Spectral inclusion and spectral exactness for singular non-self-adjoint Hamiltonian systems

We consider the effect on the spectrum of a singular non-self-adjoint Hamiltonian system of regularization by interval truncation. For problems where the deficiency indices are not maximal, there is no 'obvious' choice of boundary conditions for the problem on the truncated interval, and a wrong choice of boundary conditions can generate spurious eigenvalues ('spectral inexactness'). We present results on spectral inclusion and a test for spectral inexactness. These require the use of a Titchmarsh-Weyl M-matrix developed with respect to a different spanning set for the solution space from that normally employed. In the maximal deficiency index case, the obvious choice of boundary conditions for the regularized problems leads to spectral inclusion and exactness. In addition to the approach presented here, one can also establish this result using, for example, the results of Osborne.