Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations k α + μα † of a boson annihilation operator α and a boson creation operator α † is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, ( n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ μ/ k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators k α + k * α † to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α † are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ μ/ k as a parameter.