Flux quantization in a hollow superconducting cylinder with a solenoid inside a cavity

The behavior of a hollow superconducting cylinder with a long solenoid inside a cavity is considered within the framework of the Ginzburg--Landau theory. The general form is found of the thermodynamic potential of the system, which describes its behavior under the influence of the solenoid's field. It is shown that the magnetic field appears inside a cavity, when the solenoid's flux is not zero. The total flux in the system changes by jumps, due to the effect of flux quantization. The dependence of the superconducting order parameter on the solenoid's flux is found, the phase diagram of the superconducting states is analyzed. The hysteresis transitions in the system are investigated, as well as oscillations of the critical temperature and of the specific heat of the cylinder. The connection with the paper of G. Kunstatter, M. Revzen, and L. E. H. Trainor (Ann. Phys. (N.Y.) 145 (1983), 329), where the corresponding problem was first touched upon, is discussed