Confining quantum particles with a purely magnetic field

Abstract

We consider a Schrödinger operator with a magnetic field (and no electric field) on a domain in the Euclidean space with a compact boundary. We give sufficient conditions on the behaviour of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of view, it means that the quantum particle is confined in the domain by the magnetic field. We construct examples in the case where the boundary is smooth as well as for polytopes; these examples are highly simplified models of what is done for nuclear fusion in tokamacs. We also present some open problems.

Aharonov-Bohm effect without contact with the solenoid

“…Any self-adjoint extension of H κ is a fair candidate to model the AB effect. However, by using methods discussed in [4,5], we are able to show that condition (1) implies thatḢ κ has just one self-adjoint extension H κ , which then is the operator describing the modeling in this situation. The lack of boundary conditions at the solenoid border is interpreted as no contact of the particle in Ω a with the solenoid.…”

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

“…Next, we present a result which has also been suggested to us by O. Milatovic in a private communication. Earlier results of this type for magnetic operators in the continuum with similar kinds of proofs already appeared in [3]. …”

Eigenvalue Bound for Schrödinger Operators with Unbounded Magnetic Field

“…Apparently, for the first time such model was treated in [5], where the authors established the essential self-adjointness of H Ω (A, V ) (defined on C ∞ 0 (Ω)) under certain assumptions on the growth of B near the boundary of Ω. The obtained results are of some technical interest due to their connection with to a special kind of magnetic confinement devicestokamacs.…”