scholarly journals Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators

1993 ◽  
Vol 153 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Hiroshi Isozaki
Keyword(s):  
Asymptotic Properties ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Generalized Eigenfunctions ◽  
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].

Bloch waves and multiparameter spectral theory

1983 ◽  
Vol 95 (1-2) ◽  
pp. 73-93 ◽  
Author(s):  
B. P. Rynne ◽  
B. D. Sleeman
Keyword(s):  
Spectral Theory ◽  
Eigenfunction Expansion ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Bloch Waves ◽  
Generalized Eigenfunctions

SynopsisWe study the spectral theory of a multiparameter system of periodic Schrödinger operators. Bloch waves are generalized eigenfunctions of these operators and are used to give eigenfunction expansion theorems and to derive some properties of the spectrum of the system.

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