scholarly journals The norm estimate of the difference between the Kac operator and the Schrödinger semigroup: A unified approach to the nonrelativistic and relativistic cases

1998 ◽  
Vol 149 ◽  
pp. 53-81 ◽  
Author(s):  
Takashi Ichinose ◽  
Satoshi Takanobu
Keyword(s):  
Schrödinger Operators ◽  
Product Formula ◽  
Operator Norm ◽  
Schrodinger Operators ◽  
Unified Approach ◽  
Trotter Product Formula ◽  
Nonrelativistic Case ◽  
Norm Estimate ◽  
Schrödinger Semigroup ◽  
The Difference

Abstract.An Lp operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in the Lp operator norm. The method of the proof is probabilistic based on the Feynman-Kac formula. The problem is discussed in the relativistic as well as nonrelativistic case.

A UNIFIED APPROACH TO RESOLVENT EXPANSIONS AT THRESHOLDS

2001 ◽  
Vol 13 (06) ◽  
pp. 717-754 ◽  
Author(s):  
ARNE JENSEN ◽  
GHEORGHE NENCIU
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Unified Approach ◽  
Zero Energy ◽  
Real Potential ◽  
Factorization Technique

Results are obtained on resolvent expansions around zero energy for Schrödinger operators H=-Δ+V(x) on L2(Rm), where V(x) is a sufficiently rapidly decaying real potential. The emphasis is on a unified approach, valid in all dimensions, which does not require one to distinguish between ∫V(x)dx=0 and ∫V(x)dx≠0 in dimensions m=1,2. It is based on a factorization technique and repeated decomposition of the Lippmann–Schwinger operator. Complete results are given in dimensions m=1 and m=2.

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.

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