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

2010/01/20 by A Faraj, Faraj, Ali, Andrea Mantile +3
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph) #Quantum and electron transport phenomena #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1001.3665

openalex publication_date 2010/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Artificial interface conditions parametrized by a complex number θ₀ are introduced for 1D-Schrödinger operators. When this complex parameter equals the parameter θ∈ 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∈ \N} as h→ 0, according to θ₀.

Citations

• Exponential Estimates in Adiabatic Quantum Evolution
• A class of analytic perturbations for one-body Schrödinger Hamiltonians

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• Adiabatic theorems for quantum resonances
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