Mathematical justification of the Aharonov-Bohm hamiltonian
نویسندگان
چکیده
It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both R and R spaces. PACS: 03.65.Ta; 03.65.Db; 02.30.Sa Given a cylindrical current-carrying solenoid S of infinite length and radius a > 0, centered at the origin and axis in the z direction, there is a constant magnetic field B = (0, 0, B) confined in S◦, the interior of S, and vanishing in its exterior region S . The solenoid is considered impermeable (impenetrable), in the sense that the motion of a spinless particle (of mass m = 1/2 and electric charge q) outside the solenoid has no contact with its interior, particularly with the magnetic field B. If A is the vector potential generating this magnetic field, that is, B = ∇ ×A, the usual hamiltonian operator describing the quantum motion of this charged particle is given by (with ~ = 1) HAB = ( p− q c A )2 , p = −i∇, with Dirichlet boundary conditions, i.e., the functions ψ in the domain of HAB are supposed to vanish ψ = 0 at the solenoid boundary (the precise domain of HAB is described just before Proposition 1). Observable effects, ∗On leave of absence from Universidade Estadual de Ponta Grossa, PR, Brazil.
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