector is designed to study the full spectrum of the physics phenomena occuring in the proton-proton collisions at 14 TeV center-of-mass energy and to provide answers to the question of the origin of particle masses and of electroweak symmetry breaking. For the studies, the ATLAS muon spectrometer plays an important role. The spectrometer allows for a precise muon momentum measurement independently of other ATLAS subdetectors. The performance of the muon spectrometer depends strongly on the performance of the muon tracking detectors, the Monitored Drift Tube Chambers (MDT). Computer programs have been developed in order to test and verify the ATLAS muon spectrometer simulation, an essential ingredient for data analysis. In addition, dedicated programs for the monitoring of the quality of the data collected by the muon spectrometer have been developed and tested with data from cosmic ray muons. High-quality cosmic ray muon data have been used for the calibration of the MDT-chambers. A new calibration method, called analytical autocalibration, has been tested. The proposed method achieved the required accuracy of 20 μm in the determination of the space-to-drift-time relationship of the drift tubes of the MDT chambers with only 2000 muon tracks per chamber. Reliable muon detector simulation and calibration are essential for the study of the MSSM Higgs boson decays H/A→τ⁺τ^-→e/μ+X and H/A→μ⁺μ^- and of the corresponding background processes. The signal selection and background rejection requirements have been optimized for maximum signal significance. The following results have been obtained for different assumptions on the MSSM Higgs boson mass m_A and on the ratio tan β of the vacuum expectation values of the two MSSM Higgs boson doublets: For the decay into τ-pairs, 5σ signal significance is obtained for tan β>or similar 5 and m_A=150 GeV/c² or for tan β>or similar 25 and m_A=600GeV/c² for an integrated luminosity of 30 fb^-1. For the decay into muon pairs, 5σ signal significance is achieved for tan β>or similar 20 and m_A=200 GeV/c² or for tan β>or similar 60 and m_A=450 GeV/c² for an integrated luminosity of 30 fb^-1. In addition, the mass of the Higgs boson H/A can be measured in the decay H/A→μ⁺μ^- exploiting the high muon momentum resolution of the ATLAS detector. The measurement accuracy degrades with increasing mass. A Higgs boson mass of 200 GeV/c² can be determined with an accuracy of 2%. (orig.)