Self-adjoint realizations of higher-order squeezing operators

Higher-order squeezing captures non-Gaussian features of quantum light by probing moments of the field beyond the variance, and is associated with operators involving nonlinear combinations of creation and annihilation operators. Here we study a class of operators of the form $\xi (a^\dag)^ka^l+\xi^\ast (a^\dag)^la^k+f(a^\dag a)$, which arise naturally in the analysis of higher-order quantum fluctuations. The operators are defined on the linear span of Fock states. We show that the essential self-adjointness of these operators depends on the asymptotics of the real-valued function $f(n)$ at infinity. In particular, pure higher-order squeezing operators ($k\geq3$, $l=0$, and $f(n)=0$) are not essentially self-adjoint, but adding a properly chosen term $f(a^\dag a)$, like a Kerr term, can have a regularizing effect and restore essential self-adjointness. In the non-self-adjoint regime, we compute the deficiency indices and classify all self-adjoint extensions. Our results provide a rigorous operator-theoretic foundation for modeling and interpreting higher-order squeezing in quantum optics, and reveal interesting connections with the Birkhoff-Trjitzinsky theory of asymptotic expansions for recurrence relations.