We develop an analytic approach to two-dimensional (2D) holes in a magnetic field that allows us to gain insight into physics of measuring the parameters of holes, such as cyclotron resonance, Shubnikov-de Haas effect and spin resonance. We derive hole energies, cyclotron masses, and the $g$ factors in the semiclassical regime analytically, as well as analyze numerical results outside the semiclassical range of parameters, qualitatively explaining the experimentally observed magnetic field dependence of the cyclotron mass. In the semiclassical regime with large Landau level indices, and for size quantization energy much bigger than the cyclotron energy, the cyclotron mass coincides with the in-plane effective mass, calculated in the absence of a magnetic field. The hole $g$ factor in a magnetic field perpendicular to the 2D plane is defined not only by the constant of direct coupling of the angular momentum of the holes to the magnetic field, but also by the Luttinger constants defining the effective masses of holes. We find that the $g$ factor for quasi-2D holes with heavy mass in the [001] growth direction in GaAs quantum well is $g=4.05$ in the semiclasssical regime. Outside the semiclassical range of parameters, holes behave as a species completely different from electrons. Spectra for size- and magnetic-field-quantized holes are nonequidistant, not fanlike, and exhibit multiple crossings, including crossing in the ground level. We calculate the effect of Dresselhaus terms, which transform some of the crossings into anticrossings, and the effects of the anisotropy of the Luttinger Hamiltonian on the 2D hole spectra. Dresselhaus terms of different symmetries are taken into account, and a regularization procedure is developed for the ${k}_{z}^{3}$ Dresselhaus terms. Control of the nonequidistant levels and crossing structure by the magnetic field can be used to control the Landau level mixing in hole systems, and thereby control hole-hole interactions in the magnetic field.
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