Kang, Yang; Klein, Abel The Liouville equation for singular ergodic magnetic Schrödinger operators. (English) Zbl 1309.81354 J. Math. Phys. 51, No. 3, 032105, 21 p. (2010). Summary: We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation and solve a Liouville equation in an appropriate Hilbert space.{©2010 American Institute of Physics} MSC: 81V70 Many-body theory; quantum Hall effect 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q41 Time-dependent Schrödinger equations and Dirac equations 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics PDFBibTeX XMLCite \textit{Y. Kang} and \textit{A. Klein}, J. Math. Phys. 51, No. 3, 032105, 21 p. (2010; Zbl 1309.81354) Full Text: DOI arXiv References: [1] Bouclet, J. M.; Germinet, F.; Klein, A.; Schenker, J. H., Linear response theory for magnetic Schrödinger operators in disordered media, J. Funct. Anal., 226, 301 (2005) · Zbl 1088.82013 · doi:10.1016/j.jfa.2005.02.002 [2] Faris, W. G., Self-Adjoint Operators, 433 (1975) · Zbl 0317.47016 [3] Germinet, F.; Klein, A., New characterizations of the region of complete localization for random Schrödinger operators, J. Stat. Phys., 122, 73 (2006) · Zbl 1127.82031 · doi:10.1007/s10955-005-8068-9 [4] Germinet, F.; Klein, A., Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Am. Math. Soc., 131, 911 (2003) · Zbl 1013.81009 · doi:10.1090/S0002-9939-02-06578-4 [5] Kisyński, J., Sur les opérateurs de Green des problèms de Cauchy abstraits, Stud. Math., 23, 285 (1964) · Zbl 0117.10202 [6] Leinfelder, H.; Simader, C. G., Schrödinger operators with singular magnetic potentials, Math. Z., 176, 1 (1981) · Zbl 0468.35038 · doi:10.1007/BF01258900 [7] Reed, M.; Simon, B., Methods of Modern Mathematical Physics I: Functional Analysis (1980) · Zbl 0459.46001 [8] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (1975) · Zbl 0308.47002 [9] Simon, B., Maximal and minimal Schrödinger forms, J. Oper. Theory, 1, 37 (1979) · Zbl 0446.35035 [10] Simon, B., Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (1971) · Zbl 0232.47053 [11] Yosida, K., Functional Analysis (1980) · Zbl 0126.11504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.