Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\widehat H$ of the formal Hamiltonian $H+A^{*}+A$ with $D(H)\cap D(\widehat H)=\{0\}$. We give the explicit characterization of $D(\widehat H)$ and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$. Moreover, we consider the problem of the description of $\widehat H$ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$, where the $A_{n}\!$'s are regularized operators approximating $A$ and the $E_{n}$'s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Krein's resolvent formula and the nonperturbative theory of renormalizable models in Quantum Field Theory.

中文翻译：

---

关于H+A*+A的自伴随性

令 $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ 为自伴并令 $A:D(H)\to{\mathscr F}$（扮演歼灭者运算符）以 $H$ 为界。假设对 $A$ 有一些额外的假设（因此创建算子 $A^{*}$ 是 $H$ 的奇异扰动），通过求解 Krein 类型公式的双重应用，我们构建了自伴随实现 $ \widehat H$ 的形式化哈密顿量 $H+A^{*}+A$ 与 $D(H)\cap D(\widehat H)=\{0\}$。我们给出了 $D(\widehat H)$ 的显式表征，并提供了求解差值 $(-\widehat H+z)^{-1}-(-H+z)^{-1}$ 的公式。此外，我们将 $\widehat H$ 的描述问题视为 $H+A^{*}_{n}+A_{n}+E_{n}$ 类型序列的（范数解析）极限，其中 $A_{n}\!$'s 是近似 $A$ 和 $E_{n}$' 的正则化运算符 s 是合适的重整化有界运算符。这些结果表明，通过 Krein 的求解公式构建自伴随算子的奇异摄动与量子场论中可重整化模型的非摄动理论之间存在联系。