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We investigate the second-order Zeeman effect in a magnetic field using irreducible representations of an algebra with the Karasev Novikova quadratic commutation relations. To each such representation there corresponds a spectral cluster near the energy level of the unperturbed hydrogen atom. Using this model as an example, we describe a general method for constructing asymptotic solutions near the boundaries of spectral clusters based on a new integral representation. We also study the problem of computing quantum averages near the lower boundaries of clusters.

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters