The paper discusses the spectrum of Toeplitz operators in Bargmann spaces.
Our Toeplitz operators have real symbols with a variable sign and a compact
support. A class of examples is considered where the asymptotics of the
eigenvalues of such operators can be computed. These examples show that this
asymptotics depends on the geometry of the supports of the positive and
negative parts of the symbol. Applications to the perturbed Landau Hamiltonian
are given.