Asymptotics for the Number of Negative Eigenvalues of Three-Body Schrödinger Operators with Efimov Effect

2018 ◽  
Author(s):  
Hideo Tamura
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Efimov Effect ◽  
Negative Eigenvalues ◽  
Three Body

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 Negative Eigenvalues of Two-Dimensional Schrödinger Operators

2015 ◽  
Vol 217 (3) ◽  
pp. 975-1028 ◽  
Author(s):  
Alexander Grigor’yan ◽  
Nikolai Nadirashvili
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Two Dimensional ◽  
Negative Eigenvalues
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