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 efimov effect of three-body schrödinger operators: Asymptotics for the number of negative eigenvalues

1993 ◽  
Vol 130 ◽  
pp. 55-83 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Infinite Number ◽  
Schrödinger Operators ◽  
Related Result ◽  
Schrodinger Operators ◽  
Rigorous Proof ◽  
Body System ◽  
Zero Energy ◽  
Efimov Effect ◽  
Negative Eigenvalues ◽  
Three Body

The Efimov effect is one of the most remarkable results in the spectral theory for three-body Schrödinger operators. Roughly speaking, the effect will be explained as follows: If all three two-body subsystems have no negative eigenvalues and if at least two of these two-body subsystems have resonance states at zero energy, then the three-body system under consideration has an infinite number of negative eigenvalues accumulating at zero. This remarkable spectral property was first discovered by Efimov [1] and the problem has been discussed in several physical journals. For related references, see, for example, the book [3]. The mathematically rigorous proof of the result has been given by the works [4, 8, 9]. The aim of the present work is to study the asymptotic distribution of these negative eigenvalues below zero (bottom of essential spectrum). Denote by N(E), E > 0, the number of negative eigenvalues less than – E. Then the main result obtained here is, somewhat loosely stating, that N(E) behaves like | log E | as E → 0. We first formulate precisely the main theorem and then make a brief comment on the recent related result obtained by Sobolev [7].

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 The Computation of Thresholds for Schrödinger Operators

1999 ◽  
Vol 2 ◽  
pp. 139-154
Author(s):  
E.B. Davies
Keyword(s):  
Schrödinger Operators ◽  
Negative Energy ◽  
Schrodinger Operators ◽  
Zero Energy ◽  
Energy Eigenvalues ◽  
Energy Thresholds

AbstractThe paper describes an approach to the computation of the zero energy thresholds for the appearance of negative energy eigenvalues of Schrödinger operators.

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