scholarly journals A new phase space localization technique with application to the sum of negative eigenvalues of Schrödinger operators

1991 ◽  
Vol 24 (2) ◽  
pp. 215-225 ◽  
Author(s):  
Heinz Siedentop ◽  
Rudi Weikard
Keyword(s):  
Phase Space ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Localization Technique ◽  
Negative Eigenvalues ◽  
Space Localization ◽  
New Phase

scholarly journals The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as an emergent phenomenon

2021 ◽  
Author(s):  
Valter Moretti ◽  
Christiaan J. F. van de Ven
Keyword(s):  
Phase Space ◽  
Probability Measure ◽  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Classical Limit ◽  
Schrödinger Operators ◽  
Algebraic Approach ◽  
Schrodinger Operators ◽  
Berezin Quantization ◽  
Emergent Phenomenon

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off [Formula: see text]. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.

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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