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We analyse the ground state of a particle in a double-well potential, with a cylindrical symmetry, and in the presence of a magnetic field. We find that the azimuthal quantum number m takes the values m = 0,1,2... when we increase the magnetic field. At critical values of the magnetic field, the ground state is twice degenerate. The magnetisation shows an oscillatory behaviour and jumps at critical values of the magnetic field. This phase transition could be seen in the condensate of a dilute gas of charged bosons confined by a double-well potential., Comment: 9 pages, 8 figures