研究目的
Investigating the dynamics of a quantum particle in a periodic potential under an homogeneous external field, focusing on the existence of families of stationary (metastable) states with associated energies displaced on regular ladders, the so-called Stark–Wannier ladders, and the wave function performing Bloch oscillations.
研究成果
The paper demonstrates that in the limit of large periodic potential, the Stark–Wannier ladders of the linear equation become a dense energy spectrum due to a cascade of bifurcations of stationary solutions. This occurs when the ratio between the effective nonlinearity strength and the tilt of the external field increases.
研究不足
The study assumes a smooth, real-valued, periodic and nonnegative function for the periodic potential and a smooth real-valued function for the Stark-type potential with compact support. The analysis is limited to the semiclassical regime of small h.
1：Experimental Design and Method Selection:
The study involves the nonlinear one-dimensional Schr¨odinger equation with a periodic potential and a Stark-type perturbation. The methodology includes semiclassical approximation and the derivation of a discrete time-independent nonlinear Schr¨odinger equation.
2：Sample Selection and Data Sources:
The study is theoretical, focusing on the mathematical analysis of the Schr¨odinger equation under specified conditions.
3：List of Experimental Equipment and Materials:
The study is theoretical and does not involve physical experiments or equipment.
4：Experimental Procedures and Operational Workflow:
The paper outlines the mathematical procedures for analyzing the nonlinear Schr¨odinger equation, including the use of Bloch functions, Wannier functions, and semiclassical construction.
5：Data Analysis Methods:
The analysis involves mathematical techniques for solving the nonlinear Schr¨odinger equation, including the use of spectral properties of the Bloch operator and the construction of stationary solutions.
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