  The boundary integral method (BIM) is a formulation of Helmholtz equation in
the form of an integral equation suitable for numerical discretization to solve
the quantum billiard. This paper is an extensive numerical survey of BIM in a
variety of quantum billiards, integrable (circle, rectangle), KAM systems
(Robnik billiard) and fully chaotic (ergodic, such as stadium, Sinai billiard
and cardioid billiard). On the theoretical side we point out some serious flaws
in the derivation of BIM in the literature and show how the final formula
(which nevertheless was correct) should be derived in a sound way and we also
argue that a simple minded application of BIM in nonconvex geometries presents
serious difficulties or even fails. On the numerical side we have analyzed the
scaling of the averaged absolute value of the systematic error $\Delta E$ of
the eigenenergy in units of mean level spacing with the density of
discretization ($b$ = number of numerical nodes on the boundary within one de
Broglie wavelength), and we find that in all cases the error obeys a power law
$ <|\Delta E|> = A b^{-\alpha}$, where $ \alpha $ (and also $A$) varies from
case to case (it is not universal), and is affected strongly by the existence
of exterior chords in nonconvex geometries, whereas the degree of the classical
chaos seems to be practically irrelevant. We comment on the semiclassical limit
of BIM and make suggestions about a proper formulation with correct
semiclassical limit in nonconvex geometries.
