Title: Infinite-dimensional analyticity in quantum physics Show Chinese title

Title: 量子物理中的无限维解析性

Abstract: A study is made, of families of Hamiltonians parameterized over open subsets of Banach spaces in a way which renders many interesting properties of eigenstates and thermal states analytic functions of the parameter. Examples of such properties are charge/current densities. The apparatus can be considered a generalization of Kato's theory of analytic families of type B insofar as the parameterizing spaces are infinite dimensional. It is based on the general theory of holomorphy in Banach spaces and an identification of suitable classes of sesquilinear forms with operator spaces associated with Hilbert riggings. The conditions of lower-boundedness and reality appropriate to proper Hamiltonians is thus relaxed to sectoriality, so that holomorphy can be used. Convenient criteria are given to show that a parameterization $x \mapsto {\mathsf{h}}_x$ of sesquilinear forms is of the required sort ({\it regular sectorial families}). The key maps ${\mathcal R}(\zeta,x) = (\zeta - H_x)^{-1}$ and ${\mathcal E}(\beta,x) = e^{-\beta H_x}$, where $H_x$ is the closed sectorial operator associated to ${\mathsf {h}}_x$, are shown to be analytic. These mediate analyticity of the variety of state properties mentioned above. A detailed study is made of nonrelativistic quantum mechanical Hamiltonians parameterized by scalar- and vector-potential fields and two-body interactions. Show Chinese abstract

Abstract: 对在巴拿赫空间开子集上参数化的哈密顿量族进行了研究，这种方式使得许多本征态和热态的性质成为参数的解析函数。 这类性质的例子包括电荷/电流密度。 该装置可以看作是卡托关于类型B的解析族理论的推广，因为参数空间是无限维的。 它基于巴拿赫空间中的全纯性一般理论，并将合适的共轭双线性形式类与与希尔伯特配对相关的算子空间进行识别。 因此，适用于适当哈密顿量的下界性和实性条件被放松为扇形性，以便可以使用全纯性。 给出了方便的准则来表明一个共轭双线性形式的参数化$x \mapsto {\mathsf{h}}_x$是所需类型（{\it 正则扇形族}）。 关键映射${\mathcal R}(\zeta,x) = (\zeta - H_x)^{-1}$和${\mathcal E}(\beta,x) = e^{-\beta H_x}$，其中$H_x$是与${\mathsf {h}}_x$相关的闭扇形算子，被证明是解析的。 这些映射传达了上述各种状态性质的解析性。 详细研究了由标量势场和矢量势场以及两体相互作用参数化的非相对论量子力学哈密顿量。