scholarly journals Scattering theory for repulsive Schrödinger operators and applications to the limit circle problem

2021 ◽  
Vol 14 (7) ◽  
pp. 2101-2122
Author(s):  
Kouichi Taira
Keyword(s):  
Scattering Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Limit Circle ◽  
Circle Problem

scholarly journals Reflection Probabilities of One-Dimensional Schrödinger Operators and Scattering Theory

2017 ◽  
Vol 18 (6) ◽  
pp. 2075-2085 ◽  
Author(s):  
Benjamin Landon ◽  
Annalisa Panati ◽  
Jane Panangaden ◽  
Justine Zwicker
Keyword(s):  
Scattering Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
One Dimensional

scholarly journals SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION

1999 ◽  
Vol 11 (02) ◽  
pp. 187-242 ◽  
Author(s):  
V. KOSTRYKIN ◽  
R. SCHRADER
Keyword(s):  
Lyapunov Exponent ◽  
Scattering Theory ◽  
Schrödinger Operators ◽  
Spectral Shift ◽  
One Dimension ◽  
Random Schrödinger Operators ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Theory Approach ◽  
Spectral Shift Function

Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz–Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent.

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