Density of states near the Anderson transition in the $(4-\varepsilon)$-dimensional space

The calculation of the density of states for the Schrödinger equation with a Gaussian random potential is equivalent to the problem of a second-order transition with a ‘wrong’ sign of the coefficient of the quartic term in the Ginzburg–Landau Hamiltonian. The special role of the dimension $d=4$ for such Hamiltonian can be seen from different viewpoints but fundamentally is determined by the renormalizability of the theory. Construction of $\varepsilon$-expansion in direct analogy with the phase transitions theory gives rise to a problem of a ‘spurious’ pole. To solve this problem, the proper treatment of the factorial divergency of the perturbation series is necessary. In $(4-\varepsilon)$-dimensional theory, the terms of the leading order in $1/\varepsilon$ should be retained for $N\sim1$ ($N$ is an order of the perturbation theory) while all degrees of $1/\varepsilon$ are essential for large $N$ in view of the fast growth of their coefficients. The latter are calculated in the leading order in $N$ from the Callan–Symanzik equation with results of Lipatov method using as boundary conditions. The qualitative effect consists in shifting of the phase transition point to the complex plane. This results in elimination of the ‘spurious’ pole and in regularity of the density of states for all energies.