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 $\exp ({za}^{†k}-z^{\ast }{a}^k)$ 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 → ∞ . This implies that for any k ≥ 3 the states are normalizable. For a given k, the expectation values of operators of the form ${\left(a^{†}a\right)}^j$ are finite for positive integer j < (k/2 − 1) but diverge for integer j ≥ (k/2 − 1). For k = 3 we also give an explicit form of these states in the momentum representation in terms of Bessel functions. (paper)