polaron of a small radius.

In the present paper, within the framework of the effective three-band model we study the spectral density function $A_p({\bf k},\omega )$ of a single small polaron,i.e., an excitation which takes into account from the very beginning a local hole-spin coupling . It is well known that the simplest candidate for such a small polaron is an analog of the so-called Zhang-Rice singlet in CuO$_4$ plaquette [@ZhR],[@bmui]. The mean-field spectrum $\Omega _{{\bf k}}$ of this excitation is well studied [@bmui] and it will be used as the zero approximation in our treatment. We shall consider the coupling of small polaron to spin-wave excitations within the self-consistent Born approximation (SCBA) for the corresponding two-time retarded Green’s function $G({\bf k},\omega )$.

Our motivations to study $A_p({\bf k},\omega )$ and the corresponding quasiparticle band are the following. First, it may be easily shown for the one-hole problem that the mean-field energy of the polaron $\Omega _{{\bf k}}$ represents the center of gravity of the spectral function

$$\label{cg1}\Omega _k=\int_{-\infty }^\infty \omega A_p(k,\omega 
)d\omega$$

This means that the minimum of $\Omega _{{\bf k}}$ is the upper bound of the actual position for the quasiparticle band bottom. The SCBA, based on a 