  The density of states of a particle in a 2D area is independent both of the
energy and form of the area only at the region of large values of energy. If
energy is small, the density of states in the rectangular potential well
essentially depends on the form of the area. If the bottom of the potential
well has a potential relief, it can define the small eigenvalues as the
discrete levels. In this case, dimensions and form of the area would not have
any importance. If the conservation of zero value of the angular momentum is
taken into account, the effective one-particle Hamiltonian for the 2D electron
gas in the magnetic field in the circle is the Hamiltonian with the parabolic
potential and the reflecting bounds. It is supposed that in the square, the
Hamiltonian has the same view. The 2D density of states in the square can be
computed as the convolution of 1D densities. The density of one-particle states
for 2D electron gas in the magnetic field is obtained. It consists of three
regions. There is a discrete spectrum at the smallest energy. In the
intervening region the density of states is the sum of the piecewise continuous
function and the density of the discrete spectrum. At great energies, the
density of states is a continuous function. The Fermi energy dependence on the
magnetic field is obtained when the field is small and the Fermi energy is
located in the region of continuous spectrum. The Fermi energy has the
oscillating correction and in the average it increases proportionally to the
square of the magnetic induction. Total energy of electron gas in magnetic
field also oscillates and increases when the magnetic field increases
monotonously.
