On the derivation of exact eigenstates of the generalized squeezing operator

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 $ntoinfty$. This implies that for any $kgeq3$ 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 $jgeq (k/2-1)$. For $k=3$ we also give an explicit form of these states in the momentum representation in terms of Bessel functions.