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 Dependence of the Density of States on the Probability Distribution for Discrete Random Schrödinger Operators

2018 ◽  
Vol 2020 (17) ◽  
pp. 5279-5341 ◽  
Author(s):  
Peter D Hislop ◽  
Christoph A Marx
Keyword(s):  
Probability Measure ◽  
Density Of States ◽  
Anderson Model ◽  
Schrödinger Operators ◽  
Bethe Lattice ◽  
Single Site ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
Wide Range

Abstract We prove that the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^{\ast }$ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general compactly supported single-site probability measure, without any further assumptions. The few previously available results for the disorder dependence of the IDS valid for dimensions $d \geqslant 2$ assumed absolute continuity of the single-site measure and thus excluded the Bernoulli–Anderson model. As a further application of our main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for $d=1$ in the probability measure with respect to the weak-$^{\ast }$ topology.

scholarly journals Counting eigenvalues using coherent states with an application to Dirac and Schrödinger operators in the semi-classical limit

1996 ◽  
Vol 34 (2) ◽  
pp. 265-283 ◽  
Author(s):  
William Desmond Evans ◽  
Roger T. Lewis ◽  
Heinz Siedentop ◽  
Jan Philip Solovej
Keyword(s):  
Classical Limit ◽  
Coherent States ◽  
Schrödinger Operators ◽  
Schrodinger Operators

scholarly journals Spontaneous symmetry breaking in the phase space

2021 ◽  
Author(s):  
Yiannis Contoyiannis ◽  
Stavros G. Stavrinides ◽  
Myron Kampitakis ◽  
Michael Hanias ◽  
Stelios M Potirakis ◽  
...  
Keyword(s):  
Phase Space ◽  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking

SEMI-CLASSICAL ANALYSIS FOR TOTAL CROSS-SECTIONS OF MAGNETIC SCHRÖDINGER OPERATORS IN TWO DIMENSIONS

1995 ◽  
Vol 07 (03) ◽  
pp. 443-480 ◽  
Author(s):  
HIDEO TAMURA
Keyword(s):  
Cross Sections ◽  
Classical Limit ◽  
Schrödinger Operators ◽  
Two Dimensions ◽  
Schrodinger Operators ◽  
Classical Analysis ◽  
Total Cross Sections ◽  
Total Scattering ◽  
Scattering Cross ◽  
Scattering Cross Sections

We define total scattering cross-sections for magnetic Schrödinger operators in two dimensions and prove the shadow scattering (the quantum total cross-sections double the classical ones in the semi-classical limit) under some assumptions.

scholarly journals Spectral Analysis of Schrödinger Operators with Unusual Semiclassical Behavior

10.14311/1801 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Pavel Exner ◽  
Diana Barseghyan
Keyword(s):  
Spectral Analysis ◽  
Phase Space ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Schrödinger Operators ◽  
Critical Value ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators ◽  
Two Dimensional ◽  
Eigenvalue Bounds

In this paper we discuss several examples of Schrödinger operators describing a particle confined to a region with thin cusp-shaped ‘channels’, given either by a potential or by a Dirichlet boundary; we focus on cases when the allowed phase space is infinite but the operator still has a discrete spectrum. First we analyze two-dimensional operators with the potential |xy|p - ?(x2 + y2)p/(p+2)where p?1 and ??0. We show that there is a critical value of ? such that the spectrum for ??crit it is unbounded from below. In the subcriticalcase we prove upper and lower bounds for the eigenvalue sums. The second part of work is devoted toestimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions havinginfinite cusps which are geometrically nontrivial being either curved or twisted; we are going to showhow these geometric properties enter the eigenvalue bounds.

scholarly journals Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

2019 ◽  
Vol 9 (4) ◽  
pp. 1287-1325
Author(s):  
Michele Correggi ◽  
Marco Falconi ◽  
Marco Olivieri
Keyword(s):  
Classical Limit ◽  
Schrödinger Operators ◽  
Schrodinger Operators
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