scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

scholarly journals New integrable two-centre problem on sphere in Dirac magnetic field

2020 ◽  
Vol 110 (11) ◽  
pp. 3105-3119 ◽  
Author(s):  
A. P. Veselov ◽  
Y. Ye
Keyword(s):  
Magnetic Field ◽  
Magnetic Monopole ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
New Family ◽  
Arbitrary Charge

Abstract We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.

scholarly journals Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

2019 ◽  
Vol 125 (2) ◽  
pp. 239-269 ◽  
Author(s):  
Ari Laptev ◽  
Michael Ruzhansky ◽  
Nurgissa Yessirkegenov
Keyword(s):  
Magnetic Field ◽  
Quadratic Form ◽  
Type Inequality ◽  
Hardy Inequalities ◽  
Landau Hamiltonian ◽  
Grushin Operator ◽  
Aharonov Bohm ◽  
The Magnetic Field

In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian type magnetic field. Moreover, we obtain a Poincaré type inequality and inequalities with more general families of weights. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of the variables. The corresponding uncertainty type principles are also obtained.

scholarly journals Cantor Type Basic Sets of Surface $A$-endomorphisms

2021 ◽  
Vol 17 (3) ◽  
pp. 335-345
Author(s):  
V. Z. Grines ◽  
◽  
E. V. Zhuzhoma ◽  
Keyword(s):  
Closed Surface ◽  
Orientable Surface ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Closed Orientable Surface ◽  
One Dimensional ◽  
Basic Set ◽  
Basic Sets ◽  
Nonwandering Set

The paper is devoted to an investigation of the genus of an orientable closed surface $M^{2}$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_{r}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^{2}$ is a torus or a sphere, then $M^{2}$ admits such an endomorphism. We also show that, if $\Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^{2}\to M^{2}$ of a closed orientable surface $M^{2}$ and $f$ is not a diffeomorphism, then $\Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^{2}\to M^{2}$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_{r}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_{r}$ is regular, then $M^{2}$ is a two-dimensional torus $\mathbb{T}^{2}$ or a two-dimensional sphere $\mathbb{S}^{2}$.

scholarly journals A quantum ring in a nanosphere

2019 ◽  
Vol 16 (11) ◽  
pp. 1950167 ◽  
Author(s):  
A. L. Silva Netto ◽  
B. Farias ◽  
J. Carvalho ◽  
C. Furtado
Keyword(s):  
Magnetic Field ◽  
Quantum System ◽  
Quantum Dynamics ◽  
Uniform Magnetic Field ◽  
Quantum Ring ◽  
Two Dimensional ◽  
Spherical Space ◽  
Electron Hole ◽  
Confining Potential ◽  
Aharonov Bohm

In this paper, we study the quantum dynamics of an electron/hole in a two-dimensional quantum ring within a spherical space. For this geometry, we consider a harmonic confining potential. Suggesting that the quantum ring is affected by the presence of an Aharonov–Bohm flux and a uniform magnetic field, we solve the Schrödinger equation for this problem and obtain exactly the eigenvalues of energy and corresponding eigenfunctions for this nanometric quantum system. Afterwards, we calculate the magnetization and persistent current are calculated, and discuss influence of curvature of space on these values.

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