Results on Certain Non-Hermitian Hamiltonians

We present a few results on the spectral properties of a class of physically reasonable non-Hermitian Hamiltonians. These theorems relate the spectral properties of a non-self-adjoint operator (of the aforementioned class) in terms of that of a self-adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrödinger equation has to be solved numerically, a knowledge of the spectral properties of the non-Hermitian Hamiltonian would insure when the eigensolutions exist.