Exact Solutions of the Dirac Hamiltonian on the Sphere under Hyperbolic Magnetic Fields

Figure 1

The graphs of (a) component of the vector potential 𝐴_𝑢(𝑤) in (19), effective models (b) 𝑉_eff,1 (22) and (c) 𝑉_eff,2 (23) for the values of 𝐶₁=0.4, 𝐶₂=0.5, 𝑅=1, 𝐶₃=𝑘+1, and 𝑘=2, and (d) the energies ±𝐸_1,𝑛 in (27) for 𝑘=200. There are bound states only for 𝑛=0,1,2 and 𝑉_eff,2 is an increasing function in the positive domain while 𝑉_eff,1 remains nearly constant. The scalar component of the vector potential 𝐴_𝑢 is similar to 𝑉_eff,1.

(a)

(b)

(c)

(d)