On few-parameter generic families of vector fields on the two-dimensional sphere

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

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Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques

10.1117/12.312933 ◽

1998 ◽

Author(s):

Takahiro Aoki ◽

Toshihiro Sotomaru ◽

Yoko Miyamoto ◽

Mitsuo Takeda

Keyword(s):

Vector Fields ◽

Phase Unwrapping ◽

Two Dimensional ◽

Rotational Vector

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Melnikov–s Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields

Introduction to Applied Nonlinear Dynamical Systems and Chaos - Texts in Applied Mathematics ◽

10.1007/0-387-21749-5_29 ◽

2006 ◽

pp. 687-725

Keyword(s):

Vector Fields ◽

Homoclinic Orbits ◽

Two Dimensional ◽

Time Periodic

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Homoclinic tangles and transport in two-dimensional, time-quasiperiodic vector fields

Global Dynamics, Phase Space Transport, Orbits Homoclinic to Resonances, and Applications - Fields Institute Monographs ◽

10.1090/fim/001/04 ◽

1993 ◽

pp. 29-46

Keyword(s):

Vector Fields ◽

Two Dimensional ◽

Homoclinic Tangles

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Class of Stable Connectivity of Source-Sink Diffeomorphism on Two-Dimensional Sphere

Journal of Mathematical Sciences ◽

10.1007/s10958-020-05001-x ◽

2020 ◽

Vol 250 (1) ◽

pp. 94-108

Author(s):

E. V. Nozdrinova ◽

O. V. Pochinka

Keyword(s):

Dimensional Sphere ◽

Two Dimensional ◽

Source Sink

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

Reviews in Mathematical Physics ◽

10.1142/s0129055x21500069 ◽

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].

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