On the derivation of exact eigenstates of the generalized squeezing operator

TL;DR

This paper explicitly constructs and analyzes the properties of eigenstates invariant under a generalized squeezing operator, revealing their normalizability and behavior in number and momentum representations for arbitrary positive integer k.

Contribution

The paper derives explicit forms of eigenstates for the generalized squeezing operator for any positive integer k, including their asymptotic behavior and normalizability conditions.

Findings

States are normalizable for all k ≥ 3.

Explicit momentum representation for k=3 using Bessel functions.

Expectation values of certain operators are finite or divergent depending on k and j.

Abstract

We construct the states that are invariant under the action of the generalized squeezing operator $\exp{(z{a^{\dagger k}}-z^*a^k)}$ for arbitrary positive integer $k$. The states are given explicitly in the number representation. We find that for a given value of $k$ there are $k$ such states. We show that the states behave as $n^{-k/4}$ when occupation number $n\to\infty$. This implies that for any $k\geq3$ the states are normalizable. For a given $k$, the expectation values of operators of the form $(a^{\dagger} a)^j$ are finite for positive integer $j < (k/2-1)$ but diverge for integer $j\geq (k/2-1)$. For $k=3$ we also give an explicit form of these states in the momentum representation in terms of Bessel functions.