On the Mourre Estimates for Three Body Schrödinger Operators in a Constant Magnetic Field

2007 ◽  
Vol 50 (2) ◽  
pp. 339-370
Author(s):  
Tadayoshi Adachi
Keyword(s):  
Magnetic Field ◽  
Constant Magnetic Field ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Three Body

scholarly journals Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields II

1997 ◽  
Vol 145 ◽  
pp. 69-98
Author(s):  
Tetsuya Hattori
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Discrete Spectrum ◽  
Bound States ◽  
Atomic System ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Many Body ◽  
Discrete Energy

This paper is continuation from [10], in which we studied the discrete spectrum of atomic Hamiltonians with non-constant magnetic fields and, more precisely, we showed that any atomic system has only finitely many bound states, corresponding to the discrete energy levels, in a suitable magnetic field. In this paper we show another phenomenon in non-constant magnetic fields that any atomic system has infinitely many bound states in a suitable magnetic field.

NORM RESOLVENT CONVERGENCE TO MAGNETIC SCHRÖDINGER OPERATORS WITH POINT INTERACTIONS

2001 ◽  
Vol 13 (04) ◽  
pp. 465-511 ◽  
Author(s):  
HIDEO TAMURA
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Two Dimensions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Point Interactions ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence

The Schrödinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schrödinger operators with magnetic fields of small support and study the norm resolvent convergence to Schrödinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schrödinger operators with magnetic potentials slowly falling off at infinity.

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].

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