Spectral shift function for perturbed periodic Schrödinger operators. The large-coupling constant limit case

2011 ◽  
Vol 75 (3-4) ◽  
pp. 233-250
Author(s):  
Mouez Dimassi ◽  
Maher Zerzeri
Keyword(s):  
Schrödinger Operators ◽  
Spectral Shift ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Limit Case ◽  
Spectral Shift Function ◽  
Coupling Constant ◽  
Large Coupling ◽  
Large Coupling Constant ◽  
Perturbed Periodic

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.

scholarly journals THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS

2000 ◽  
Vol 12 (06) ◽  
pp. 807-847 ◽  
Author(s):  
VADIM KOSTRYKIN ◽  
ROBERT SCHRADER
Keyword(s):  
Density Of States ◽  
Arbitrary Dimension ◽  
Schrödinger Operators ◽  
Spectral Shift ◽  
Random Schrödinger Operators ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Spectral Shift Function ◽  
Arbitrary Dimensions ◽  
The Difference

In this article we continue our analysis of Schrödinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.

Cluster Properties of One Particle Schrödinger Operators. II

1998 ◽  
Vol 10 (05) ◽  
pp. 627-683 ◽  
Author(s):  
V. Kostrykin ◽  
R. Schrader
Keyword(s):  
Explicit Form ◽  
Scattering Amplitude ◽  
Schrödinger Operators ◽  
Spectral Shift ◽  
Shift Function ◽  
Schrodinger Operators ◽  
Spectral Shift Function ◽  
Oscillatory Behaviour ◽  
Cluster Properties ◽  
Math Phys

We continue the study of cluster properties of spectral and scattering characteristics of Schrödinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833–853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of the wells tends to infinity. In this article we determine the explicit form of the subleading contributions, which in particular show strong oscillatory behaviour. Also we apply our methods to the critical and subcritical double well problems.

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