Essentially singular limits of Jacobi operators and applications to higher-order squeezing

TL;DR

This paper investigates the limiting behavior of Jacobi operators with a coupling parameter approaching zero, revealing how different self-adjoint extensions emerge and applying these findings to quantum optics squeezing operators.

Contribution

It introduces a novel analysis of essentially singular limits of Jacobi operators and connects these limits to the behavior of higher-order squeezing operators in quantum optics.

Findings

Uniform bounds for eigenvectors in small-λ regime

Convergence to self-adjoint extensions along subsequences

Different extensions are selected depending on the sequence approaching zero

Abstract

We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $\lambda\geq0$. Under suitable conditions, the operator is self-adjoint for every $\lambda>0$, while the formal limit at $\lambda=0$ is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-$\lambda$ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior $\lambda\to0$ in the strong resolvent sense, proving that for every sequence $\lambda_j\to0$ one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.