Induced vacuum energy density of quantum charged scalar matter in the background of an impenetrable magnetic tube with the Neumann boundary condition

We consider vacuum polarization of charged scalar matter field outside the tube with magnetic flux inside. The tube is impenetrable for quantum matter and the perfectly rigid (Neumann) boundary condition is imposed at its surface. We write expressions for induced vacuum energy density for the case of a space of arbitrary dimension and for an arbitrary value of the magnetic flux. We do the numerical computation for the case of half-integer flux value in the London flux units and (2+1)-dimensional space-time. We show that the induced vacuum energy of the charged scalar matter field is induced if the Compton wavelength of the matter field exceeds the transverse size of the tube considerably. We show that vacuum energy is periodic in the value of the magnetic flux of the tube, providing a quantum-field-theoretical manifestation of the Aharonov-Bohm effect. The dependencies of the induced vacuum energy upon the distance from the center of the tube under the different values of its thickness were obtained. Obtained results are compared to the results obtained earlier in the case of the perfectly reflecting (Dirichlet) boundary condition. It is shown that the value of the induced vacuum energy density in the case of the Neumann boundary condition is greater than in the case of the Dirichlet boundary condition.