论文标题
具有差距的操作员的自我伴侣扩展和特征值。申请Dirac-Coulomb操作员
Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators
论文作者
论文摘要
我们考虑了希尔伯特空间中的线性对称算子，该算子既不是从上方也不是从下方界定的，它承认与希尔伯特空间正交分裂相对应的块分解，并且具有与块分解相关的变异间隙属性。一个典型的示例是在$ c^\ infty_c（\ Mathbb r^3 \ setMinus \ {0 \}，\ Mathbb C^4）$上定义的Dirac-Coulomb操作员。在本文中，我们用光谱差距定义了一个杰出的自我参与扩展，并通过最小的最大原则在该差距中表征其特征值。在技术条件下，这是在过去做到的。在这里，我们使用不同的几何策略来实现该目标，仅做出最小的假设。我们的结果适用于狄拉克 - 库仑样哈米顿人，涵盖了签名的电位以及分子，其原子数小于或等于137的原子数的核数量是任意数量的。
We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac-Coulomb operator defined on $C^\infty_c(\mathbb R^3\setminus\{0\}, \mathbb C^4)$. In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac-Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137.
