From ODLRO to the Meissner Effect and Flux Quantization

It has been shown that the electron system with ODLRO in the reduced density matrix $\rho _2$ can not support a uniform magnetic field, i.e., ODLRO\ in $% \rho _2$ implies the Meissner effect \cite{2}\cite{3}, furthermore, the magnetic field trapped in the system is quantized\cite{3}. This note extends above results in two cases. We show that (1) the system with ODLRO in $\rho _2$ can not support a non uniform, cylindrically symmetric magnetic field; (2) the system with ODLRO in $\rho _2$ can not support a magnetic field slowly varying in space, and the magnetic flux trapped in it is quantized.