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The density of states for the Schroedinger equation with a Gaussian randompotential is determined by the functional integral corresponding to the phi^4theory with a `wrong' sign of the interaction constant. The special role of thedimension d=4 for such a problem can be seen from different viewpoints but isfundamentally determined by the renormalizability of the theory. Theconstruction of an epsilon-expansion in direct analogy with thephase-transition theory gives rise to the problem of a `spurious' pole. Tosolve this problem, a proper treatment of the factorial divergency of theperturbation series is necessary. Simplifications arising in high dimensionscan be used for the development of a (4-epsilon)-dimensional theory, but thisrequires successive consideration of four types of theories: anonrenormalizable theory for d>4, nonrenormalizable and renormalizable theoriesin the logarithmic situation (d=4), and a super-renormalizable theory for d<4.An approximation is found for each type of theory giving asymptotically exactresults. The qualitative effect is the same in all four cases and consists in ashifting of the phase transition point in the complex plane. This results inthe elimination of the `spurious' pole and in regularity of the density ofstates for all energies. A discussion is given of the calculation of highorders of perturbation theory and a perspective of the epsilon-expansion forthe problem of conductivity near the Anderson transition.Comment: 27 pages, PD

Development of a (4–ε)-dimensional theory for the density of states of a disordered system near the Anderson transition