On a general nonlinear Schrödinger equation admitting diffusion currents

The authors recently derived a family of nonlinear Schrodinger equations on R3 from fundamental considerations of generalized symmetry: ih(cross) delta t psi =-(h(cross)2/2m) Del 2 psi +F( psi , psi ) psi +ih(cross)D( Del 2 psi +( mod Del psi mod 2/ mod psi mod 2) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the group Diff(M) parametrized by D, where M=R3 is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.

https://iopscience.iop.org/article/10.1088/0305-4470/27/5/036/pdf

Properties of nonlinear Schrodinger equations associated with diffeomorphism group representations

On the spin-1/2 Aharonov-Bohm problem in conical space: bound states, scattering and helicity nonconservation

In this work bound states for the Aharonov-Casher problem are considered. According to Hagen's work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $\boldsymbol{\nabla}\cdot\mathbf{E}$ term cannot be neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Aharonov-Casher problem. As an application, we consider the Aharonov-Casher plus a two-dimensional harmonic oscillator. We derive the expression for the harmonic oscillator energies and compare it with the expression obtained in the case without singularity. At the end, an approach for determination of the self-adjoint extension parameter is given. In our approach, the parameter is obtained essentially in terms of physics of the problem.

On Aharonov-Casher bound states

The effect of singular potentials on the harmonic oscillator

Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold M as its configuration space is based on a quantization method, which, in general, reflects global properties of M, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on M. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold M, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on M, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self‐adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self‐adjoint extensions of the energy operator. An experimental setup...

W. F. Heidenreich

Papers

On topological effects in quantum mechanics: The harmonic oscillator in the pointed plane