论文标题
介质系统中环形偶极子操作员的特征值和特征值
The eigenvalues and eigenfunctions of the toroidal dipole operator in a mesoscopic system
论文作者
论文摘要
我们在一个由固定在薄膜中的粒子组成的系统中，为$ z $轴的$ z $ - 轴投影提供了$ \ hat {t} _3 $的特征值的分析表达式。我们找到了特征值的量化规则，这对于描述$ \ hat {t} _3 $的测量至关重要。本征函数不是正方形的，因此它们不属于波浪函数的希尔伯特空间，而是可以用索具希尔伯特空间的形式主义解释为分布的内核。尽管由于奇异性，这些内核乍一看似乎是有问题的，但实际上它们可以用于实际计算中。为了说明这一点，我们明确规定了他们的行动，还提供了一个归一化程序。
We give analytical expressions for the eigenvalues and generalized eigenfunctions of $\hat{T}_3$, the $z$-axis projection of the toroidal dipole operator, in a system consisting of a particle confined in a thin film bent into a torus shape. We find the quantization rules for the eigenvalues, which are essential for describing measurements of $\hat{T}_3$. The eigenfunctions are not square-integrable, so they do not belong to the Hilbert space of wave functions, but they can be interpreted in the formalism of rigged Hilbert spaces as kernels of distributions. While these kernels appear to be problematic at first glance due to singularities, they can actually be used in practical computations. In order to illustrate this, we prescribe their action explicitly and we also provide a normalization procedure.
