About Scattering on the Ring

Abstract

The mathematical model of a simplest quasi-one-dimensional quantum network constructed of relatively narrow waveguides (the width of the waveguide is less than the de Broghlie wavelength of the electron in the material) is developed. This model allows to reduce the problem of calculating the current through the quantum network to the construction of scattered waves for some Schrödinger equation on the corresponding one-dimensional graph. We consider a graph consisting of a compact part and few semiinfinite rays attached to it via some boundary condi-tion depending on a parameter β (analog of the inverse exponential “mass” e -bH of a potential barrier H separating the rays from the compact part of the graph). This parameter regulates the connection between the rays and the compact part. Spectral properties of the Schrödinger operator on this graph are described with a special emphasis on the resonance case when the Fermi level in the rays coincides with one of eigenvalues of the nonperturbed Schrödinger operator on the ring. An explicit expression is obtained for the scattering matrix in the resonance case for weakening connection β → 0 between the rays and the compact part. KeywordsGreen FunctionTransmission CoefficientScattered WaveSchrodinger EquationResonance CaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.