On normal forms in a neighborhood of two-dimensional resonance tori for a multidimensional anharmonic oscillator. (Q556587)

The paper studies the problem of constructing a normal form near a two-dimensional invariant isotropic torus for an \(n+2\)-dimensional anharmonic oscillator given by a Hamiltonian of the form \[ H=\frac{P_1^2+\omega_1^2Q_1^2}{2}+ \frac{P_2^2+\omega_2^2Q_2^2}{2}+ \frac{1}{2}\langle r,\Gamma (Q_1,Q_2)r\rangle , \quad r=^t(q,p)\in\mathbb R^{2n}_{qp}, \] where \(\Gamma\) is a \(2n\times 2n\)-matrix whose entries smoothly depend on \(Q_1\) and \(Q_2\) and \(\langle\cdot ,\cdot\rangle\) stands for the inner product. The frequencies \(\omega_1\) and \(\omega_2\) are assumed to be constant and can be either incommensurable or commensurable. The second (resonance) case, i.e., \(\omega_1/\omega_2=m_1/m_2\), \(m_1,m_2\in\mathbb Z\), is mainly studied. For this case, under some nonresonance relations, a fourth-order normal form with respect to harmonic-oscillator-type variables is constructed. It is shown that in the presence of resonances, the dependence of the normal form on the action-type variables becomes nonpolynomial. The results obtained can be useful in certain spectral problems of quantum mechanics.