Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

Details Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

2010-01-20

arxiv.org/abs/1001.3665v4

Mathematics

Analysis of PDEs

60 pages, 13 figures

Scientific paper

Artificial interface conditions parametrized by a complex number $\theta_{0}$ are introduced for 1D-Schr\"odinger operators. When this complex parameter equals the parameter $\theta\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $\theta_{0}$.

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