Estimates and asymptotics for eigenvalues of Birman-Schwinger-type operator with singular measure in the critical case

The Birman-Schwinger operator, which plays an important role in the spectral theory and the scattering theory, is of the form $T_V=(1-\Delta)^{-l/2}P(1-\Delta)^{-l/2}$ in $\mathbb{R}^d$ with $P$ being a measure on $\mathbb{R}^d$. Estimates for eigenvalues is a traditional object of investigation. In particular, for $2l>d$ it is known that the singular part of the measure $P$ gives a weaker contribution to the spectral estimates than that of the absolutely continuous part. We found out that this is not the case if $2l=d$. For a wide class of singular measures we obtain the spectral estimates. For the measures with support on Lipschitz surfaces of arbitrary codimension, we find an asymptotics of the spectrum. The order and the coefficitient of the asymptotics are independent of the surface dimension and the dimension of the ambient space. If we have time, I will explain a connection with non-commutative measure integration.