On the derivation of exact eigenstates of the generalized squeezing operator

Abstract

We construct the states that are invariant under the action of the generalized squeezing operator $\exp (z{a}^{†k}-z^{\ast }{a}^k)$ 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 $k\ge 3$ the states are normalizable. For a given $k$, the expectation values of operators of the form $(a^{†}a{)}^j$ are finite for positive integer $j<(k/2-1)$ but diverge for integer $j\ge (k/2-1)$. For $k=3$ we also give an explicit form of these states in the momentum representation in terms of Bessel functions.