We introduce a boundary integral method for two-dimensional quantum billiar
ds subjected to a constant magnetic field, it allows us to calculate spectr
a and wavefunctions, in particular at strong fields and semiclassical value
s of the magnetic length. The method is presented for interior and exterior
problems with general boundary conditions. We explain why the magnetic ana
logues of the field-free single- and double-layer equations exhibit an infi
nity of spurious solutions and how these can be eliminated at the expense o
f dealing with (hyper-)singular operators. The high efficiency of the metho
d is demonstrated by numerical calculations in the extreme semiclassical re
gime.