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Localization on a quantum graph with a random potential on the edges. (English) Zbl 1147.82018

The authors prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential.For a particular case of a special cubic lattice graph that can be embedded in \(\mathbb R^d\) they define the family of random Schrödinger operators that exhibits deterministic spectrum. It is proved that in some neighbor hood the operators exhibit pure point spectrum with exponentially decaying eigenfunctions almost surely (Anderson localization).

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81V99 Applications of quantum theory to specific physical systems
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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