  k:th power (amplitude-)squeezed states are defined as the normalized states
giving equality in the Schroedinger-Robertson uncertainty relation for the real
and imaginary parts of the k:th power of the one-mode annihilation operator.
Equivalently they are the set of normalized eigenstates (for all possible
complex eigenvalues) of the Bogolubov transformed "k:th power annihilation
operators". Expressed in the number representation the eigenvalue equation
leads to a three term recursion relation for the expansion coefficients, which
can be explicitly solved in the cases k = 1, 2. The solutions are essentially
Hermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary
squeezed states, i.e. displaced squeezed vacua. For k equal to or larger than
three, where no explicit solution has been found, the recursion relation for
the symmetric operator given by the real part of the k:th power of the
annihilation operator defines a Jacobi matrix corresponding to a classical
Hamburger moment problem, which is undetermined. This implies that the operator
has an infinity of self-adjoint extensions, all with disjoint discrete spectra.
The corresponding squeezed states are well-defined, however.
