标题： 广义的Birman-Schwinger原理及其在具有分布势的一维Schrödinger算子中的应用 显示英文标题

标题： A generalized Birman-Schwinger principle and applications to one-dimensional Schrödinger operators with distributional potentials

摘要： 给定一个在复希尔伯特空间$\mathcal{H}$中下有界的自伴算子$H_0$，相应的空间尺度$\mathcal{H}_{+1}(H_0) \subset \mathcal{H} \subset \mathcal{H}_{-1}(H_0) = [\mathcal{H}_{+1}(H_0)]^*$，以及一个固定的$V\in \mathcal{B}(\mathcal{H}_{+1}(H_0),\mathcal{H}_{-1}(H_0))$，我们通过\[ A_V(z):=-\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}V\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}\in \mathcal{B}(\mathcal{H}),\quad z\in \rho(H_0), \]定义一个算子值映射$A_V(\,\cdot\,):\rho(H_0)\to \mathcal{B}(\mathcal{H})$，其中$\rho(H_0)$表示$H_0$的预解集。 假设对于某个$z=z_0\in \rho(H_0)$，$A_V(z)$是紧的，并且对于某个$z=E_0\in (-\infty,0)$，其范数严格小于一，我们采用 Tiktopoulos 公式的抽象版本来定义一个在$\mathcal{H}$中的算子$H$，它形式上是$H_0$和$V$的和。 我们然后为$H$建立一个 Birman-Schwinger 原理，其中$A_V(\,\cdot\,)$起到 Birman-Schwinger 算子的作用：$\lambda_0\in \rho(H_0)$是$H$的特征值当且仅当$1$是$A_V(\lambda_0)$的特征值。 此外，$\lambda_0$和$1$作为$H$和$A_V(\lambda_0)$的特征值的几何（但不一定是代数）重数是相同的。 作为一个具体的例子，我们考虑具有$H^{-1}(\mathbb{R})$分布势的一维薛定谔算子。 显示英文摘要

摘要： Given a self-adjoint operator $H_0$ bounded from below in a complex Hilbert space $\mathcal{H}$, the corresponding scale of spaces $\mathcal{H}_{+1}(H_0) \subset \mathcal{H} \subset \mathcal{H}_{-1}(H_0) = [\mathcal{H}_{+1}(H_0)]^*$, and a fixed $V\in \mathcal{B}(\mathcal{H}_{+1}(H_0),\mathcal{H}_{-1}(H_0))$, we define the operator-valued map $A_V(\,\cdot\,):\rho(H_0)\to \mathcal{B}(\mathcal{H})$ by \[ A_V(z):=-\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}V\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}\in \mathcal{B}(\mathcal{H}),\quad z\in \rho(H_0), \] where $\rho(H_0)$ denotes the resolvent set of $H_0$. Assuming that $A_V(z)$ is compact for some $z=z_0\in \rho(H_0)$ and has norm strictly less than one for some $z=E_0\in (-\infty,0)$, we employ an abstract version of Tiktopoulos' formula to define an operator $H$ in $\mathcal{H}$ that is formally realized as the sum of $H_0$ and $V$. We then establish a Birman-Schwinger principle for $H$ in which $A_V(\,\cdot\,)$ plays the role of the Birman-Schwinger operator: $\lambda_0\in \rho(H_0)$ is an eigenvalue of $H$ if and only if $1$ is an eigenvalue of $A_V(\lambda_0)$. Furthermore, the geometric (but not necessarily the algebraic) multiplicities of $\lambda_0$ and $1$ as eigenvalues of $H$ and $A_V(\lambda_0)$, respectively, coincide. As a concrete application, we consider one-dimensional Schr\"odinger operators with $H^{-1}(\mathbb{R})$ distributional potentials.