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

Abstract

AbstractThe eigenvalue problem for arbitrary linear combinations kα + 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 kα + 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.