On the singularities of the spectral shift function for some tight-binding models

We consider perturbed discrete tight-binding models in $\ell^2(\mathbb{Z_h},\mathcal{G})$ describing union of quantum particles with localized interactions, where $\mathbb{Z_h}$ is the 1D lattice $h\mathbb{Z_h}$, $h > 0$, and $\mathcal G$ is a separable Hilbert space. The perturbations play the role of self-adjoint relatively compact (matrix-valued) electric potentials with $\mathcal B(\mathcal G)$-valued coefficients decaying polynomially at infinity. We analyze the Spectral Shift Function (SSF) associated to the pair of the perturbed and the unperturbed operators. On the one hand, we show that the SSF is bounded near the spectral thresholds of the essential spectrum if $\dim(\mathcal G) < +\infty$. On the other hand, if $\dim(\mathcal G) = +\infty$, we show that it may have singularities at some thresholds points $\mu$ of the essential spectrum. In particular, new mechanisms allowing the SSF to have singularities at the thresholds are exhibited, based on the degeneracy of the spectrum of the unperturbed operator. Moreover, we give the main terms of the asymptotic behaviors of the SSF near $\mu$ described in terms of some explicit effective Berezin-Toeplitz type operators. These results are completed by Levinson type formulas and examples of eigenvalues asymptotics for power-like and exponential decay potentials.