Harmonic functions which vanish on coaxial cylinders

It was recently established that a function which is harmonic on an infinite cylinder and vanishes on the boundary necessarily extends to an entire harmonic function. This paper considers harmonic functions on an annular cylinder which vanish on both the inner and outer cylindrical boundary components. Such functions are shown to extend harmonically to the whole of space apart from the common axis of symmetry. One of the ingredients in the proof is a new estimate for the zeros of cross product Bessel functions.