  Eigenstates and energy levels of a square quantum billiard in a magnetic
field, or with an Aharonov-Bohm flux line, are found in quasiclassical
approximation, that is, for high enough energy. Explicit formulas for the
energy levels and wavefunctions are found. There are localized states, never
before noticed in this well studied problem, whose localization is due to phase
interference, even though there is no or negligible classical effect of the
magnetic field. These and related states account almost entirely for the
magnetic response in certain temperature ranges, and thus have a bearing on the
experiments of Levy et al.
