Calculation of electron states in disordered media requires finding
of the statistical sum
are the impurity positions, distributed randomly over the
The main problem in calculation of statistical sum (8.1) is a great
dimensionality of integration, which makes difficult numerical calculations for real
systems, even by a supercomputer. To avoid this we can make a reduced description of
the disordered system by correlation functions. For such functions we can take
correlation functions of the medium density. These functions have been much studied
for a wide range of media - liquids, spin glasses, polymer melts and polymer globules.
Proceeding from the self-consistency of the medium and the electron states, the electron
wave function can be expressed by a nonlinear differential equation of Schrodinger type
with the effective potential V(f,w,Х1,Х2,...)
on the wave function f
, pairwise interaction potential
and correlation functions of the medium Х1,Х2,...
This equation allows to calculate electron states in complex medium
taking into account the medium's molecular structure and the thermodynamic state of the medium.
The electron states in polar medium are, in essence, similar to polaron. Presently a theory
is being developed, allowing to calculate the effective potential, the absorption
spectrum, the effective mass and the mobility for polaron-like states in disordered
When the interaction forces are polar and the medium is homogeneous
this equation is reduced to the equation of the type of (3.2). In the case of short-range
potential and homogeneous medium v(r-R)=2pas(r-R)
minimization of free energy corresponding to statistical sum (8.1) leads to the differential
equation of the form (R.Friedberg, J.M.Luttinger.
electornic levels in disordered systems //Phys. Rev. B., 1975, v.12, №10, p.4460-4474) :
equation (8.3) has non-trivial solutions. Fig.1 shows the dependence of
, see also (7.1)
Fig.1. Where f(y0)=[ln(1+y20)]1/2,
The shape of the curve suggests that there exists a region of parameter values, at which
six different ground states without nodes coexist at a time. These results witness to
unusual properties of nonlinear equations of quantum mechanics.