ON THE SPECTRUM OF BARGMANN-TOEPLITZ OPERATORS WITH SYMBOLS OF A VARIABLE SIGN

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