Then, inspired from the specific form of these transfer matrices, we will define sets of transfer matrices for any discrete Hermitian operator with locally finite hopping by considering quasi-spherical partitions.

A generalization of some spectral averaging formula for Jacob operators is given and criteria for the existence and pureness of absolutely continuous spectrum are derived.

In the one-channel case this already led to several examples of existence of absolutely continuous spectrum for the Anderson models on such graphs with finite dimensional growth (of dimension $d>2$).

The method has some potential of attacking the open extended states conjecture for the Anderson model in $\mathbb{Z}^d, d\geq 3$.

I will talk about the discrete spectrum generated by complex matrix-valued perturbations for a class of 2D and 3D Pauli operators with non-constant admissible magnetic fields. We shall establish a simple criterion for the potentials to produce discrete spectrum near the low ground energy of the operators. Moreover, in case of creation of non-real eigenvalues, this criterion specifies also their location.

The celebrated Shnol theorem asserts that every polynomially bounded generalized eigenfunction for a given energy E associated with a Schrodinger operator H implies that E is in the L2-spectrum of H. Later Simon rediscorvered this result independently and proved additionally that the set of energies admiting a polynomially bounded generalized eigenfunction is dense in the spectrum. A remarkable extension of these results hold also in the Dirichlet setting. It was conjectured that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During the talk, we positively answer the conjecture indicating that the Agmon ground state describes the spectrum of the operator H. Specifically, we show that if u is a generalized eigenfunction for the eigenvalue E that is bounded by the Agmon ground state then E belongs to the L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a suitable notion of Agmon ground state is available.