scholarly journals A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants

1987 ◽  
Vol 123 (1) ◽  
pp. 181-198 ◽  
Author(s):  
F Gesztesy ◽  
H Holden
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Unified Approach ◽  
Fredholm Determinants

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 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.

scholarly journals Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta
Keyword(s):  
Density Of States ◽  
Local Density ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Density Of States ◽  
Weak Disorder ◽  
Cluster Expansions ◽  
Strong Disorder ◽  
Lifshitz Tail

AbstractWe study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

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