Spectral Analysis of Quantum Hamiltonians : Spectral Days 2010 This volume contains surveys as well as research articles broadly centered on spectral analysis. Topics range from spectral continuity for magnetic and pseudodifferential operators to localization in random media from the stability of matter to properties of Aharonov Bohm and Quantum Hall Hamiltonians from waveguides and resonances to supersymmetric models and dissipative fermion systems. This is the first of a series of volumes reporting every two years on recent progress in spectral theory. This volume contains surveys as well as research articles broadly centered on spectral analysis. Topics range from spectral continuity for magnetic and pseudodifferential operators to localization in random media from the stability of matter to properties of Aharonov Bohm and Quantum Hall Hamiltonians from waveguides and resonances to supersymmetric models and dissipative fermion systems. This is the first of a series of volumes reporting every two years on recent progress in spectral theory.ai