In this paper, we study the localization of the discrete spectrum of certain perturbations of a two-dimensional harmonic oscillator. The convergence of the expansion of the source function in terms of the eigenfunctions of a two-dimensional harmonic oscillator is investigated. A representation of Green's function of a two-dimensional harmonic oscillator is obtained. The singularities of Green's function are highlighted. The well-posed definition of the maximal operator generated by a two-dimensional harmonic oscillator on a specially extended domain of definition is given. Then, we describe everywhere solvable invertible restrictions of the maximal operator. We establish that the eigenvalues of a harmonic oscillator will also be the eigenvalues of well-posed restrictions. The results are supported by illustrative examples.

在本文中，我们研究了二维谐波振荡器某些扰动的离散频谱的定位。根据二维谐波振荡器的本征函数，研究了源函数展开的收敛性。获得了二维谐波振荡器的格林函数的表示。突出显示了格林函数的奇异之处。给出了二维谐波振荡器在定义的特殊扩展域上生成的最大算子的恰当定义。然后，我们在各处描述最大算子的可解可逆限制。我们确定谐波谐振器的特征值也将是适当约束的特征值。结果得到说明性实例的支持。