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 giv
en 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.
It is well known that the Gaussian wave packet dynamics can be written in terms of Hamilton equations in the extended phase space that is twice as large as in the corresponding classical system. We construct several generalizations of this approach t
hat include non-Gausssian wave packets. These generalizations lead to the further extension of the phase space while retaining the Hamilton structure of the equations of motion. We compare the Gaussian dynamics with these non-Gaussian extensions for a particle with the quartic potential.
For electron-phonon Hamiltonians with the couplings linear in the phonon operators we construct a class of unitary transformations that separate the normal modes into two groups. The modes in the first group interact with the electronic degrees of fr
eedom directly. The modes in the second group interact directly only with the modes in the first group but not with the electronic system. We show that for the $n$-level electronic system the minimum number of modes in the first group is $n_s=(n^2+n-2)/2$. The separation of the normal modes into two groups allows one to develop new approximation schemes. We apply one of such schemes to study exitonic relaxation in a model semiconducting molecular heterojuction.
