Applications of Self-Adjoint Extensions in Quantum Physics by Pavel Exner, Petr Seba

By Pavel Exner, Petr Seba

The shared objective during this number of papers is to use the idea of self-adjoint extensions of symmetry operators in quite a few components of physics. this permits the development of precisely solvable types in quantum mechanics, quantum box concept, excessive strength physics, solid-state physics, microelectronics and different fields. The 20 papers chosen for those lawsuits provide an summary of this box of study unparallelled within the released literature; specifically the perspectives of the best colleges are basically provided. The booklet should be a major resource for researchers and graduate scholars in mathematical physics for a few years to come back. In those complaints, researchers and graduate scholars in mathematical physics will locate how you can build precisely solvable versions in quantum mechanics, quantum box thought, excessive power physics, solid-state physics, microelectronics and different fields.

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But if we skip them, we return to the linear wave equation, restore integrability, and recover Anderson localization. So then, for that enlarged wave packet, we can again add trapping hard walls, but keep the nonlinear terms, and ask the question whether the dynamics inside the wave packet remains regular, or will be chaotic at large enough times. Again the experience of molecular dynamics tells that the dynamics will stay chaotic with high probability, but the decoherence times increase. Therefore the possible flaw in the argument when dropping the nonlinear terms is the time scale.

T/. n2 n1 / ; (58) 42 S. Flach Fig. 17 Under a kick strength of k D 5, measures for ˇ D 0:3 (blue) and ˇ D 10 (red), for both quasiperiodic sequences set by D 1 (solid line), and for random sequences (dashed line, see [23] for details). Upper row: Mean logarithms for energy < log10 E >. The clouds around the quasiperiodic sequences correspond to one standard deviation error. Lower row: finite-difference derivative of the above. Grey horizontal lines correspond to exponents for weak and strong chaos regimes.

The predicted subdiffusive exponents are controlled only by the lattice dimension, and the power of nonlinearity. So far we discussed the resulting nonlinear diffusion for uncorrelated random potentials l . For linear wave equations, a number of other correlated potentials are known to result in wave localization for a corresponding linear wave equation. 1 Subdiffusive Destruction of Aubry-Andre Localization Let us replace the uncorrelated disorder potential in Sect. 2 ˛AA l C Â/ : (54) For the linear wave equation ˇ D 0 and any irrational choice of ˛AA this results in the well-known Aubry-Andre localization [64].

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