Eigenvalue estimates and asymptotics for Toeplitz operators with compactly supported symbols (an extreme case of Birman-Yafaev results on the spectrum of the scattering matrix)
We study asymptotic properties of eigenvalues of Toeplitz operators in Bergman type spaces. For the case of the space of Herglotz solutions of the Helmholtz equation, this problem is equivalent to the study of the eigenvalue asymptotics of the scattering matrix. For the potential being almost homogeneous at infinity, this eigenvalue asymptotics was found by Birman-Yafaev in 1980, and the asymptotics proved to be power-like, with exponent determined by the homogeneity degree. We consider the case of the potential (symbol) having compact support. Depending on the choice of the main space, the asymptotics can be exponential or even super-exponential, which means that for compactly supported symbols the eigenvalues decay extremely fast. The upper estimate is obtained by elementary methods and holds without additional conditions. The lower estimate for eigenvalues is proved only for radial symbols, and it requires rather advanced instruments of complex analysis.